1&'..+0) /#6*'/#6+%#.#0#.;5+5 07/'4+%#.51.76+10#0&2#4#/'6 ... · Preface In general terms, a chemical reactor can be understood as a vessel used in trans-forming the initial chemical

TESIS DE DOCTORADO

MODELLING, MATHEMATICAL ANALYSIS,

NUMERICAL SOLUTION AND PARAMETER

IDENTIFICATION IN REACTION SYSTEMS

Noemí Esteban Rodríguez

ESCUELA DE DOCTORADO INTERNACIONAL EN CIENCIAS Y TECNOLOGÍA DE LA USC

PROGRAMA DE DOCTORADO EN MÉTODOS MATEMÁTICOS Y SIMULACIÓN

NUMÉRICA EN INGENIERÍA Y CIENCIAS APLICADAS

SANTIAGO DE COMPOSTELA

AÑO 2019

DECLARACIÓN DEL AUTOR DE LA TESIS [Modelling, mathematical analysis, numerical solution and parameter

identification in reaction systems] Dña. Noemí Esteban Rodríguez

Presento mi tesis, siguiendo el procedimiento adecuado al Reglamento, y declaro que:

1) La tesis abarca los resultados de la elaboración de mi trabajo. 2) En su caso, en la tesis se hace referencia a las colaboraciones que tuvo este trabajo. 3) La tesis es la versión definitiva presentada para su defensa y coincide con la versión enviada en

formato electrónico. 4) Confirmo que la tesis no incurre en ningún tipo de plagio de otros autores ni de trabajos

presentados por mí para la obtención de otros títulos.

En Santiago de Compostela, 21 de mayo de 2019

Fdo. Noemí Esteban Rodríguez

AUTORIZACIÓN DEL DIRECTOR / TUTOR DE LA

TESIS [Modelling, mathematical analysis, numerical solution and parameter

identification in reaction systems.] D. Alfredo Bermúdez de Castro López-Varela

Dña. Oana Teodora Chis

INFORMA/N: Que la presente tesis, corresponde con el trabajo realizado por Dña. Noemí Esteban Rodríguez, bajo

mi dirección, y autorizo su presentación, considerando que reúne l os requisitos exigidos en el

Reglamento de Estudios de Doctorado de la USC, y que como director de ésta no incurre en

las causas de abstención establecidas en Ley 40/2015.

En Santiago de Compostela, 21 de mayo de 2019

Fdo. Alfredo Bermúdez de Castro López-Varela

Fdo. Oana Teodora Chis

A mis padres,Emilio y Rosa

Agradecimientos

En los siguientes parrafos me gustarıa agradecer a todas las personas que deuna u otra forma me han ensenado, ayudado y apoyado a lo largo de estos anoscomo doctorando. Todo lo que me han aportado a contribuido a la consecucionde esta tesis.

Quisiera comenzar agradeciendoles a mis directores de tesis, Alfredo Bermu-dez de Castro Lopez-Varela y Oana Teodora Chis, su dedicacion en todo mo-mento, su sabidurıa y su empatıa y cercanıa conmigo. Me habeis abierto laspuertas al mundo de la investigacion. Me habeis transmitido la ilusion que setiene al trabajar en lo que a uno le gusta. Vuestra experiencia y conocimien-tos me han sido de mucha ayuda. Ha resultado mucho mas facil trabajar ası.Muchas gracias, Alfredo y Oana.

En segundo lugar, me gustarıa agradecerle a Jose Francisco Rodrıguez Calosu gran compromiso y dedicacion con el proyecto de la UMI Repsol-ITMATIvinculado a esta tesis. Una parte de este proyecto se gesto a partir de sus ideas.

Tambien quiero dar las gracias a todo el equipo de ITMATI (Ruben, Adri-ana, Ariana, etc.) y por supuesto a todos mis companeros (Marta, Pedro,Diego, Manuel, Gabriel, Irene, Jorge, Alex, Joaquın, Oscar, Fito, etc.) con losque he pasado muchos momentos en los que nos hemos reıdo, y disfrutado deun monton de comidas (gran parte de la gastronomıa gallega la he disfrutadoa vuestro lado), viajes, etc.

Me gustarıa hacer extensible este agradecimiento a todos los miembros delDepartamento de Matematica Aplicada de la Universidad de Santiago de Com-postela. Ha sido un privilegio poder compartir con vosotros esta experien-cia. Especialmente a Rafael Munoz por su inestimable ayuda y los valiososconocimientos aportados. Tambien a Jose Luis y Jeronimo que han formadoparte en algunos momentos de este proyecto. Sin olvidar a mis companeros dedespacho en mi primera etapa en el departamento (Cris, Jorge, Angel, Nizom,Javi, etc.)

Gracias a Ricardo, apareciste en mitad de este proyecto y has sabido com-prenderme y apoyarme en los momentos que lo he necesitado, contribuyendo aque no tirase la toalla.

i

ii

Por ultimo, me gustarıa dirigirme a mi familia para agradecerles el apoyoincondicional que me han demostrado. Gracias a mis padres Emilio y Rosapor todos los animos y el apoyo en todo momento y en todos los sentidos; amis abuelos Pilar (que en paz descanse), Angel y Tere que me han ayudado adesconectar siempre que los visitaba y por mostrarme lo orgullosos que estabaisde mı; a mi hermano Emilio por estar ahı; a mis tıas Fefa y Encar por animarmea conseguirlo. Sin todos ellos, realizar este trabajo hubiera sido imposible.

¡Muchas gracias a todos!

Contents

Preface 1

I Modelling chemical reactors 7

Introduction 9

1 Modelling stirred tank reactors 111.1 Modelling chemical reactions . . . . . . . . . . . . . . . . . . . 12

1.1.1 Chemical species and elements. Conservation relations . 121.1.2 Finite rate chemical reactions . . . . . . . . . . . . . . . 131.1.3 Reaction rate constant . . . . . . . . . . . . . . . . . . . 13

1.2 Modelling Batch Stirred Tank Reactors . . . . . . . . . . . . . 131.2.1 The transient batch STR model . . . . . . . . . . . . . . 171.2.2 Steady-state Batch STR . . . . . . . . . . . . . . . . . . 17

1.3 Modelling Semi-Batch and CSTR . . . . . . . . . . . . . . . . . 181.3.1 The transient semi-batch and CSTR model . . . . . . . 191.3.2 Steady-state CSTR model . . . . . . . . . . . . . . . . . 22

2 Convection-diffusion-reaction model 232.1 The convection-diffusion-reaction model . . . . . . . . . . . . . 23

2.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . 262.1.2 Initial conditions . . . . . . . . . . . . . . . . . . . . . . 272.1.3 The full n-dimensional model . . . . . . . . . . . . . . . 27

2.2 Modelling Plug Flow Reactors . . . . . . . . . . . . . . . . . . . 282.2.1 Transient Plug Flow Reactors . . . . . . . . . . . . . . . 282.2.2 Steady-state PFR . . . . . . . . . . . . . . . . . . . . . 32

3 Modelling catalytic fixed bed reactors 353.1 Modelling the macro-scale (fluid bulk) . . . . . . . . . . . . . . 37

3.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . 40

iii

iv CONTENTS

3.1.2 Initial conditions . . . . . . . . . . . . . . . . . . . . . . 413.2 Modelling the micro-scale . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . 423.2.2 Initial conditions . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Sources of mass and energy in the fluid bulk . . . . . . . . . . . 44

Conclusions 47

II Mathematical analysis and numerical solution 49

Introduction 51

4 Existence and uniqueness of solution 534.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Local existence of weak solution . . . . . . . . . . . . . . . . . . 56

4.2.1 Existence of solution to an auxiliary elliptic problem . . 574.2.2 Local existence of a homogeneous problem . . . . . . . . 614.2.3 Local existence of solution to problem (4.3)–(4.4) . . . . 714.2.4 The maximally defined solution . . . . . . . . . . . . . . 73

4.3 Global existence of weak solution . . . . . . . . . . . . . . . . . 744.3.1 Uniqueness of solution . . . . . . . . . . . . . . . . . . . 83

5 Numerical analysis 855.1 The semidiscrete problem . . . . . . . . . . . . . . . . . . . . . 86

5.1.1 A finite element method . . . . . . . . . . . . . . . . . . 875.1.2 Local existence and uniqueness of solution to the semidis-

crete problem . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Error estimates for the semidiscrete solution . . . . . . . . . . . 925.3 Global solution to the semidiscrete problem . . . . . . . . . . . 98

6 Numerical solution of the PFR model 1016.1 Time and spatial discretizations of the problem . . . . . . . . . 1026.2 Academic tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7 Numerical solution of the FBR model 1157.1 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.1.1 Macroscale: fluid bulk . . . . . . . . . . . . . . . . . . . 1157.1.2 Micro-scale: spherical solid particles . . . . . . . . . . . 1177.1.3 Mass conservation at steady-state . . . . . . . . . . . . 120

7.2 An academic test in steady state . . . . . . . . . . . . . . . . . 122

Conclusions 129

CONTENTS v

III Identification in reaction systems 131

Introduction 133

8 The identification problem 1358.1 Measurements and reactions scheme . . . . . . . . . . . . . . . 1358.2 Kinetic models . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368.3 The general model . . . . . . . . . . . . . . . . . . . . . . . . . 1378.4 Model selection and parameter identification . . . . . . . . . . 138

8.4.1 Initial approximation: The incremental method. . . . . 1398.4.2 Improvements in solution: The integral method. . . . . 142

8.5 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Conclusions 157

Future work 159

A Summary continuum thermomechanics 161A.1 Equations of continuum thermomechanics . . . . . . . . . . . . 161

B Abstract semilinear problems 165B.1 Operators. Spectrum and resolvent . . . . . . . . . . . . . . . . 165B.2 Sectorial operators . . . . . . . . . . . . . . . . . . . . . . . . . 166B.3 Second order differential operators . . . . . . . . . . . . . . . . 167B.4 Local existence results . . . . . . . . . . . . . . . . . . . . . . . 168B.5 The maximally defined solution . . . . . . . . . . . . . . . . . 170

C Interpolation Error Estimates 173C.1 Local interpolation operator . . . . . . . . . . . . . . . . . . . . 173C.2 Global interpolation operator . . . . . . . . . . . . . . . . . . . 175C.3 Some bounds for the interpolation operator . . . . . . . . . . . 175C.4 Some inverse inequalities . . . . . . . . . . . . . . . . . . . . . . 177

D Resumen 179

Bibliography 189

vi CONTENTS

Preface

In general terms, a chemical reactor can be understood as a vessel used in trans-forming the initial chemical species into desired final products. They can beideal reactors, for example, stirred tank reactors (STR) in the simplest cases,or more complex reactors such as fixed bed reactors (FBR). In any case, itis important that the residence time inside the reactor be sufficiently large toproduce the expected chemical reactions.

The design of the reactors involves three main fields in chemical engineer-ing: thermodynamics, kinetics and heat transfer. Thus, when a reaction occursin a batch STR reactor, a reasonable question would be “What is the maxi-mum expected conversion?”. This is a question related to thermodynamics. Ifwe want to know how long the reaction should take to convert the reactantsin the desired products, we would be asking ourselves about the kinetics (weshould know both the stoichiometry and the reaction rates). Finally, if wewant to know how much heat must be transferred to the reactor or from itto keep the isothermal condition, we are dealing with a heat transfer problemcombined with thermodynamics (we must know if the reaction is endothermicor exothermic).

In order to describe more in detail a reactor it is necessary to distinguishamong different types. In the literature there are many reactor classifications.Each one is performed according to the feature to be highlighted. Now, wedescribe the main classifications following the scheme from [29].

Operation types: This classification is related to the operational con-figuration of the reactor. This is the classification we use a priori in thepresent thesis (batch STR, semi-batch, CSTR, PFR or FBR are differentreactors according to their operational configuration).

1. Batch stirred tank reactor (Batch STR): reactants are introducedinto the reactor at the initial time only. There are no input oroutput flows along the process.

1

2 Preface

2. Semi-batch Batch (Semi-batch STR): some of the reactants are intro-duced in the reactor at the initial time; some others are continuouslyintroduced along the process.

3. Continuous (flow) stirred tank reactor (CSTR): reactants are con-tinuously introduced along the time. There is also an output flowalong the process.

4. Plug flow reactors (PFR): the plug flow is a tubular reactor in whichthe so called plug flow assumption is considered. That is, the velocityis constant on any cross-section of the pipe.

5. Fixed bed reactors (FBR): the fixed bed reactor is a single cylindricalshell with convex heads with a packed bed of catalytic particles ofuniform size, which are immobilized or fixed within the tube.

Number of phases: Reactors can also be classified by the numberof phases present in the reactor at any time. They are called homoge-neous and heterogeneous reactors. The first ones represent the reactorswith only one phase (STRs are homogeneous reactors). The second onescontain more than one phase. Several heterogeneous reactor types areavailable due to various combinations of phases (like PFRs or FBRs).

Reaction types: This classification is made considering different typeof reactions. Some of the most important are:

1. Catalytic: reactions that require the presence of a catalyst to obtainthe necessary rate conditions. An example of these reactors is theFBR.

2. Noncatalytic: reactions that do not include either a homogeneous orheterogeneous catalyst. They are the opposite to the previous ones.

3. Autocatalytic: in this reaction scheme one of the products increasesthe rate of reactions.

4. Biological : reactions that involve living cells (enzymes, bacteria,etc.).

5. Polymerization: reactions that involve formation of molecular poly-mer chains.

Finally, depending on the final destination in industry we consider a clas-sification in concordance with two different motivations:

1. Industrial reactors: simulation of its operation with the ultimate aimof optimizing it economically by modifying the operating conditions(initial conditions, temperature, ...).

Preface 3

2. Laboratory reactors/pilot plant : optimization of the reactor design:optimal geometric and optimal operating conditions for a future re-actor. The goal of the design is to determine the reactor featuressuch as pipe, valves or mixers. For example, the reactor must havesufficient volume to allow the reaction reach a level of conversion orallow the heat exchange necessary.

Reactor design requires first to establish the type and size of the reactorand then the operation type according to the chemical process, as explainedin [24]. Important considerations are related to the chemical reactions and,in any of the described reactors, reaction velocity expressions (kinetics) mustbe known. The reaction rate involves a mathematical expression. In order topredict the size of the reactor needed to obtain both the desired conversion ofreactants and a fixed output of the product, it is required information on thecomposition and temperature changes, as well as reaction rate, obtained fromthe mole and energy balance equations.

Assuming available experimental data and known stoichiometry (the reac-tions), we must search for an identification methodology for determining thebest kinetic model. There are several techniques available in the literature,such as differential, integral and incremental methods [12]. The differentialmethod compares the right-hand side of the model with the derivatives of thedata. The incremental method works with the “extent” concept, which pro-vides an analytic solution of a new decoupled system. The integral methodsolves numerically the model and compares it with the data. In any case,an optimization problem is constructed, depending on the kinetic parameters.Furthermore, if better results are desired, a combination of these methods isrecommended.

In a theoretical framework, the mathematical analysis of the models men-tioned above has driven out curiosity in the spirit of scientific inquiry. Par-ticularly, the general convection-diffusion-reaction equations have enjoyed aconsiderable amount of scientific interest. They can be studied from differentapproaches using a variety of different methods from many areas of mathemat-ics. Some of those studies are based on bifurcation and stability or semigrouptheory, or variational approach. After the theoretical analysis a natural ques-tion is related to the numerical solution of the model. It can be based on finitedifference schemes, in most of the cases, and in a few others on finite elementsmethod.

This thesis is divided in three different parts in which an extensive studyof reactors is done, from theoretical and practical point of view. It is comple-mented with three appendices describing some tools and results used through-out the thesis in order to get a self-contained work. In the following we describe

4 Preface

briefly the contents of each part.

I Modelling chemical reactors

The first part is devoted to the description of the reactors we are in-terested in. At first, we recall basic concepts about chemical speciesand reactions. We also introduce the functional form of reaction veloci-ties recalling the most important ones from the literature ([46], [34] and[31]). They play an important role in Part III. Next, we formulate themodels of the main STR reactors (batch, semi-batch and CSTR), withmole and energy balance equations, both in transient and steady state,and assuming constant density. These models are extensively used in in-dustry, specifically in kinetic identification [17]. Then, we describe thegeneral convection-diffusion-reaction model that will be applied in theanalysis of a particular reactor of this type, namely PFR. The existenceand uniqueness of solution is studied and the behaviour of the error isanalyzed when numerical methods are employed. Finally, we obtain theFBR model, describing the boundary conditions and distinguish betweentwo cases, resistance and not resistance. This reactor is the most com-plex of those considered in this work. In fact, its mathematical analysisis beyond the scope of this thesis.

II Mathematical analysis and numerical solution

In this part an extensive study is done for the convection-diffusion-reactionmodel beginning with the mathematical analysis for the n−dimensionalreactor and then numerical solution of the reactor models is designed. Theproof of local existence and uniqueness of solution is based on the semi-group theory. The global existence is proved via L∞ estimates explodingtwo properties (called (P) and (M)) described in Chapter 4. In Chap-ter 5 an error estimation is obtained following the techniques from [63].A priori we have proved the existence of solution of the semi-discretizedproblem, using the Picard-Lindelof theorem. For the numerical solutionwe use a finite difference scheme for the PFR model and a finite elementmethod for the FBR model.

III Identification in reaction systems

This last part is related to model identification. More precisely, it dealswith the identification of the best kinetic model from a list of proposedfunctional forms, and also of the values of its corresponding parameters bymeans of an optimization process. In order to solve the problem, we usea combination of an incremental and an integral method. The idea of thefirst one is to decompose the identification task into a set of subproblems,one for each kinetic model. Meanwhile, the second one is based on adirect comparison of species measurements and computed concentrations

Preface 5

via the theoretical model. Due to this, the second method is obviouslymore expensive and works better with the parameters obtained from theincremental method. Such identification processes are usually studied insystems where the phenomena of interest can be observed in isolation,without other physical phenomena. This is the case for the identificationof reaction kinetics in liquid phase, where a stirred batch or semi-batchreactor is used in the majority of cases, as explained in [17]. For thisreason, we focus on stirred tank reactors, using a set of experimental dataand the reactions taking place. A catalogue of kinetic models containingthe parameters to be identified will be provided too.

Appendices

Appendix A summarizes the general equations of continuum thermome-chanics for reacting mixtures.

Appendix B describes some basic definitions and theorems of semigrouptheory needed in the proof of local existence theorem.

Appendix C describes some useful bounds for the error estimates.

Appendix D contains a summary of this dissertation in Spanish.

6 Preface

Part I

Modelling chemicalreactors

7

Introduction

Chemical reactors are widely used by chemical engineers in industry to trans-form raw materials into final products. The main purposes are related tomaximizing benefits, which is equivalent to “design” an optimal reactor, andthus minimizing the production costs. Hence, adequate modelling and correctanalysis are essentials. Together with chemical kinetics they are the scientificbasis for the analysis of most engineering procedures, occurring in nature orrelated to synthetic processes.

An important part in reactor modelling is the reaction rate that representsthe measure of the variation in concentration of the reactants or the variationin concentration of the products per unit time. Reaction rate can be a pri-ori understood, independently of the reactor shape and length. The overallchemical process also depends on the reactor size.

Many processes have been traditionally modelled as ideal reactors: stirredtank or plug flow reactors. This type of modelling is mainly based on thereactor features and phenomena such as heat or mass transfer.

In the first chapters we formulate the mathematical modelling of stirred tankreactors (STR) and plug flow reactors (PFR). We consider both the transientand the steady-state cases. Reactors are not supposed to be either adiabaticor isotherm, so temperature as well as species concentrations have to be com-puted by the models that are obtained from the energy and mass conservationequations, respectively.

Firstly, we consider lumped parameter models (or zero-dimensional mod-els), where the thermo-mechanical magnitudes do not depend on the particularposition in the reactor. They correspond to the so-called stirred tank reac-tors. We write mathematical models for batch and semi-batch STR and alsofor continuous STR. Then we describe plug-flow reactors (PFR). In this case,the thermo-mechanical magnitudes depend on a spatial variable (as well as ontime). Hence, the mathematical models are described by partial differentialequations involving derivatives with respect to spatial variable and time.

The mathematical model for a reaction-diffusion system was initially derivedfrom the work of Alan Turing in 1952 [65] and it was used for the first time in

9

10 Introduction

chemistry. However, it can also describe dynamical processes of non-chemicalnature. Some fields of application are biology, geology, physics, epidemiology,oncology or environmental engineering processes, among others.

From a qualitative point of view, a reaction-diffusion system is a mathemat-ical model describing how the concentration of one or more substances variesover time and space under the influence of two terms: the reaction term, inwhich concentrations are generated or degenerated by interaction, and diffusionthat generates substances expansion in space.

The idea of Chapter 2 in this modelling part is to consider the model de-scribed in Chapter 1 as a general n−dimensional model which can be particu-larized to recover the PFR model. The mathematical model is a coupled systemof partial differential equations involving gradient and laplacian with respectto spatial variables and partial time derivative.

Finally, the most sophisticated reactors we consider are the fixed bed re-actors (FBR). As explained in [57], the first commercial application of thesereactors dates from 1831 when a vinegar maker developed a process for makingsulfur trioxide using air and sulfur dioxide in bed of platinum sponge previouslyheated. After that, he patented it. Since the catalyst was not consumed, thecontinuous flow of reactants was passed over the bed, without the need of re-cycling the catalyst. Nowadays, the typical application is related to the designand development of a catalyst that improves the conversion of an intermediateproduct to the final product.

These reactors are understood in this thesis as heterogeneous reaction sys-tems in which plug-flow is assumed. The model is based on the conservationlaws for mass, energy and momentum and lead to partial differential equations.We consider a multi-scale model. The bed is modelled as a continuum of smallparticles (solids) containing the catalyst and interacting with the fluid. Thisparticles are consider of spherical shape and hence spherical symmetry is as-sumed. The fluid bulk is modelled as a fluid flowing in a porous media. Wehave be to careful with the coupling between the models.

Chapter 1

Modelling stirred tankreactors

The study of reactors is initially performed by means of ideal models. Stirredtank reactors are a good example of these kind of reactors, as a perfectly mixedvolume is considered and the behaviour of the reacting system is not influencedby the fluid dynamic conditions.

In this chapter we introduce models for both transient and steady-statestirred tank reactors (STR). We focus on the three following stirred tank reac-tors types:

• Batch STR: reactants are introduced into the reactor at initial time only.There are no input/output flows along the process.

• Semi-batch STR: some of the reactants are introduced in the reactorat the initial time; some others are continuously introduced along theprocess.

• Continuous (flow) stirred tank reactor (CSTR): reactants are continu-ously introduced along the time. There is also an output flow along theprocess.

The main assumption in these reactors is that the mixture inside is perfectlyhomogeneous because of stirring, so the physico-chemical magnitudes do notdepend on position.

11

12 CHAPTER 1. MODELLING STIRRED TANK REACTORS

1.1 Modelling chemical reactions

In this section we introduce the tools needed to construct the reaction term inthe model. Let us consider a set of reacting chemical species:

S = E1, . . . , EN.

Let Mi be the molecular mass of species Ei. All species are involved in a setof L chemical reactions

νl1E1 + ...+ νlNEN → λl1E1 + ...+ λlNEN , 1 ≤ l ≤ L,

where νli and λli, i = 1, . . . , N , l = 1, · · · , L are called stoichiometric coeffi-cients.

1.1.1 Chemical species and elements. Conservation rela-tions

Let us suppose that the species are formed by K different chemical elements,named Hk, 1 ≤ k ≤ K. Let the formula of species Ei be

Ei = (H1)h1i· · · (HK)hKi , 1 ≤ i ≤ N. (1.1)

Since atoms are conserved in chemical reactions, we have

N∑i=1

hkiνli =

N∑i=1

hkiλli, k = 1, · · · ,K, l = 1, · · · , L.

Matrix (H)ki = hki is called element (or atomic) matrix . The aboverelations can be written in a more compact form as follows

HA = 0,

where A is the stoichiometric matrix:

(A)il := λli − νli , i = 1, · · · , N, l = 1, · · · , L.

From the mass conservation in the l-th chemical reaction

N∑i=1

Miλli =

N∑i=1

Miνli , l = 1, · · · , L

we can easily deduce thatAtM = 0, (1.2)

where M is the column vector of the molecular masses of species.Hence,

dim(ker(At)) ≥ 1.

1.2. MODELLING BATCH STIRRED TANK REACTORS 13

1.1.2 Finite rate chemical reactions

For elementary reactions, the law of mass action (C.M. Guldberg and P. Waagein [44]) yields the following expression for the velocity of the l-th reaction:

δl(θ, y1, . . . , yN ) = kl(θ)

N∏j=1

yνljj . (1.3)

Along this work we will focus on these reaction rate functional forms andassume that coefficients νlj are positive integer numbers.

Many other reactions can be modelled by similar functions, but with ex-ponents that are different from the stoichiometric coefficients on the left-handside of the reaction, namely,

δl(θ, y1, . . . , yN ) = kl(θ)

N∏j=1

yαljj . (1.4)

In general, δl can be any function. Examples of kinetics different from the massaction law are the Hill [34] or Michaelis-Menten kinetics [46].

1.1.3 Reaction rate constant

Factor kl = kl(θ) is called reaction rate constant of the l-th reaction. As thenotation indicates, it is not constant but a function of the reactor temperatureθ through the

Arrhenius law

kl(θ) = Blexp(−EalRθ

), (1.5)

where Bl is the pre-exponential factor, Eal is the activation energy of the l-threaction and R is the universal gas constant.

1.2 Modelling Batch Stirred Tank Reactors

The batch reactors are typically used in industry in many processes such as dis-solution of solids, mixing reactants, chemical reactions, polymerization, amongothers.

These reactors consist of a container with a mixer. Their dimensions canvary in a large range from less than 1 litre to more than 10000 litres. Theyare usually built with steel, glass-lined steel or glass. Reactants are usuallyintroduced at the top of the reactor at the beginning of the process. We cansee the illustration of the reactor below:


y(t)θ(t)

y0

Figure 1.1: Batch Stirred Tank Reactor

We consider the case where the mixture density can be assumed to be con-stant. In order to derive the full model, we need to compute the composition,temperature and density of the reacting mixture along time, so several ODEsmust be introduced.

The species conservation equations

In transient Batch STRs time evolution of concentrations yi (kmol/m3), of thechemical species Ei, i = 1, . . . , N, satisfy the following system of ODEs:

d(V yi)

dt= V

L∑l=1

(λli − νli)δl(θ, y1, . . . , yN ),

where function δl is the velocity of the l-th reaction and V (t) (m3) is the volumeoccupied by the mixture at time t.

Notice that the mole (mol) is the SI unit for the amount of a chemicalsubstance. However, we use kmol due to the practical application we do ofthese models later.

In order to get a well-posed problem, initial conditions must be prescribed;more precisely, the initial concentration of each species y0,i must be given.Finally, the model can be written in matrix form as follows:

d(V y)

dt= V Aδ(θ,y), (1.6)

y(0) = y0. (1.7)


The energy equation

Sometimes the temperature is given. This is the case, for instance, if the reactoris isothermal. Otherwise, the temperature evolution must be computed by amodel that arises from the energy conservation principle.

Let us assume that the reactor exchanges heat with its surroundings. Wedenote by θext(t) the outside temperature along time and by g(t) (W/K) theheat transfer coefficient between the reactor and its surroundings. If e(t) (J/kg)denotes the specific internal energy of the mixture and ρ(t) its density at timet, then the total internal energy is given by V (t)ρ(t)e(t) and the energy con-servation principle yields the ordinary differential equation

d(V ρe)

dt= g(θext − θ). (1.8)

We want to eliminate e from this equation in terms of θ. Let us assume that foreach species there is a function ei(θ) giving internal energy from temperature(this is usually true for liquids and also for perfect gases):

ei(t) = ei(θ(t)),

with

ei(θ) := e∗i +

∫ θ

θ∗cvi(s) ds,

where e∗i is the internal energy of formation of the i-th species at the referencetemperature θ∗ (usually, θ∗ is taken to be the so-called standard temperature,i.e., 25oC) and ci = cvi(θ) is the specific heat of the i-th species (J/(kgK)).For the mixture,

e(t) =

N∑i=1

Yi(t)ei(t),

where Yi(t) denotes the mass fraction of species Ei at time t which is relatedto the concentration by

Yi =Miyiρ

.

Hence,

ρe =

N∑i=1

ρYiei =

N∑i=1

Miyiei

and then

d(V ρe)

dt=

N∑i=1

Mi

(d(V yi)

dtei + V yi

deidt

)= V

N∑i=1

Mi

(ei(Aδ(θ,y)

)i+ yicvi(θ)

dθ

dt

).

(1.9)


Let us define the specific heat of the mixture c by

c :=

N∑i=1

Yici.

Then,

ρc =N∑i=1

Miyici =

N∑i=1

w′i(θ)yi = w′(θ) · y, (1.10)

where the components of vector w(θ) ∈ RN are the molar internal energiesdefined by

wi(θ) :=Miei(θ) (J/mol).

By using (1.9) and (1.10), the energy equation (1.8) becomes

ρcdθ

dt= −w(θ) ·Aδ(θ,y) +

g

V(θext − θ)

= −Atw(θ) · δ(y, θ) +g

V(θext − θ)

= −∆H(θ) · δ(θ,y) +g

V(θext − θ),

(1.11)

where the components of the L-dimensional vector

∆H(θ) := Atw(θ) (J/mol)

are the molar heats of reaction (absorbed by the reaction) at temperatureθ. Hence,

∆Hl(θ) =

N∑i=1

(λli − νli)Miei(θ).

From the computational point of view, it is convenient to divide equation (1.11)by ρc,

dθ

dt= − 1

ρc

(∆H(θ) · δ(y, θ)− g

V(θext − θ)

).

By using (1.10) we finally get

dθ

dt= −

∆H(θ) · δ(θ,y)− g

V(θext − θ)

w′(θ) · y. (1.12)

Moreover, an initial condition is needed:

θ(0) = θ0. (1.13)


1.2.1 The transient batch STR model

Let us assume that the volume occupied by the mixture is constant along timeand given (for instance, equal to the reactor volume). Then the density ofthe mixture is also constant and the mathematical model for the batch STRbecomes the following initial-value problem:

dy

dt= Aδ(θ,y),

dθ

dt= −

∆H(θ) · δ(θ,y)− g

V(θext − θ)

w′(θ) · y,

y(0) = y0,

θ(0) = θ0.

Remark 1.2.1. Since the total mass of the mixture is conserved in a BatchSTR, if the volume does not change, then the mixture density is also constantalong time. In the case of a mixture of perfect gases, the pressure in the reactorchanges along time and is given by

p(t) = ρR(t)θ(t),

where R(t) is the gas constant of the mixture at time t which is given by

R(t) =RM(t)

,

being R the universal gas constant and M(t) is the molar mass of the mixtureat time t defined by

1

M(t)=

N∑i=1

Yi(t)

Mi.

1.2.2 Steady-state Batch STR

Usually, when the outside temperature θext is time independent from a certaintime onwards, the solution of the batch STR model above tends to a steady-state solution (i.e., time independent) as the time increases. This solution is atriple (y, θ, V ), being y a vector in RN and θ and V real numbers.

In the constant density case V is given. Then, the model for a steady-statebatch STR is the numerical non-linear system,


Aδ(θ,y) = 0,

∆H(θ) · δ(θ,y)− g

V(θext − θ) = 0.

We notice that the fact that the above systems have the same number of equa-tions as unknowns does not imply that they have a unique solution.

1.3 Modelling Semi-Batch and Continuous Sti-rred Tank Reactors

Continuous stirred tank reactors, also called ideal stirred tank reactorsor CSTR are, maybe, the most used reactors in chemical industry. In mostcases they operate at steady state and are considered as homogeneous reactorsdue to their well mixing properties.

These reactors have also vessel form, but there exist a continuous inputflow and, in the case of CSTR, there is also a continuous output flow. For thesake of completeness, let us assume that the input flow is obtained by mixingseveral streams, each of them characterized by the following magnitudes thatcan be function of time:

• Flow rate (m3/s)

• Composition (in terms of concentrations, mol/m3)

• Temperature (K)

Usually, for semi-batch STR there are only input streams, so the volume ofthe mixture must change along the time. In the case of CSTR there is also anoutput stream, so this volume could be constant.

Let P be the number of input streams. The above magnitudes are denotedby

u1(t), · · · , uP (t) (m3/s),

W1(t), · · · ,WP (t) ∈ RN (mol/m3),

θ1(t), · · · , θP (t) (K).

Regarding the outflow, we assume there is only one stream. Its compositionand temperature at time t are those of the mixture in the reactor at that time.We are interested in considering the case where the output flow rate can bedifferent from the sum of the input flow rates. Therefore, the volume of the

1.3. MODELLING SEMI-BATCH AND CSTR 19

mixture in the reactor may change along time. In fact, the volume can alsochange due to changes in composition and/or temperature as in batch STR.

Let us show the images of both semi-batch and CSTR in order to illustratethe above description:

y(t)

W (t) u(t)θ(t)

V (t)

y0

Figure 1.2: Semi-batch STR

y(t)

W (t) u(t)uout(t)θ(t)

V (t)

y0

Figure 1.3: CSTR

1.3.1 The transient semi-batch and CSTR model

Firstly, the mass conservation equation (1.6) has to be replaced by

d(V y)

dt= V Aδ(t, θ,y) +Wu− yuout, (1.14)

where W (t) denotes the matrix whose columns are the vectors W1(t), · · · ,WP (t). We notice that the output flow rate uout is null for semi-batch STR.


Let us also notice that(W (t)u(t)

)i

is the number of moles per second of speciesi entering the reactor at time t, so it is a molar flow rate. Since the density

may change, we cannot assume that uout =

P∑p=1

up.

The equation for temperature (1.12) also needs to be modified in order toaccount for the convective energy flows.

Firstly, the total convective internal energy flow (W ) entering the reactorthrough the P input streams is given by

P∑p=1

ρpepup =

P∑p=1

ρp( N∑i=1

Y pi ei(θp))up =

P∑p=1

( N∑i=1

MiWpi ei(θ

p))up,

because ρpY pi = MiWpi . Similarly, the convective energy flow exiting the

reactor (W ) is given by

ρeuout = uout

N∑i=1

Miyiei(θ).

Secondly, (1.9) has to be modified as follows:

d(V ρe)

dt=

N∑i=1

Mi

(d(V yi)

dtei + V yi

deidt

)= V

N∑i=1

Mi

(ei[Aδ(θ,y) +

1

VWu− 1

Vyuout

]i+ yicvi(θ)

dθ

dt

),

(1.15)

where ei = ei(θ). Thirdly, equation (1.8) has to be replaced by

d(V ρe)

dt= g(θext−θ)+

P∑p=1

( N∑i=1

MiWpi ei(θ

p))up−uout

N∑i=1

Miyiei(θ). (1.16)

By subtracting (1.15) from (1.16) and then dividing by V , we get the equationreplacing (1.11):

ρcdθ

dt= −∆H(θ) · δ(θ,y) +

g

V(θext − θ) +

1

V

P∑p=1

( N∑i=1

MiWpi

(ei(θ

p)− ei(θ)))up.

Dividing by ρc we finally get

dθ

dt= −

∆H(θ) · δ(θ,y)− gV (θext − θ)− 1

V

∑Pp=1

(∑Ni=1MiW

pi

(ei(θ

p)− ei(θ)))up

w′(θ) · y.

1.3. MODELLING SEMI-BATCH AND CSTR 21

Let us notice that if there is no change of state then

ei(θp)− ei(θ) =

∫ θp

θ

cvi(s) ds.

Now, let us assume that the density of the reacting mixture does not dependeither on the composition or on the temperature. Then the volume only changesdue to input and output flows and the equations giving the volume are thefollowing:

dV

dt(t) =

P∑p=1

up(t)− uout(t), (1.17)

V (0) = V0. (1.18)

In this case we have

d(V y)

dt= V

dy

dt+dV

dty = V

dy

dt+ y

( P∑p=1

up(t)− uout(t))

and since we use equations (1.17) for the volume, we can write the model interms of y as follows:

dV

dt=

P∑p=1

up − uout,

dy

dt= Aδ(θ,y) +

1

VWu− 1

Vy

P∑p=1

up,

dθ

dt= −

∆H(θ) · δ(θ,y)− gV (θext − θ)− 1

V

P∑p=1

( N∑i=1

MiWpi

(ei(θ

p)− ei(θ)))up

w′(θ) · y,

V (0) = V0,

y(0) = y0,

θ(0) = θ0.


1.3.2 Steady-state CSTR model

Let us assume that the outside temperature and the input flows are time inde-pendent. Then the CSTR may attain a steady-state.

Assuming also that the mixture density is independent of composition andtemperature, then the volume of the mixture is constant and equal to the initialone which is supposed to be given. Then the model is the following non-linearsystem of numerical equations:

Aδ(θ,y) +1

V(Wu− uouty) = 0,

∆H(θ) · δ(θ,y)− g

V(θext − θ)−

1

V

P∑p=1

( N∑i=1

MiWpi

(ei(θ

p)− ei(θ)))up = 0.

Remark 1.3.1. Let us notice that the energy equation has the same form forthe constant and variable density cases. Moreover, by using (1.16) it can alsobe written as

g(θext − θ) +

P∑p=1

( N∑i=1

MiWpi ei(θ

p))up − uout

N∑i=1

Miyiei(θ) = 0.

Chapter 2

Modelling n−dimensionalconvection-diffusion-reactionsystems

Mathematical models for heat and mass transfer in this type of reactors areusually called in the literature convection-reaction-diffusion systems. In thissense, the term convection-reaction-diffusion systems refers to models whichproduce locally transformations of chemical species by chemical reactions andat the same time are transported in the reactor by convection and diffusion.They appear obviously in chemical engineering, but they have been used in thestudy of different phenomena in biology, geology or physics.

We study this type of reactors in a general mathematical framework. In thenext chapters we will first prove a theorem of existence of solution and then wewill address the numerical analysis by using finite element methods which willinclude error estimates.

2.1 Modelling the convection-diffusion-reactionsystem

Let us consider Ω an open bounded set in Rn with smooth boundary ∂Ω, asrepresented in Figure 2.1, and let ν be the outward unit normal vector to ∂Ω.Moreover, let Γ1 denote the reactor inlet boundary, Γ2 the reactor outlet andΓ3 the reactor wall.

23

24 CHAPTER 2. CONVECTION-DIFFUSION-REACTION MODEL

Ω

1

2

3

Figure 2.1: n−dimensional domain with bounds

Let y(t, x) (mol/m3) be the vector of species concentrations involved in thereactions, where (t, x) ∈ (0, T )×Ω. Then, the mass balance of species leads tothe following system of partial differential equations (see (A.1)):

∂y

∂t+∇yv −D∆y = ϕ, (2.1)

where

• v (m/s) is the (given) velocity. We consider the case of an incompressiblefluid and hence div v = 0 . We also assume that v is time independent.

• D (m2/s) is the diagonal matrix of the diffusion coefficients of specieswhich are assumed constant and strictly positive.

• ϕ(t, x,y, θ) (mol/(m3 s)) denotes the source term also called reactionterm. More precisely, it represents the vector of reaction rates describedin Section 1.1.2 and corresponding to the law of mass action, multipliedto the left by the stoichiometric matrix A.

Similarly, the general energy conservation equation for the fluid bulk canbe written as follows (see (A.4)):

∂(ρe)

∂t+ div(ρev) + divq = 0, (2.2)

where

• ρ (kg/m3) is density of the mixture (it may be variable along the time),

• e (J/kg) is the specific internal energy of the bulk fluid,

2.1. THE CONVECTION-DIFFUSION-REACTION MODEL 25

• q (W/(m2) is the heat flux vector given by Fourier’s law q = −k grad θ,

• k (W/(mK)) is the effective coefficient of thermal conductivity (is a pos-itive constant).

Let us recall that

e =

N∑i=1

Yiei,

where Yi denotes the mass fraction of species Ei which is related to its concen-tration by

ρYi =Miyi

and the specific internal energy of species Ei depends on temperature: ei =ei(θ). Hence,

ρe =

N∑i=1

ρYiei =

N∑i=1

Miyiei

and then

∂(ρe)

∂t=

N∑i=1

Mi

(ei∂yi∂t

+ yi∂ei∂t

). (2.3)

Similarly,

div(ρev) = ρedivv + v · ∇(ρe) = v · ∇(ρe) =

N∑i=1

Mi (ei∇yi · v + yi∇ei · v) .

(2.4)

By adding (2.3) and (2.4) we obtain

∂(ρe)

∂t+∇(ρe) · v

=

N∑i=1

Miei

(∂yi∂t

+∇yi · v)

+

N∑i=1

Miyi

(∂ei∂t

+∇ei · v)

=

N∑i=1

wi(θ)

(∂yi∂t

+∇yi · v)

+

N∑i=1

Miyicvi(θ)

(∂θ

∂t+∇θ · v

),

(2.5)

where cvi(θ) is the specific heat at constant volume of the i-th species and wi(θ)is defined in (2.22). Let us recall that the specific heat at constant volume ofthe mixture is defined by

cv(θ) =

N∑i=1

Yicvi(θ)


and hence,N∑i=1

Miyicvi(θ) = ρ

N∑i=1

Yicvi(θ) = ρcv(θ).

By using (2.1) and the previous equality we get

∂(ρe)

∂t+∇(ρe) · v = w(θ) · [D∆y +ϕ] + ρcv(θ)

(∂θ

∂t+∇θ · v

). (2.6)

The mass diffusion term in the previous equation can be neglected againstthe source terms ϕ. Thus, the energy equation (2.2) can be finally written as

ρcv(θ)(∂θ∂t

+∇θ · v)− k∆θ = −w(θ) ·ϕ. (2.7)

2.1.1 Boundary conditions

The boundary conditions are imposed on the boundary of the reactor.

• Reactor entrance (Γ1).

1. Mass:

D∂y

∂ν(t, x)−(v(x)·ν(x))y(t, x) = g(t, x), (t, x) ∈ (0, T )×Γ1, (2.8)

where g (mol/(m2s) is an enough smooth function, representing themolar flux os species entering the reactor at time t and position x.

2. Energy:

θ(t, x) = θin(t), (t, x) ∈ (0, T )× Γ1. (2.9)

• Reactor exit (Γ2).

1. Mass:

D∂y

∂ν(t, x) = 0, (t, x) ∈ (0, T )× Γ2. (2.10)

2. Energy:

k∂θ

∂ν(t, x) = 0, (t, x) ∈ (0, T )× Γ2. (2.11)

• Reactor walls (Γ3).

1. Mass:

D∂y

∂ν(t, x) = 0, (t, x) ∈ (0, T )× Γ3. (2.12)

2.1. THE CONVECTION-DIFFUSION-REACTION MODEL 27

2. Energy:

k∂θ

∂ν(t, x) = hext

(θext(t)− θ(t, x)

), (t, x) ∈ (0, T )× Γ3, (2.13)

where hext (W/(m2K)) is a heat transfer coefficient between thereactants and the exterior of the reactor and θext denotes the tem-perature of the latter.

2.1.2 Initial conditions

1. Mass:

y(0, x) = y0(x). (2.14)

2. Energy:

θ(0, x) = θ0(x). (2.15)

2.1.3 The full n-dimensional model

Finally, the full model for a n− dimensional convection-diffusion-reaction sys-tem can be writen as:

∂y

∂t+∇yv −D∆y = ϕ,

ρcv(θ)(∂θ∂t

+∇θ · v)− k∆θ = −w(θ) ·ϕ,

D∂y

∂ν(t, x)− (v(x) · ν(x))y(t, x) = g(t, x), (t, x) ∈ (0, T )× Γ1,

θ(t, x) = θin(t), (t, x) ∈ (0, T )× Γ1,

D∂y

∂ν(t, x) = 0, (t, x) ∈ (0, T )× Γ2,

k∂θ

∂ν(t, x) = 0, (t, x) ∈ (0, T )× Γ2,

D∂y

∂ν(t, x) = 0, (t, x) ∈ (0, T )× Γ3,

k∂θ

∂ν(t, x) = hext

(θext(t)− θ(t, x)

), (t, x) ∈ (0, T )× Γ3,

y(0, x) = y0(x), θ(0, x) = θ0(x).


2.2 A particular one-dimensional reactor: thePlug Flow Reactor (PFR)

Plug flow reactors are also called continuous tubular reactors and pistonflow reactors. What does plug flow mean? In principle, due to viscosity, thevelocity of the flow in a pipe is null on the wall. Moreover, in the laminarregime, the velocity profile is parabolic with the maximum at the central axisof the pipe. The plug flow is a simple model where the velocity is assumed tobe constant on any cross-section of the pipe (but it may depend on time).

In order to write a model for a transient PFR the conservation equations(see Appendix A) are integrated on the cross-sections of the pipe leading to aone-dimensional model. Let z be the axial coordinate of the reactor of lengthL. Then z ∈ [0, L].

We make the following assumptions:

• All thermodynamic magnitudes depend only on z and t.

• Velocity: v = vzez with vz independent of x and y (plug flow assump-tion). In what follows we drop subscript z from vz.

• The conductive term can be neglected, namely, term Tv · D is droppedout of the equations (where Tv is the viscous stress tensor and D =12 (gradv + gradvt) is the strain rate).

• There is no external volumetric heat source, i.e., f = 0.

2.2.1 Transient Plug Flow Reactors

In general, the mixture is not incompressible and the velocity may depend onz and t. But we work with constant density. In this case we assume thatthe initial density of the mixture in the reactor is a constant function (i.e.,homogeneous) and equal to the density of the input mixture which is thenalso constant along the time. From the mass conservation equation (A.2) wededuce that the mixture is incompressible, i.e., divv = 0 which in the present

case means∂vz∂z

= 0, then vz is independent of z, thus vz = v(t) (in what

follows we drop subscript z for the sake of simplicity). Therefore, in this casev(t) is supposed to be given (in fact, it is the velocity of the input current atthe inlet of the reactor which can be obtained from its volumetric flow ratedividing by the area of the reactor cross-section). Hence, neither the motionequation nor the state equation are needed.

We can see the illustration of the reactor below:

2.2. MODELLING PLUG FLOW REACTORS 29

y(0, t)θ(0, t)Input Output

z

y(z, t)θ(z, t)

Figure 2.2: Plug Flow Reactor

Species mass conservation

By dividing by the molecular mass Mi and introducing the concentration ofspecies,

yi =ρYiMi

,

equations (A.1) can be rewritten. In reactors of this type, the diffusion isusually neglected. However, in industrial reactors although the convectionterm predominate, diffusion also occurs, so we have

∂yi∂t

+∂(yiv)

∂z− di

∂2yi∂z2

=

L∑j=1

aijδj(θ,y), i = 1, ..., N, (2.16)

because fields only depend on z and t. (2.16) can be written in a more compact

form as∂y

∂t+∂(yv)

∂z−D∂

2y

∂z2= Aδ(θ,y). (2.17)

Energy equation

Firstly, the specific (i.e., per unit mass) internal energy of the i-th species isgiven by

ei(z, t) = ei(θ(z, t)),

with

ei(θ) := e∗i +

∫ θ

θ∗cvi(s) ds,


where e∗i is the internal energy of formation of the i-th species at temperature θ∗

and ci = cvi(θ) is the specific heat of the i-th species (J/(kgK)) at temperatureθ. For the mixture we have

e =

N∑i=1

Yiei

and hence,

ρe =

N∑i=1

ρYiei =

N∑i=1

Miyiei.

Therefore,

∂(ρe)

∂t+∂(ρve)

∂z=

N∑i=1

Mi

(∂(yiei)

∂t+∂(yieiv)

∂z

)=

N∑i=1

Miei(∂yi∂t

+∂(yiv)

∂z

)+

N∑i=1

Miyi(∂ei∂t

+ v∂ei∂z

).

(2.18)

From the definition of specific heat, the chain rule yields

∂ei∂t

= ci∂θ

∂t,

∂ei∂z

= ci∂θ

∂z.

By using these equalities and (2.16) we get

∂(ρe)

∂t+∂(ρve)

∂z=

N∑i=1

Miei(Aδ(θ,y) +D

∂2y

∂z2

)i+

N∑i=1

Miyici(∂θ∂t

+ v∂θ

∂z

).

(2.19)

Moreover, let c be the specific heat of the mixture defined by

c :=

N∑i=1

Yici.

We have

ρc =

N∑i=1

Miyici. (2.20)

Let us assume that the reactor is adiabatic, i.e., there is no heat exchangewith the exterior. Thus, by using (2.19), the energy equation (A.4) becomes

ρc∂θ

∂t+ρcv

∂θ

∂z−k∂

2θ

∂z2= −w(θ)·Aδ(θ,y) = −Atw(θ)·δ(θ,y) = −∆H(θ)·δ(θ,y),

(2.21)


where the components of vector w(θ) ∈ RN are defined by

wi(θ) =Miei(θ) (J/mol) (2.22)

and the components of the L-dimensional vector

∆H(θ) := Atw(θ) (J/mol)

are the heat of the reactions at temperature θ. We have,

∆Hl(θ) =

N∑i=1

(λli − νli)Miei(θ).

Summarizing, the energy conservation equation for an adiabatic transientPFR is the following nonlinear partial differential equation system:

ρc(∂θ∂t

+ v∂θ

∂z

)− k∂

2θ

∂z2= −∆H(θ) · δ(θ,y), (2.23)

with

ρc = w′(θ) · y. (2.24)

Boundary conditions

Let us notice that the above partial differential equations are of first orderin time and second order in space. Hence, boundary conditions has to beprescribed for each of them:

y(0, t) and y(L, t) given, t ∈ (0, T ),

θ(0, t) and θ(L, t) given, t ∈ (0, T ).

Initial conditions

The following initial conditions has to be given:

• y(z, 0), z ∈ (0, L),

• θ(z, 0), z ∈ (0, L).

The non-adiabatic case

In the above models we have assumed that the reactor is thermally isolated, sothere is no heat exchange with the exterior. In this case we say that the PFRis adiabatic.


Now, let us consider the case where the reactor exchanges heat with itssurroundings according to the Newton convection law. This means that thereactor heat loss per unit surface and time is given by

h (θ − θext) (W/m2), (2.25)

where h (W/(m2K)) is a convective heat transfer coefficient depending on theoutside cooling (e.g., natural convection, forced convection, ...) and θext(z, t)(K) is the outside temperature. If the reactor is a cylinder with radius R, thenthe energy equation (2.23) must be replaced by the following one:

ρc(∂θ∂t

+ v∂θ

∂z

)− k∂

2θ

∂z2= −∆H(θ) · δ(θ,y) +

2πRh

πR2(θext − θ). (2.26)

Thus, the full model for a non-adiabatic PFR in transient state with con-stant density is

∂y

∂t+ v

∂y

∂z−D∂

2y

∂z2= Aδ(θ,y),

(w′(θ) · y)

(∂θ

∂t+ v

∂θ

∂z

)− k∂

2θ

∂z2= −∆H(θ) · δ(θ,y) +

2h

R(θext − θ),

y(0, t) and θ(0, t) are given,

D∂y

∂z(t, L) = 0,

k∂θ

∂z(t, L) = 0,

y(z, 0) = y0(z), θ(z, 0) = θ0(z).

2.2.2 Steady-state PFR

In this case the thermodynamic magnitudes do not depend on time, so par-tial derivatives with respect to time disappear from model. We also assumeconstant density.


vdy

dz−D∂

2y

∂z2= Aδ(θ,y),

(w′(θ) · y)v∂θ

∂z− k∂

2θ

∂z2= −∆H(θ) · δ(θ,y) +

2h

R(θext − θ),

y(0), y(L), θ(0) and θ(L) are given.

Remark 2.2.1. Notice that if the diffusion terms are neglected, the abovemodel is similar to the one corresponding to a batch STR. Indeed, by makingthe change of variable,

t =z

v,

d

dz=

1

v

d

dt,

the above equations and boundary conditions yield the following initial-valueproblem:

dy

dt= Aδ(θ,y),

dθ

dt= − 1

w′(θ) · y

(−∆H(θ) · δ(θ,y) +

2h

R(θ∞ − θ)

),

y(0) and θ(0) are given.


Chapter 3

Modelling catalytic fixedbed reactors (FBR)

In this chapter we derive the model for fixed bed reactors, also called packedbed reactors (PBR) or packed bed catalytic reactors. We focus the studyin the continuum models which are frequently used in important industrial pro-cesses. Some of them are the ethylene oxidation and the oxidation of methanolto formaldehyde. Despite of the existence of newer type of reactors such asfluidized bed reactors, the packed bed reactors are extensively used for bothlarge scale processing in petroleum and basic chemical industry.

In fact, in industry a bundle of tubes filled with catalyst is considered,usually arranged within a large reactor shell. In these terms, it is assumedthat the temperature in the tube remains constant and that the conditionsare equal in each tube (there is a fluid around the tubes to keep an adequatetemperature). But this does not happen in practice, where the reactions mayhave a significant effect on the reactor.

In our framework, the term “packed bed reactor” is related to a singlecylindrical shell with convex heads with a packed bed of catalytic particles ofuniform size, which are immobilized or fixed within the tube. A fluid mixtureof reactants is introduced at the reactor entrance which moves along the reactorand interacts with the catalytic active particles; the reactions usually produceheat exchanges. If it is necessary, the temperature is regulated through thewall of the tube.

We consider FBRs as heterogeneous reactors. Plug-flow is assumed, i.e.,v = vez, where z is the axial direction.

The model for these reactors is based on the conservation laws for mass,energy and momentum, and leads to partial differential equations. Due to thecomplexity of the system, the description of packed bed reactors must be sim-

35

36 CHAPTER 3. MODELLING CATALYTIC FIXED BED REACTORS

plified. For this reason, there are different valid packed bed reactor models. Infact, each problem should be analyzed to do the adequate simplifying assump-tions. In some cases, the reactor can be considered as pseudo-homogeneous. Ifthe differences between the fluid and solid phases are significant, heterogeneousmodels have to be considered. Moreover, sometimes intra-particle resistanceshould be taken into account.

We consider a multi-scale model. The bed is modelled at the micro-scalelevel as a continuum of small particles of solid material containing the catalystand interacting with the fluid. In what follows we will assume these parti-cles spherical, but other geometries as cylinder or slab can be considered bystraightforward modifications in the model. The fluid bulk is modelled at themacro-scale level as a fluid flowing in a porous media. For the macro-scalemodel, the effect of the micro-scale is represented by source terms both in thespecies concentration equations and in the energy equation. In its turn, themicro-scale model is coupled to the macro-scale magnitudes through boundaryconditions.

In what follows, we denote with superscript f the magnitudes for the macro-scale and by superscript s those for the micro-scale. We assume that all fieldshave cylindrical symmetry in the macro-scale. Thus, we write the equationsin cylindrical coordinates in order to exploit this fact by reducing the spatialdimension. We denote by r the radial coordinate and by z the axial one. Wealso assume that inside the spherical particles all fields have spherical symmetry,i.e., their spatial dependence is only through the radial variable which will bedenoted by rs. Summarizing, we propose below a heterogeneous model whichassumes that the chemical reactions take place both in the fluid bulk and insidethe particles at the micro-scale level.

In order to visualize the above description of the reactor we show a diagramin Figure 3.1.

yf

θf

ys

θsy

fin

θfin

Catalytic bed

Input Output

Figure 3.1: Fixed bed reactor

3.1. MODELLING THE MACRO-SCALE (FLUID BULK) 37

3.1 Modelling the macro-scale (fluid bulk)

We assume cylindrical symmetry in the macro-scale. Then, the following do-main is considered:

Figure 3.2: Macro-scale domain

Let us denote by εf (r, z, t) the (given) bed porosity at point (r, z) in thereactor and at time t, i.e., the volume occupied by the fluid per unit reac-tor volume. If the field of concentrations in the fluid is yf (r, z, t) (mol/m3),then the concentrations with respect to the total volume of the reactor will beεf (r, z, t)yf (r, z, t) (mol/m3) and the mass conservation equations of speciesbecome (see (A.1)),

∂

∂t(εfyf ) +

∂

∂z(εfyfv)− 1

r

∂

∂r

(Dfr r

∂

∂r(εfyf )

)− ∂

∂z

(Dfz

∂

∂z(εfyf )

)= Afδf (θf ,yf ) + g,

(3.1)

where

• v (m/s) is the (given) axial velocity. We consider the case of an incom-

pressible fluid so, divv =∂v

∂z= 0 and hence v cannot depend on z:

v = v(r, t)).

• Dfr and Df

z (m2/s) are diagonal matrices containing the diffusion coeffi-cients of species (this case corresponds to a particular orthotropic diffu-sion being r and z the principal directions, but other anisotropic casescould be easily considered).

• g(r, z, t) (mol/(m3 s)) denotes the amount of substance of species perunit of reactor volume and time provided by the solid phase to the liquid


bulk at point (r, z) and time t. It will be computed from the micro-scalemodel for the solid phase.

Similarly, neglecting viscous dissipation, the general energy conservationequation for the fluid bulk can be written as follows (see (A.4)):

∂(εfρfef )

∂t+∂(εfρfefv)

∂z+ divqf = f, (3.2)

where

• ρf (kg/m3) is the density of the bulk fluid,

• ef (J/kg) is the specific internal energy of the bulk fluid,

• qf (W/(m2) is the heat flux vector given by Fourier’s law qf = −kfgrad θf ,

• kf (W/(mK)) is the diagonal matrix of effective coefficient of thermalconductivities in radial and axial directions, kfr and kfz , respectively. Thesame remark as for mass diffusion can be made.

• f (W/m3) denotes the heat per unit of reactor volume and time providedby the solid phase to the liquid bulk. It will be computed from themicro-scale model.

Let us write (3.2) in cylindrical coordinates. Assuming axisymmetry itbecomes

∂(εfρfef )

∂t+∂(εfρfefv)

∂z− 1

r

∂

∂r

(kfr r

∂θf

∂r

)− ∂

∂z

(kfz∂θf

∂z

)= f. (3.3)

Let us recall that

ef =

N∑i=1

Y fi efi ,

where Y fi denotes the mass fraction of species Ei which is related to its con-centration by

ρfY fi =Miyfi

and the specific internal energy of species Ei depends on temperature: efi =ei(θ

f ). Hence,

ρfef =

N∑i=1

ρfY fi efi =

N∑i=1

Miyfi efi

and then

∂(εfρfef )

∂t=

N∑i=1

Mi

(efi∂(εfyfi )

∂t+ εfyfi

∂efi∂t

). (3.4)

3.1. MODELLING THE MACRO-SCALE (FLUID BULK) 39

Similarly

∂(εfρfefv)

∂z=

N∑i=1

Mi

(efi∂(εfyfi v)

∂z+ εfvyfi

∂efi∂z

). (3.5)

By adding (3.4) and (3.5) we obtain

∂(εfρfef )

∂t+∂(εfρfefv)

∂z

=

N∑i=1

Miefi

(∂(εfyfi )

∂t+∂(εfyfi v)

∂z

)+ εf

N∑i=1

Miyfi

(∂efi∂t

+ v∂efi∂z

)=

N∑i=1

wi(θf )(∂(εfyfi )

∂t+∂(εfyfi v)

∂z

)+ εf

N∑i=1

Miyfi cfvi(θ

f )(∂θf∂t

+ v∂θf

∂z

),

(3.6)where cvi(θ

f ) is the specific heat at constant volume of the i-th species andwi(θ

f ) has been defined in (2.22). Let us recall that the specific heat at constantvolume of the mixture is defined by

cfv (θf ) =

N∑i=1

Y fi cvi(θf )

and hence,

N∑i=1

Miyfi cvi(θ

f ) = ρfN∑i=1

Y fi cvi(θf ) = ρf cfv (θf ).

By using (3.1) and the previous equality we get

∂(εfρfef )

∂t+∂(εfρfefv)

∂z

= w(θf ) ·[

1

r

∂

∂r

(Dfr r

∂

∂r(εfyf )

)+

∂

∂z

(Dfz

∂

∂z(εfyf ))

)+Afδf + g

]+ εfρf cfv (θf )

(∂θf∂t

+ v∂θf

∂z

).

(3.7)

Usually, in FBRs the diffusion term in the previous equation can be ne-glected against source terms Afδf and g. Thus, the energy equation (3.2) canbe finally written as

εfρf cfv (θf )(∂θf∂t

+ v∂θf

∂z

)− 1

r

∂

∂r

(kfr r

∂θf

∂r

)− ∂

∂z

(kfz∂θf

∂z

)= f − w(θf ) ·

(Afδf (θf ,yf ) + g

).

(3.8)



For the fluid bulk the boundary conditions are imposed on the boundary of thereactor.

• Reactor entrance (z = 0).

1. Mass:

−Dfz

∂

∂z(εfyf )(r, 0, t) + vεfyf (r, 0, t) = vεfyfin(t), (3.9)

where yfin (mol/m3) is the vector of species concentrations in thebulk fluid entering into the reactor. The right-hand side is the vec-tor of species mass flux (mol/(m2 s)) (with respect to the entrancesurface of the reactor) entering the reactor at time t, at any point(r, 0).

2. Energy:θf (r, 0, t) = θfin(t). (3.10)

• Reactor exit (z = L).

1. Mass:∂

∂z(εfyf )(r, L, t) = 0. (3.11)

2. Energy:∂θf

∂z(r, L, t) = 0. (3.12)

• Reactor walls (r = R).

1. Mass:∂

∂r(εfyf )(R, z, t) = 0. (3.13)

2. Energy:

kf∂θf

∂r(R, z, t) = hext

(θext(t)− θf (R, z, t)

), (3.14)

where hext (W/(m2K)) is a heat transfer coefficient between the fluidbulk and the exterior of the reactor and θext denotes the temperatureof the latter.

• Reactor axis: (r = 0).

1. Mass:∂εfyf

∂r(0, z, t) = 0. (3.15)

2. Energy:∂θf

∂r(0, z, t) = 0. (3.16)

3.2. MODELLING THE MICRO-SCALE 41


1. Mass:yf (r, z, 0) = yf0 (r, z). (3.17)

2. Energy:θf (r, z, 0) = θf0 (r, z). (3.18)

3.2 Modelling the micro-scale

We assume spherical coordinates for the micro-scale. The domain shown inFigure 3.3 is considered:

Figure 3.3: Micro-scale domain

We assume that the catalytic solid consists, at the micro-scale level, ofporous spherical particles with (given) porosity εs(rs, r, z, t) (ratio of particlepore volume to particle volume) in which the species diffuse and react. Thus, ateach point of the reactor (r, z) there is a particle representative of the porousbed, interacting with the fluid located at this point. Let us write a modelfor this particular particle. We assume that all fields inside the particle havespherical symmetry which means that they only depend on time and radialvariable rs. Thus, the species mass conservation equations read as follows

∂εsys

∂t− 1

r2s

∂

∂rs

(Dsr2

s

∂εsys

∂rs

)= Asδs(θs,ys), (3.19)

where ys(rs, r, z, t) (mol/m3) denotes the vector of species concentrations in thefluid occupying the intraparticle pores of the particle located at point (r, z) of


the reactor and Ds (m2/s) is the diagonal matrix containing the mass diffusioncoefficients of species in the solid bed.

On the other hand, the energy conservation equation is

∂(ρsεses)

∂t− 1

r2s

∂

∂rs

(ksr2

s

∂θs

∂rs

)= 0. (3.20)

Similar to the fluid bulk, we have

es =

N∑i=1

Y si esi ,

where Y si denotes the mass fraction of species Ei. Since

ρsY si =Miysi ,

then

εsρses = εsN∑i=1

ρsY si esi = εs

N∑i=1

Miysi esi

and, by using (3.19),

∂(εsρses)

∂t=

N∑i=1

Mi

(∂(εsysi )

∂tesi + εsysi

∂esi∂t

)=

N∑i=1

wi(θs)∂(εsysi )

∂t+ εs

N∑i=1

Miysi cvi(θ

s)∂θs

∂t

= w(θs) ·[ 1

r2s

∂

∂rs

(Dsr2

s

∂(εsys)

∂rs

)+Asδ

s(θs,ys)

]+ εsρscsv(θ

s)∂θs

∂t.

(3.21)

In practical cases, the diffusion term in the previous equation can be ne-glected against the reaction term. Thus, the energy equation (3.20) can befinally written as

εsρscsv(θs)∂θs

∂t− 1

r2s

∂

∂rs

(ksr2

s

∂θs

∂rs

)= −w(θs) ·Asδs(θs,ys). (3.22)


Let ds(r, z, t) be the diameter at time t of the particle located at point (r, z)and Rs(r, z, t) = ds(r, z, t)/2 its radius.

3.2. MODELLING THE MICRO-SCALE 43

• Center of the particle (rs = 0). Symmetry condition.

Mass:∂εsys

∂rs(0, r, z, t) = 0. (3.23)

Energy:

∂θs

∂rs(0, r, z, t) = 0. (3.24)

• Surface of the particle (rs = Rs(r, z, t)). Different possibilities can beconsidered:

– Dirichlet condition (no resistance):

Mass:

ys(Rs(r, z, t), r, z, t) = yf (r, z, t). (3.25)

Energy:

θs(Rs(r, z, t), r, z, t) = θf (r, z, t). (3.26)

– Robin boundary conditions (resistance):

Mass:

Ds ∂εsys

∂rs(Rs(r, z, t), r, z, t) = ηfs(r, z, t)

(yf (r, z, t)−ys(Rs(r, z, t), r, z, t)

).

(3.27)Energy:

ks∂θs

∂rs(Rs(r, z, t), r, z, t) = hfs(r, z, t)

(θf (r, z, t)−θs(Rs(r, z, t), r, z, t)

),

(3.28)where ηfs and hfs are given coefficients


1. Mass:

ys(rs, r, z, 0) = ys0(rs, r, z). (3.29)

2. Energy:

θs(rs, r, z, 0) = θs0(rs, r, z). (3.30)


3.3 Sources of mass and energy in the fluid bulk

In order to complete the model, we only need to compute the source terms inthe fluid bulk equations, namely, g(r, z, t)) (respectively, f(r, z, t)), which is themass flow rate (respectively, the heat flow rate) per unit of volume provided bythe solid particles to the fluid bulk at point (r, z) at time t. For this purpose,we consider

Ds ∂εsys

∂rs(Rs(r, z, t), r, z, t)

which is the species mass flux (i.e., the rate of mass flow rate per unit area,kg/(m2s)) entering the particle across its surface. Similarly,

ks∂θs

∂rs(Rs(r, z, t), r, z, t)

is the heat flux (i.e., the rate of heat flow per unit area, J/(m2 s) ) enteringthe particle across its surface.

Let us denote by a(r, z, t) (m−1) the external surface area of the particlesper unit reactor volume at point (r, z) and time t. Since we are assumingspherical particles, we have

a(r, z, t) =4πR2

s43πR

3s

(1− εf (r, z, t)

)=

3(1− εf (r, z, t)

)Rs

. (3.31)

Then the mass flow rates of species per unit reactor volume (mol/(m3 s))supplied by the particles to the fluid bulk at point (r, z) and time t is given by

g(r, z, t) := −a(r, z, t)Ds ∂εsys

∂rs(Rs(r, z, t), r, z, t). (3.32)

Moreover, in the case of resistance

g(r, z, t) = a(r, z, t)ηfs(r, z, t)(ys(Rs(r, z, t), r, z, t

)− yf (r, z, t)

). (3.33)

In a similar way, the rate of internal energy per unit reactor volume suppliedby the particles to the fluid bulk at point (r, z) and time t is

f(r, z, t) = −a(r, z, t)ks∂θs

∂rs(Rs(r, z, t), r, z, t) + w(θf ) · g, (3.34)

where the first term on the right-hand side represents the flow rate of internalenergy per unit reactor volume (W/m3) from the solid particles to the liquid.In the case of resistance, f becomes

f(r, z, t) = a(r, z, t)hfs(r, z, t)(θs(Rs(r, z, t), r, z, t

)− θf (r, z, t)

)+ w(θf ) · g.

(3.35)

3.3. SOURCES OF MASS AND ENERGY IN THE FLUID BULK 45

The full model of an FBR for the case of resistance can be written as follows:

∂

∂t(εfyf ) +

∂

∂z(εfyfv)− 1

r

∂

∂r

(Dfr r

∂

∂r(εfyf )

)− ∂

∂z

(Dfz

∂

∂z(εfyf )

)= Afδf (θf ,yf ) + aηfs

(ys(Rs)− yf

),

εfρf cfv (θf )(∂θf∂t

+ v∂θf

∂z

)− 1

r

∂

∂r

(kfr r

∂θf

∂r

)− ∂

∂z

(kfz∂θf

∂z

)= ahfs

(θs(Rs)− θf

)− w(θf ) ·Afδf (θf ,yf ),

∂

∂t(εsys)− 1

r2s

∂

∂rs

(Dsr2

s

∂εsys

∂rs

)= Asδs(θs,ys),


∂t− 1

r2s

∂

∂rs

(ksr2

s

∂θs

∂rs

)= −w(θs) ·Asδs(θs,ys),

−Dfz

∂

∂z(εfyf )(r, 0, t) + vεfyf (r, 0, t) = vεfyfin(t),

θf (r, 0, t) = θfin(t),

∂εfyf

∂z(r, L, t) = 0,

∂θf

∂z(r, L, t) = 0,

∂εfyf

∂r(R, z, t) = 0,

kfr∂θf

∂r(R, z, t) = hext

(θext(t)− θf (R, z, t)

),

∂εfyf

∂r(0, z, t) = 0,

∂θf

∂r(0, z, t) = 0,

∂(εsys)

∂rs(0, r, z, t) = 0,

∂θs

∂rs(0, r, z, t) = 0,

Ds ∂εsys

∂rs(Rs(r, z, t), r, z, t) = ηfs(r, z, t)

(yf (r, z, t)− ys(Rs(r, z, t), r, z, t)

),

ks∂θs

∂rs(Rs(r, z, t), r, z, t) = hfs(r, z, t)

(θf (r, z, t)− θs(Rs(r, z, t), r, z, t)

),

yf (r, z, 0) = yf0 (r, z), θf (r, z, 0) = θf0 (r, z),

ys(rs, r, z, 0) = ys0(rs, r, z), θs(rs, r, z, 0) = θs0(rs, r, z).


Conclusions

In this part, we have constructed the mathematical model of stirred tank reac-tors (batch and semi-batch STR and also for continuous STR) and plug flowreactors. We have considered both the transient and the steady-state cases.Reactors are not supposed to be either adiabatic or isotherm, so temperatureas well as species concentrations have to be computed by the models that areobtained from the energy and mass conservation equations, respectively.

We have modelled the general n−dimensional model which has been par-ticularized to the PFR model. The mathematical model is a coupled system ofpartial differential equations involving gradient and Laplacian with respect tospatial variables and partial derivative with respect to time.

Finally, we have obtained the model of the FBR which has been understoodas a heterogeneous reaction system in which plug-flow is assumed. The modelis based on the conservation laws for mass, energy and momentum, and leadsto partial differential equations. We have considered the bed as a continuumof small particles (solids spheres) containing the catalyst and interacting withthe fluid. Accordingly, spherical symmetry has been assumed. The fluid bulkhas been modelled as flowing in a porous media.

47

48 Conclusions

Part II

Mathematical analysis andnumerical solution

49

Introduction

At the beginning of this part of the thesis we focus on the n−dimensionalreactor described in Chapter 2. As we have already mentioned, equations inreaction-diffusion systems have been studied by different approaches using avariety of methods.

The objective here is to prove global existence of solution for convection-diffusion-reaction systems. The topic is classical and studied along time. Evenif the analysis on this problem has been done over two centuries, no comprehen-sive mathematical theory has been established, on the contrary, the literatureis full of challenging open problems. In several space dimensions, not eventhe global existence of solutions is presently known in any significant degreeof generality. Until now, most of the analysis has been concerned with theone-dimensional case.

The proof of the global theorem is based on the techniques in [55]. In thisarticle, the local existence of the reaction–diffusion systems is provided via thesemigroup theory by considering the semilinear parabolic problem. However,we combine this theory with the variational approach. In that case we haveto sacrifice regularity of the solution. Of course, this solution is understood inthe weak sense. Going back to the global solution, properties (P) and (M),that will hold because of the form of our particular reaction term (the law ofmass action), play an important role in the existence proof. The variables inour problem represent species concentration so, their positivity is a naturalproperty (helpful to verify (P)).

Control and optimization problems in chemical engineering and their ap-plications often require many numerical simulations of large-scale dynamicalsystems with different conditions. If a fast or real-time control strategy is de-sired, the direct numerical simulation does not work well. It is important toknow how the error behaves. In Chapter 5 an error estimation is obtainedby following the techniques in [63]. The proposed approach approximates thenonlinear function by its Lagrange interpolant. To make sure that we can dothese estimations we need previously to prove the existence of solution of thesemidiscretized problem we use in the estimations. Once this study has been

51

52 Introduction

carried out, it is necessary to compute a numerical solution of the model thatinterests us from the practical point of view. We focus on PFR and FBR mod-els. For the first one we use a finite difference scheme and for the second onethe finite element method is applied.

Chapter 4

Existence and uniquenessof solution in convection-diffusion-reactionsystems

The study of existence and uniqueness of solution in convection-diffusion-reactionsystems is an interesting topic which represents a challenging task due to thenon-linearity of the source term, the coupling of the equations and, sometimes,even the existence of non-linear diffusion terms. During the last decades, thisproblem was treated from different points of view. Some authors studied onlythe local existence. Others treated the problem through weak formulationand some of them worked with classical solutions and particular assumptionson the source term and/or the initial conditions. Different boundary condi-tions can be considered. The existing results in the literature are focusedon two-dimensional models, but in many situations extension to more generaln−dimensional case can be done.

Some authors use approaches that involve Lyapunov functions, but the useof semigroup theory still has a great impact in this research field. Other in-teresting approaches use the concepts of upper and lower solutions, or thepositivity of the solution and the control of mass. These techniques are brieflydescribed in the following paragraphs.

In the paper of Amman [1] a semilinear parabolic system is considered andinterpreted as an abstract evolution system. A theory of existence, regularity

53

54 CHAPTER 4. EXISTENCE AND UNIQUENESS OF SOLUTION

and continuous dependence is developed, but only local existence is proved.

The global existence for the two-dimensional model was first demonstratedin [35] using semigroup theory and the n-dimensional model with convection-diffusion terms was studied in [60]. Two important references for this approachare [40] and [54].

The technique of upper and lower solution is developed by Pao in [53] forcoupled parabolic systems with Robin conditions.

The case of coupled or cross-diffusion terms is treated in the thesis [15]. Inaddition, for two-dimensional models, a technique that employs the sign func-tion is used to prove existence of a classical solution, see for example [16].

Lyapunov functions are used in [25] to prove existence of weak solutionsglobal in time for more general diffusion terms. Lyapunov structure is alsoapplied to a general class of Lotka-Volterra systems in [28] or with Lp-bounds,p ≥ 1 [38]. The one-dimensional case with convection-diffusion terms andDirichlet boundary conditions is studied in [50] via Lyapunov type conditions.A general domain is considered in [48] and [49] for existence and boundednessof global solution of a diffusion-reaction problem.

Global existence in time of solutions for reaction-diffusion systems relies ontwo essential properties: positivity of solution along time and boundedness oftotal mass in [61]. These properties are also used in [55] and [39], where weakand strong formulations of the problem are described, and some representa-tive examples are given. Similar methodology for polynomial growth of sourceterm is presented in [45] for two-dimensional systems. Besides, Lp, p ≥ 1,and L∞-approaches also appear in these references. L∞-blow up may occurin reaction-diffusion systems even if they seem simple. To avoid it, strong as-sumptions on the source term are required. For more details see, for example,[56]. Let us end this introduction by reproducing the abstract of the interestingsurvey by M. Pierre [55]:

The goal of this paper is to describe the state of the art on the question ofglobal existence of solutions to reaction-diffusion systems for which two mainproperties hold: on the one hand, the positivity of the solutions is preserved forall time; on the other hand, the total mass of the components is uniformly con-trolled in time. This uniform control on the mass (or –in mathematical terms–on the L1-norm of the solution) suggests that no blow up should occur in finitetime. It turns out that the situation is not so simple. This explains why somany partial results in different directions are found in the literature on this

4.1. THE MODEL 55

topic, and why also the general question of global existence is still open, whilelots of systems arise in applications with these two natural properties.

Throughout this chapter we will prove global existence of solution for someconvection-diffusion-reaction systems. The proof of such theorem is basedon some properties described in [55] but, first, the techniques we employ areslightly different (they combine semigroup theory and weak formulation tech-niques) and, second, we include a convection term which is mandatory to modelmost of industrial chemical reactors. The properties mentioned in the previousparagraph will be verified for our particular reaction term which correspondsto the law of mass action. The variables in our problem represent species con-centration, so their positivity is a natural property. Firstly, we will prove localexistence of solution by using semigroup theory and then we will prove thatit is bounded in L∞, which will allow us to conclude the existence of a globalsolution. For the document to be self-contained we detail the proofs of all theresults, although some of them have been previously obtained or use techniquesthat are standard in the mathematical analysis of partial differential equations.

4.1 The model

We focus on n−dimensional convection-diffusion-reaction models for chemicalreactors. The mathematical modelling of these reactors was described in detailin Chapter 2. In this chapter we only consider the system related to chemicalspecies, so we assume that temperature is known. We adopt this simplificationbecause the property of boundedness of total mass needed for the global exis-tence cannot be shown for the energy equation.

Let us recall and establish the assumptions we consider:

• Robin boundary conditions at the entrance of the reactor.

• Homogeneous Neumann boundary condition at the exit of the reactor.

• Null mass flux through the wall of the reactor.

• The reaction term is given by the law of mass action and the Arrheniuslaw.

• The diffusion coefficients are all equal to d > 0.

Remark 4.1.1. Considering equal diffusion coefficients is not an innocentassumption as it has been noticed in [55].


Summarizing, in the present chapter, the following convection-diffusion-reaction system will be studied:

(P )

∂y

∂t(t, x) +∇y(t, x)v − d∆y(t, x) = ϕ(t, x,y(t, x)), (t, x) ∈ (0, T )× Ω,

y(0, x) = y0(x), x ∈ Ω,

d∂y


d∂y

∂ν(t, x) = 0, (t, x) ∈ (0, T )× (Γ2 ∪ Γ3),

where Ω is a bounded domain in Rn with smooth boundary ∂Ω, as representedin Figure 2.1 and ν is the outward unit normal vector to ∂Ω. Recall that Γ1

denotes the reactor entrance, Γ2 the reactor exit and Γ3 the reactor wall. Werecall that y(t, x) ∈ RN , N being the number of species. Moreover, for now weassume that

ϕ : (t, x,w) ∈ (0, T )× Ω× RN → ϕ(t, x,w) ∈ RN

is a Caratheodory function, i.e., it is measurable with respect to (t, x) andcontinuous with respect to w.

Let us recall that if ϕ corresponds to the law of mass action with theArrhenius law, it can be written as

ϕ(t, x,w) = Aδ(t, x,w), (t, x,w) ∈ [0, T ]× Ω× RN ,

with δ given by (1.4) and (1.5) with non-negative integer exponents αli, i =1, . . . , N, l = 1, . . . L, which represent the coefficients of the i−th reactant inthe l−th reaction.

4.2 Local existence of weak solution

In this section we study the existence and uniqueness of solution of problem(P ) in an open interval (0, δ), for some δ > 0. This local solution is intendedin a “weak sense” via the variational formulation of the problem that is givenbelow. In what follows τ denotes a positive real number and 〈., .〉 the dualitybetween spaces (H1(Ω))′ and H1(Ω)).

Let us define the bilinear continuous form a : H1(Ω)×H1(Ω)→ R by

a(y,w) :=

∫Ω

∇y(x)v(x)·w(x)+

∫Ω

d∇y(x) : ∇w(x)−∫

Γ1

(v(x)·ν(x))y(x)·w(x)

(4.1)

4.2. LOCAL EXISTENCE OF WEAK SOLUTION 57

and the linear continuous form l(t) : H1(Ω)→ R by

l(t)(w) :=

∫Γ1

g(t, x) ·w(x). (4.2)

Definition 4.2.1. Let v ∈ L∞(Ω) with div v = 0 and v · ν ∈ L∞(Γ1), g ∈L2(0, τ ; L2(Γ1)) and y0 ∈ L2(Ω). A function y ∈ L2((0, τ); H1(Ω))such that∂y

∂t∈ L2(0, τ ;

(H1(Ω))′) and ϕ(t, x,y(t, x)) ∈ L2(0, T ; L2(Ω)) is said a weak

solution of problem (4.3)-(4.4) if and only if

〈∂y

∂t(t),w〉+ a(y(t),w) =

∫Ω

ϕ(t, x,y(t, x)) ·w(x) + l(t)(w) ∀w ∈ H1(Ω),

(4.3)

y(0) = y0.(4.4)

We work with a first order in time semi-linear parabolic partial differen-tial equation (PDE) system. A first attempt to prove a local existence resultfor our system would consists in building an adequate contraction in orderto use the Banach fixed point theorem (it is described in [3], also known in[36] as the Banach Contraction Principle). Unfortunately, we are not able toprove the Lipschitz property in the Banach space where the mapping is defined.

This is why we use the semigroup theory where the local existence theory fora first order in time parabolic PDE system is seen as an extension of the ODEstheory. In this framework some hypotheses such as Holder condition and lo-cally Lipschitz property of the reaction term are needed. One of the difficultieswhen applying semigroup theory is that only homogeneous boundary condi-tions are allowed (in our case g should be null). Thus, we proceed in two stepsto build the solution of the non-homogeneous problem. Firstly, a translation isintroduced to get homogeneous boundary condition and then semigroup theoryis applied. Finally, the two steps provides us with a unique continuous solution.

4.2.1 Existence of solution to an auxiliary elliptic prob-lem

In order to make a translation in the weak problem (4.3)–(4.4), we introducea family of linear elliptic problems with Robin boundary condition g at the


entrance Γ1, parametrized by the time variable t, namely,

(PG)

∇G(t, x)v(x)− d∆G(t, x) = 0, (t, x) ∈ (0, T )× Ω,

d∂G

∂ν(t, x)− (v(x) · ν(x))G(t, x) = g(t, x), (t, x) ∈ (0, T )× Γ1,

d∂G

∂ν(t, x) = 0, (t, x) ∈ (0, T )× (Γ2 ∪ Γ3),

and its weak formulation: find G(t) ∈ H1(Ω) such that

a(G(t),w) = l(t)(w) for w ∈ H1(Ω). (4.5)

We notice that time t is a parameter rather than an independent variable be-cause there are no time derivatives in (4.5). Now, let us prove that, under someassumptions, this problem has a unique weak solution that is also continuous.

Proposition 4.2.1. Let n ≥ 2. Under the assumptions,

• v ∈ L∞(Ω),

• divv = 0,

• v · ν ∈ L∞(∂Ω),

• v · ν ≤ 0 on Γ1, v · ν ≥ 0 on Γ2, v · ν = 0 on Γ3,

• there exists S ⊂ Γ2 with non null surface measure and α > 0 such thatv · ν ≥ α on S,

• g ∈W1,1(0, T ; Lp(Γ1)) with p = n−1+ε for n > 2 or g ∈W1,1(0, T ; L2(Γ1))for n = 2.

Then, problem (4.5) has a unique solution G ∈ W1,1(0, T ; H1(Ω) ∩ C0,γ(Ω))where γ is a suitable number, γ ∈ (0, 1), and hence, in particular, G ∈ C([0, T ]×Ω).

Proof.Firstly, for t ∈ [0, τ ] the Lax-Milgram lemma provides us a unique solution

G(t, .) ∈ H1(Ω). Indeed, we have

1) l(t) is linear continuous in H1(Ω):

|l(t)(w)| =∣∣∣ ∫

Γ1

g(t) ·w∣∣∣ ≤ ∫

Γ1

|g(t) ·w|.

By applying Holder inequality we have

|l(t)(w)| ≤ ‖g(t)‖Lp(Γ1)‖w‖Lq(Γ1),


where1

p+

1

q= 1. For n > 2 we take p = n− 1 + ε > 2 and then q < 2.

If n = 2 and g ∈W1,1(0, τ ; L2(Γ1)), then q = 2.

Finally, by using a trace theorem (see, for instance, [51, Theor. 1.2]), wehave

|l(t)(w)| ≤ C‖g(t)‖Lp(Γ1)‖w‖H1(Ω).

2) a is bilinear continuous in H1(Ω)×H1(Ω):

|a(G,w)| =∣∣∣ ∫

Ω

∇Gv ·w +

∫Ω

d∇G : ∇w −∫

Γ1

(v · ν)G ·w∣∣∣

≤∣∣∣ ∫

Ω

∇Gv ·w∣∣∣+∣∣∣ ∫

Ω

d∇G : ∇w∣∣∣+∣∣∣ ∫

Γ1

(v · ν)G ·w∣∣∣

≤∫

Ω

|∇Gv ·w|+ d

∫Ω

|∇G : ∇w|+∫

Γ1

|(v · ν)G ·w|.

By using Holder’s inequality we obtain,

|a(G,w)| ≤(∫

Ω

‖∇G‖2‖v‖2) 1

2

‖w‖L2(Ω) + d‖∇G‖L2(Ω)‖∇w‖L2(Ω)

+ ‖v · ν‖L∞(Γ1)

∫Γ1

|G ·w|

and by applying Holder’s inequality we get, for ‖v‖L∞(Ω) := ‖|v|‖L∞(Ω),

|a(G,w)| ≤ ‖v‖L∞(Ω)‖∇G‖L2(Ω)‖w‖L2(Ω) + d‖∇G‖L2(Ω)‖∇w‖L2(Ω)

+ ‖v · ν‖L∞(Γ1)

∫∂Ω

|G ·w|

≤√

2 max‖v‖L∞(Ω), d‖∇G‖L2(Ω)‖w‖H1(Ω)

+ ‖v · ν‖L∞(Γ1)‖G‖L2(∂Ω)‖w‖L2(∂Ω).

Finally, by using again the trace theorem in [51], we conclude that

|a(G,w)| ≤ C(v)‖G‖H1(Ω)‖w‖H1(Ω),

w ∈ H1(Ω).

3) a is coercive

Indeed, firstly,

a(G,G) =

∫Ω

∇Gv ·G +

∫Ω

d∇G : ∇G−∫

Γ1

(v · ν)‖G‖2.


Moreover, we have

∇Gv ·G = v · (∇G)TG =1

2v · ∇(G ·G) =

1

2v · ∇(‖G‖2).

Then, using that divv = 0 we have

a(G,G) =1

2

∫∂Ω

(v · ν)‖G‖2 +

∫Ω

d∇G : ∇G−∫

Γ1

(v · ν)‖G‖2

=1

2

∫Γ2

(v · ν)‖G‖2 +

∫Ω

d‖∇G‖2 − 1

2

∫Γ1

(v · ν)‖G‖2.

Notice that, on the inlet boundary Γ1, v · ν ≤ 0 and on subset S of theoutlet boundary Γ2, v · ν ≥ α. Hence,

a(G,G) =1

2

∫∂Ω

(v · ν)‖G‖2 +

∫Ω

d‖∇G‖2 −∫

Γ1

(v · ν)‖G‖2

≥ 1

2

∫Γ2

(v · ν)‖G‖2 +

∫Ω

d‖∇G‖2 ≥ α

2

∫S

‖G‖2 +

∫Ω

d‖∇G‖2

≥ β‖G‖2H1(Ω),

where the last inequality is a consequence of Friedrichs inequality (see,for instance, [51, Theor. 1.1.9]).

Now, by the Lax-Milgram lemma (see, for instance, [27, Lemma 2.2]), thereexists a unique G(t, .) ∈ H1(Ω) satisfying a(G(t),w) = l(t)(w) ∀w ∈ H1(Ω).After that, we obtain the claimed additional regularity by using Theorem 3.14in [52]. Notice that the hypothesis of g ∈W1,1(0, τ ; Lp(Γ1)) with p = n−1+ε,which is included in Theorem 3.14., is also satisfied if n = 2 because we haveg ∈ W1,1(0, τ ; L2(Γ1)) ⊂ W1,1(0, τ ; L1+ε(Γ1)). Finally, this result has beenproved for scalar PDEs, but it does not matter because our vector boundary-value problem is fully decoupled into scalar ones:

(PG)i

v · ∇Gi(t, x)− d∆Gi(t, x) = 0, (t, x) ∈ (0, τ)× Ω,

d∂Gi∂ν

(t, x)− (v · ν)Gi(t, x) = gi(t, x), (t, x) ∈ (0, τ)× Γ1,

d∂Gi∂ν

(t, x) = 0, (t, x) ∈ (0, τ)× Γ2 ∪ Γ3.

Thus, G(t, .) ∈ C0,γ(Ω) for some γ > 0 and fulfills the following inequality:

‖G(t)‖C(Ω) ≤ ‖G(t)‖C0,γ(Ω) ≤ C‖g(t)‖Ln−1+ε(Γ1). (4.6)


By integrating in time we deduce that G ∈ L1(0, τ ;C(Ω)).Now, let us denote by T the bounded linear operator (from Lp(Γ1) in

H1(Ω)) mapping g ∈ Lp(Γ1) into the solution of problem (PG), G ∈ H1(Ω) ∩C(Ω). Since 1 < p < ∞ then Lp(Γ1) is a reflexive Banach space and we have(see [20, Cor. A2]),

g(t) = g(0) +

∫ t

0

dg

ds(s)ds.

Hence,

G(t) = Tg(t) = Tg(0) +

∫ t

0

Tdg

ds(s)ds = G(0) +

∫ t

0

Tdg

ds(s)ds

which impliesdG

dt(t) = T

dg

dt(t) ∈ H1(Ω) ∩ C(Ω).

Therefore, ∥∥∥∥∥dGdt (t)

∥∥∥∥∥C(Ω)

≤ C

∥∥∥∥∥dgdt (t)

∥∥∥∥∥Ln−1+ε(Γ1)

(4.7)

and then∂G

∂t∈ L1(0, τ ;C(Ω)). Finally, from [41, Lemma 1.2.], we deduce that

G ∈ C([0, τ ];C(Ω)) = C([0, τ ]× Ω).

4.2.2 Local existence of a homogeneous problem

In this section, in order to get a local solution to problem (4.3)-(4.4) of the formy = u + G we apply the semigroup theory to the following auxiliary nonlinearhomogeneous parabolic problem:

(Pu)

∂u

∂t(t, x) +∇u(t, x)v(x)− d∆u(t, x) = ϕ(t, x,u(t, x)), (t, x) ∈ (0, τ)× Ω,

d∂u

∂ν(t, x)− (v · ν)u(t, x) = 0, (t, x) ∈ (0, τ)× Γ1,

d∂u

∂ν(t, x) = 0, (t, x) ∈ (0, τ)× Γ2 ∪ Γ3,

u(0, x) = u0(x), x ∈ Ω,

where ϕ : [0, T ]× Ω× RN → RN is defined a.e. in (0, T )× Ω× RN by

ϕ(t, x,u) = ϕ (t, x,u + G(t, x))− ∂G

∂t(t, x) (4.8)


and u0(x) := y0(x)−G(0, x). We notice that ϕ is also a Caratheodory function.Now, in order to apply the semigroup theory, we rewrite problem (Pu) in termsof a second order strongly elliptic operator, to be called A, and a differentialoperator of order one acting on the boundary, to be called B.

The operator A contains the diffusion and convection terms in the systemdescribed above and it is defined by components in the following way:

Ai(x,D) = d

n∑k=1

Dkk −n∑k=1

vk(x)Dk ∀i = 1, ..., N and x ∈ Ω, (4.9)

where Dk = ∂/∂xk and Dkk = ∂2/∂x2k. The differential operator B is defined

also by components by

Bj(x,D) = b0(x)I +

n∑k=1

b1k(x)Dk ∀j = 1, ..., N and x ∈ ∂Ω (4.10)

with

b0(x) =

− v(x) · ν(x), x ∈ Γ1 ∪ Γ3,

0, x ∈ Γ2,(4.11)

b1k(x) = dνk, k = 1, ..., n and x ∈ ∂Ω. (4.12)

Notice that boundary conditions were considered of Robin type in Γ1 and ofNeumann type in Γ2 ∪Γ3. However, we can rewrite them as a Robin conditionin all the boundary using coefficients b0 and b1, where

b1 = (b11, . . . , b1n)T .

Now problem (Pu) is written as follows

(Pu)

∂ui∂t

(t, x) = Ai(x,D)ui(t, x) + ϕi(t, x,u(t, x)), (t, x) ∈ (0, τ)× Ω,

Bi(x,D)ui(t, x) = 0, t ∈ (0, τ), x ∈ ∂Ω,

ui(0, x) = u0i(x), x ∈ Ω,

∀i = 1, ..., N.

In order to use the abstract theory in the following paragraphs, we noticethat problem (Pu) can be seen as a semilinear parabolic problem, as the non-linearity depends on the unknown vector u, but not on its derivatives.


Let X := C(Ω;RN ) be endowed with the norm

‖u‖X := maxx∈Ω‖u(x)‖RN ,

where ‖.‖RN is a suitable norm in RN . Since all norms are equivalent we willdrop the subscript RN . Similarly, if there is no ambiguity we will suppress thesubscript X in the above norm in the normed space X.

Let the nonlinear mapping f : [0, T ]×X→ X be defined by

f(t,u)(x) := ϕ(t, x,u(x)) ∀x ∈ Ω. (4.13)

We introduce the linear unbounded operator A : D(A) ⊂ X → X to becalled the realization of A(·, D) in X as

Au := A(·, D)u ∀u ∈ D(A),

where the domain D(A) is given by

D(A) = u ∈⋂

1≤p<∞

W2,p(Ω) : (A1u1, ...,ANuN )T ∈ X,

(B1u1, ...,BNuN )T |∂Ω = 0. (4.14)

Remark 4.2.1. Homogeneous boundary conditions are needed because D(A)must be a vector space in order to apply semigroup theory. This is why we haveintroduced vector function G in order to make a translation.

We notice that for p >n

2, the Sobolev embedding theorem (see, for instance,

[51, Theorem 3.8, Chap. 2]) implies W2,p(Ω) ⊂ C(Ω). Therefore, D(A) ⊂ C(Ω)in any spatial dimension.

Now, let us introduce the problem

(Pu)

du

dt= Au + f(t,u), t > 0,

u(0) = u0.(4.15)

In the next paragraphs, we describe some hypotheses for operators A andB and mapping ϕ. These hypotheses must be fulfilled in order to prove alocal existence theorem for problem (Pu) by applying the theorems describedin Appendix B.

Hypothesis on the domain Ω

(D1) The domain Ω is a bounded open set with C2 boundary ∂Ω.


Hypotheses on the elliptic differential operator A

(A1) The diffusion coefficient d is positive.

(A2) v ∈ C1(Ω).

These assumptions imply the hypotheses for the operator A that are needed inAppendix B.

Hypotheses on the differential operator B

We can prove all hypotheses regarding the boundary operator B in Appendix Bwithout any additional assumption. Indeed,

(B1) b0 ∈ C1(Ω)Firstly, we have b0 = −v · ν ∈ C1(∂Ω) because both v and ν belong tothis space and, since v(x) ·ν(x) = 0 for x ∈ Γ3, we can replace v(x) ·ν(x)by 0 on Γ2 while preserving the membership to C1(∂Ω). Finally, b0 can beextended to the whole domain Ω in such a way that the resulting functionbelongs to C1(Ω) by using a result from Ladyzenskaya et al. [40, page 10].Thus, b0 ∈ C1(Ω).

(B2) b1 ∈ C1(Ω)From (B1), the normal vector ν belongs to C1(∂Ω) and so b1(x) = dν(x)can be extended to the whole Ω with C1(Ω) regularity.

(B3) Transversality condition:

b1(x) · ν(x) = d‖ν(x)‖2 = d > 0 ∀x ∈ ∂Ω.

Remark 4.2.2. We notice that the case of a cylindrical-3D reactor does notsatisfy assumption (B1) as its boundary is not smooth enough (think on thecircles where the bases and the lateral boundary meet).

Hypothesis of function ϕ

Next, we will prove locally Lipschitz and Holder properties of function ϕ inproblem (Pu) that will be stated in next Lemma 4.2.3 in a single inequality.They are needed in the local existence Theorem 4.2.1 below. Firstly, we needto prove two inequalities regarding locally Lipschitz ( Lemma 4.2.1) and Holder(Lemma 4.2.2) properties of function ϕ in problem (P ).

Lemma 4.2.1. In problem (P ), function ϕ corresponding to the law of massaction satisfies the following locally Lipschitz property with respect to its thirdvariable:


For all t ∈ [0, T ], x ∈ Ω, y0 ∈ RN and R > 0 there exists M(t, x,y0, R) > 0such that

‖ϕ(t, x,y1)−ϕ(t, x,y2)‖ ≤M(t, x,y0, R)‖y1 − y2‖ ∀y1,y2 ∈ B(y0, R).

Furthermore, if θ ∈ C([0, T ]) × Ω) and θ(t, x) 6= 0 ∀(t, x) ∈ [0, T ] × Ω, Mdoes not depend either on t or on x.

Proof.Let y0 ∈ RN , R > 0 and y1,y2 ∈ B(y0, R).

Then,

‖ϕ(t, x,y1)−ϕ(t, x,y2)‖ = ‖Aδ(t, x,y1)−Aδ(t, x,y2)‖≤ ‖A‖‖δ(t, x,y1)− δ(t, x,y2)‖.

We define the following composition of functions

s ∈ [0, 1]ζ−→ ζ(s) := sy1+(1−s)y2 ∈ B(y1, R)

δ(t,x,.)−−−−→ δ(t, x, sy1+(1−s)y2) ∈ RL.

Notice that δ(t, x,y1) = δ(t, x, ζ(1)) and δ(t, x,y2) = δ(t, x, ζ(0)). Then,

‖ϕ(t, x,y1)−ϕ(t, x,y2)‖ ≤ ‖A‖‖δ(t, x, ζ(1))− δ(t, x, ζ(0))‖. (4.16)

Now, applying the Barrow Rule and the Chain Rule we get the following in-equality:

‖ϕ(t, x,y1)−ϕ(t, x,y2)‖ ≤ ‖A‖∥∥∥∥ ∫ 1

0

(δ(t, x, .) ζ)′(s)ds

∥∥∥∥= ‖A‖

∥∥∥∥∫ 1

0

Dyδ(t, x, sy1 + (1− s)y2)(y1 − y2)ds

∥∥∥∥. (4.17)

By operating with the second term on the left-hand side we obtain∥∥∥∥∫ 1

0

Dyδ(t, x, sy1 + (1− s)y2)(y1 − y2)ds

∥∥∥∥≤∫ 1

0

‖Dyδ(t, x, sy1 + (1− s)y2)(y1 − y2)‖ds

≤∫ 1

0

‖Dyδ(t, x, sy1 + (1− s)y2)‖‖y1 − y2‖ds

≤(∫ 1

0

‖Dyδ(t, x, sy1 + (1− s)y2)‖ds)‖y1 − y2‖

≤(

sup0≤s≤1

‖Dyδ(t, x, sy1 + (1− s)y2)‖)‖y1 − y2‖.

(4.18)


As y1,y2 ∈ B(y0, R) and any ball is convex, then sy1 + (1− s)y2 ∈ B(y0, R).Since Dyδ is continuous with respect to its third variable, by using Weirestrass’theorem we deduce that it is bounded on the ball and so,

‖ϕ(t, x,y1)−ϕ(t, x,y2)‖ ≤M(t, x,y0, R)‖y1 − y2‖. (4.19)

Moreover, if θ satisfies the above assumptions, the Arrhenius terms are boundedand hence,

L(y0, R) = sup(t,x)∈[0,τ ]×Ω

sup0≤s≤1

‖Dyδ(t, x, sy1 + (1− s)y2)‖ <∞.

Finally,‖ϕ(t, x,y1)−ϕ(t, x,y2)‖ ≤ L(y0, R)‖y1 − y2‖. (4.20)

Lemma 4.2.2. Let us assume that the temperature in the Arrhenius law be-longs to the space C1([0, T ])×Ω) and it is non-null ∀(t, x) ∈ [0, T ]×Ω. Then,function ϕ in problem (P ) corresponding to law of mass action with Arrheniuslaw satisfies the following Holder condition with respect to its first variable:there exists a constant C such that

‖ϕ(t, x,y)−ϕ(s, x,y)‖ ≤ C‖A‖‖h(y)‖|t− s| ∀s, t ∈ [0, T ] ∀y ∈ RN ,

where hl(y) :=

N∏i=1

yαili , l = 1, ..., L.

Proof.Let 0 ≤ s < t ≤ τ . Let us introduce the L× L diagonal matrix

E(t, x) :=

exp

(− Ea1Rθ(t,x)

). . .

exp(− EaLRθ(t,x)

) .

Then we can writeδ(s, x,y) = E(t, x)h(y),

Therefore,

‖ϕ(t, x,y)−ϕ(s, x,y)‖ = ‖AE(t, x)h(y)−AE(s, x)h(y)‖≤ ‖A‖‖E(t, x)− E(s, x)‖‖h(y)‖.

Moreover, it is easy to prove the following inequality via the Mean Value The-orem. By using the fact that θ and its derivative are continuous and boundedin a compact set and θ is not null, we have


‖E(t, x)− E(s, x)‖ ≤ C|t− s| ∀t, s ∈ [0, T ], x ∈ Ω.

Indeed, let us introduce the notation θ′(t, x) =∂θ

∂t(x, t). Then, by the Mean

Value Theorem we have

Eii(t, x)− Eii(s, x) = (t− s) exp

(− EaiRθ(ζi, x)

)Eai

θ′(ζi, x)

Rθ2(ζi, x),

for some ζi between s and t, ∀x ∈ Ω and i = 1, ...L. Using that θ ∈ C1([0, T ]×Ω)and θ 6= 0, the absolute value of the the fraction

θ′(ζi, x)

Rθ2(ζi, x)

can be bounded by taking the minimum value of θ2 in the denominator andthe maximum value of the absolute value of θ′ in the numerator. Obviouslythe exponential is also bounded.

Now, it is the turn to study locally Lipschitz and Holder properties for ϕ.

Lemma 4.2.3. Under the assumptions

• g ∈W2,r(0, T ; Lp(Γ1)) with p = n−1+ε for n > 2 or g ∈W2,r(0, T ; L2(Γ1))for n = 2, and r > 1.

• temperature θ ∈ C1([0, T ])× Ω) and θ(t, x) 6= 0 ∀(t, x) ∈ [0, T ]× Ω,

function ϕ in problem (Pu) satisfies the following locally Lipschitz and Holderproperty:

For all u0 ∈ RN and R > 0, there exists P (u0,G, R) > 0 such that,

‖ϕ(t, x,u1)− ϕ(s, x,u2)‖ ≤ P (uo,G, R)(‖u1 − u2‖+ |t− s|

r−1r

), (4.21)

for all u1,u2 ∈ B(u0, R) and for all s, t ∈ [0, T ].

Proof. Firstly, let

mG = max(t,x)∈[0,τ ]×Ω

‖G(t, x)‖,

u0 ∈ RN and R be any positive real number. Then,

u1 + G(t, x), u2 + G(t, x) ∈ B(u0, R+mG)


for all u1,u2 ∈ B(u0, R) and for all (t, x) ∈ [0, τ ]× Ω.

Moreover, for ϕ defined by (4.8) we have

‖ϕ(t, x,u1)− ϕ(s, x,u2)‖ = ‖ϕ(t, x,u1)− ϕ(t, x,u2) + ϕ(t, x,u2)− ϕ(s, x,u2)‖≤ ‖ϕ(t, x,u1)− ϕ(t, x,u2)‖+ ‖ϕ(t, x,u2)− ϕ(s, x,u2)‖

≤ ‖ϕ(t, x,u1 + G(t, x))−ϕ(t, x,u2 + G(t, x))‖+‖ϕ(t, x,u2 + G(t, x))−ϕ(s, x,u2 + G(s, x))‖

+

∥∥∥∥∂G

∂t(t, x)− ∂G

∂t(s, x)

∥∥∥∥≤ ‖ϕ(t, x,u1 + G(t, x))−ϕ(t, x,u2 + G(t, x))‖+‖ϕ(t, x,u2 + G(t, x))−ϕ(t, x,u2 + G(s, x))‖+‖ϕ(t, x,u2 + G(s, x))−ϕ(s, x,u2 + G(s, x))‖

+

∥∥∥∥∂G

∂t(t, x)− ∂G

∂t(s, x)

∥∥∥∥.

By using Lemma 4.2.1 and Lemma 4.2.2 we get

‖ϕ(t, x,u1)− ϕ(s, x,u2)‖ ≤ L(u0, R+mG)(‖u1 − u2‖+ ‖G(t, x)−G(s, x)‖

)+C‖A‖‖h(u2 + G(s, x))‖|t− s|+

∥∥∥∥∂G

∂t(t, x)− ∂G

∂t(s, x)

∥∥∥∥≤ L(u0, R+mG)

(‖u1 − u2‖+ ‖G(t, x)−G(s, x)‖

)+Q(u0, R+mG)|t− s|+

∥∥∥∥∂G

∂t(t, x)− ∂G

∂t(s, x)

∥∥∥∥,(4.22)

where

Q(u0, R+mG) := C‖A‖max‖h(u)‖ : u ∈ B(u0, R+mG).

Notice that G and∂G

∂tare in C

([0, τ ];C(Ω)

). Hence G is continuously

differentiable and then Lipschitz-continuous with respect to t. Moreover, if we

repeat the argument in the proof of Proposition 4.2.1 for∂2G

∂t2, we obtain that

∂G

∂t∈W1,r(0, T ;C(Ω)) and then it is Holder with respect to t, with exponent


r − 1

r. Indeed, we have

∥∥∥∂G

∂t(t2)− ∂G

∂t(t1)

∥∥∥C(Ω)

=

∥∥∥∥∥∫ t2

t1

∂2G

∂t2(t) dt

∥∥∥∥∥C(Ω)

≤∫ t2

t1

∥∥∥∂2G

∂t2(t)∥∥∥C(Ω)

dt

≤ (t2 − t1)r−1r

(∫ t2

t1

∥∥∥∂2G

∂t2(t)∥∥∥rC(Ω)

dt)1/r

≤ C(t2 − t1)r−1r .

Finally, this inequality and (4.22) yield (4.21).

Now, we state the local existence theorem for problem (Pu).

Theorem 4.2.1. Assume that hypotheses (D1), (A1), (A2), as well as thosein Lemma 4.2.3, hold. Let u0 ∈ C(Ω). Then,

(i) there exists τ = τ(u0) > 0 such that problem (Pu) has a unique classicalsolution u : [0, τ ]× Ω→ RN with u ∈ C([0, τ ]× Ω).

(ii) u can be extended to a maximally defined solution

u : I(u0)× Ω→ RN ,

I(u0) being an interval starting in 0 and relatively open in [0, T ], i.e.,either I(u0) = [0, τ) or I(u0) = [0, τ ] with τ = T .

Besides, u ∈ C((0, τ);D(A)) where D(A) is defined in (4.14), u : (0, τ)→

C(Ω) is derivable and∂u

∂tand Au are continuous in (0, τ)× Ω.

Proof. It follows from results in Appendix B and it is divided in four steps:

First step. A is sectorial in X = C(Ω)The realization Ai : D(Ai)→ X of operator Ai in X = C(Ω) with domain

D(Ai) = u ∈⋂

1≤p<∞

W2,p(Ω) : Aiu ∈ Xi, Biu|∂Ω = 0

is sectorial in X thanks to Theorem B.3.1.This means that, for each Ai there are constants ωi ∈ R,

π

2< θi < π and

Mi > 0 such that

(i) ρ(Ai) ⊃ Sθi,ωi = λ ∈ C : λ 6= ωi, |arg(λ− ωi)| < θi;

(ii) ‖R(λ,Ai)‖L(X) ≤Mi

|λ− ωi|∀λ ∈ Sθi,ωi .


On the one hand, if we choose ω = maxi=1,...,N

ωi and θ = mini=1,...,N

θi for each

λ ∈ Sθ,ω we have that λ ∈ ρ(Ai) ∀i = 1, ..., N and hence, λ ∈ ρ(A).

On the other hand, from the definition of ω we have

|λ− ω| ≤ Ci|λ− ωi|

for some constant Ci, i = 1, . . . , N . Indeed, it is enough to take Ci as the realnumber

Ci = supλ∈Sθ,ω

∣∣∣∣∣ λ− ωλ− ωi

∣∣∣∣∣.We notice that Ci exists because function

λ ∈ Sθ,ω −→λ− ωλ− ωi

∈ C

is continuous and lim|λ|→∞

λ− ωλ− ωi

= 1.

Then, we define M = maxi=1,...,N

CiMi and thus ‖R(λ,A)‖L(X) ≤M

|λ− ω|∀λ ∈ Sθ,ω.

Second step. Mapping f in (4.13) is well-defined, continuous and for allR > 0 and u0 ∈ X there exists L > 0 satisfying,

‖f(t,u1)− f(t,u2)‖X ≤ L‖u1−u2‖X ∀u1,u2 ∈ BX(u0, R) and for 0 ≤ t ≤ T.

Indeed, firstly for t ∈ [0, T ] and u ∈ C(Ω) the function x ∈ Ω→ ϕ(t, x,u(x)) is

continuous and hence f is well-defined. Moreover, the continuity in t for fixedu follows from Lemma 4.2.3.

Let u1,u2 ∈ BX(u0, R) for some R > 0. Then,

u1(x), u2(x) ∈ B(0 , R+ ‖u0‖) ∀x ∈ Ω

and using Lemma 4.2.3 we get

‖f(t,u1)− f(t,u2)‖X = maxx∈Ω‖f(t,u1)(x)− f(t,u2)(x)‖

= maxx∈Ω‖ϕ(t, x,u1(x))− ϕ(t, x,u2(x))‖ ≤ max

x∈ΩP (0 ,G, R+ ‖u0‖)‖u1(x)− u2(x)‖

= P (0 ,G, R+ ‖u0‖)‖u1 − u2‖X.

Third step. D(A) is dense in X. This is a consequence of Theorem B.3.1.

Then, by using Theorem B.4.1, problem (Pu) has a unique mild solution.


Finally, by making an additional step we are able to prove that this solutionis a classical solution.

Fourth step. There exists α ∈ (0, 1) such that for all R we have

‖f(t,u)− f(s,u)‖ ≤ C(R)(t− s)α, 0 ≤ s < t ≤ T,

for some constant depending on R, C(R) and for all u ∈ X such that ‖u‖X ≤ R.Let u ∈ X. Then, from Lemma 4.2.3

‖f(t,u)− f(s,u)‖X = maxx∈Ω‖f(t,u)(x)− f(s,u)(x)‖

= maxx∈Ω‖ϕ(t, x,u(x))− ϕ(s, x,u(x))‖

≤ maxx∈Ω

P (0 ,G, R)(‖u(x)− u(x)‖+ (t− s)

r−1r

)= P (0 ,G, R)(t− s)

r−1r .

Therefore, by applying Theorem B.5.1 the mild solution of (Pu) is also aclassical solution of our problem (Pu) which, in particular, implies the claimedregularity results.

The second part of the theorem easily follows from Proposition B.5.1.

The regularity obtained for the solution will be necessary in the next section

to prove the existence of a global solution in [0, T ].

4.2.3 Local existence of solution to problem (4.3)–(4.4)

By using the above results we can prove a local existence theorem for theoriginal problem (4.3)–(4.4).

Theorem 4.2.2 (Local existence of solution). Assume that hypotheses(D1), (A1), (A2), as well as those in Lemma 4.2.3, hold. Let y0 ∈ C(Ω).

Then, there exists a unique weak solution of problem (4.3)–(4.4) that iscontinuous, y ∈ C([0, τ ]× Ω), where τ < sup I(u0) with u0 = y0 −G(0).

Proof. On the one hand, let us recall that G ∈ C([0, T ]×Ω) is the uniquesolution of

∫Ω

∇Gv ·w+

∫Ω

d∇G : ∇w−∫

Γ1

(v ·ν)G ·w =

∫Γ1

g ·w w ∈ H1(Ω) (4.23)

(see Proposition 4.2.1).On the other hand, let u ∈ C(I(u0) × Ω) ∩ C(I(u0) \ 0; W2,p(Ω)) be the


unique maximal solution of (Pu). Multiplying scalarly the PDE system in (Pu)by w ∈ H1(Ω) and integrating in Ω we obtain

d

dt

∫Ω

u ·w +

∫Ω

∇uv ·w −∫

Ω

d∆u ·w =

∫Ω

ϕ(t, x,u) ·w ∀w ∈ H1(Ω).

Now, using a Green’s formula and the definition of ϕ we have,

d

dt

∫Ω

u ·w +

∫Ω

∇uv ·w +

∫Ω

d∇u : ∇w −∫∂Ω

d∂u

∂ν·w

=

∫Ω

ϕ(t, x,u + G) ·w − d

dt

∫Ω

G ·w.

Replacing boundary conditions and rearranging terms we obtain

d

dt

∫Ω

(u+G)·w+

∫Ω

∇uv·w+

∫Ω

d∇u : ∇w−∫

Γ1

(v·ν)u·w =

∫Ω

ϕ(t, x,u+G)·w.

(4.24)Finally, by adding both variational formulations (4.23) and (4.24), we have

d

dt

∫Ω

(u + G) ·w +

∫Ω

∇(u + G)v ·w +

∫Ω

d∇(u + G) : ∇w

−∫

Γ1

(v · ν)(u + G) ·w =

∫Ω

ϕ(t, x,u + G) ·w +

∫Γ1

g ·w.

Since u + G is bounded in [0, τ ] × Ω, for τ < sup I(u0), the same is true forϕ(t, x,u + G). Then, by using also standard energy estimates, we can prove

that y := u+G ∈ C([0, τ ]×Ω)∩L2((0, τ); H1(Ω)) with∂y

∂t∈ L2((0, τ ; H1(Ω)′)

and it is a solution in (0, τ) of the weak problem

〈∂y

∂t,w〉+ a(y,w) =

∫Ω

ϕ(t, x,y) ·w +

∫Γ1

g ·w w ∈ H1(Ω),

y(0) = y0.

Now, let us prove uniqueness. Assume there exist two local solutions yi ∈C([0, τi] × Ω) ∩ L2((0, τi); H

1(Ω)), i = 1, 2, of (4.3). Then, we subtract theabove weak formulation for y1 and y2 and use w = y1− y2 as test function toobtain

〈∂(y1 − y2)

∂t,y1−y2〉+a(y1−y2,y1−y2) =

∫Ω

(ϕ(t, x,y1)−ϕ(t, x,y2))·(y1−y2)

for all t ∈ [0, τ ], where, τ = minτ1, τ2. Equivalently,


1

2

d

dt

∫Ω

‖y1−y2‖2+a(y1−y2,y1−y2) =

∫Ω

(ϕ(t, x,y1)−ϕ(t, x,y2))·(y1−y2).

Due to the coerciveness of the bilinear form a that has been shown in theproof of Proposition 4.2.1 and since ϕ is locally Lipschitz with respect to itsthird variable (see Lemma 4.2.1), we can write

1

2

d

dt

∫Ω

‖y1 − y2‖2 + β‖y1 − y2‖2H1(Ω) ≤ L(0 ,m)‖y1 − y2‖2L2(Ω),

wherem = max‖y1‖C([0,τ ]×Ω), ‖y2‖C([0,τ ]×Ω).

Hence,1

2

d

dt

∫Ω

‖y1 − y2‖2 − L(0 ,m)‖y1 − y2‖2L2(Ω) ≤ 0.

Multiplying this equality by 2e−2L(0 ,m)t we deduce

d

dt

(e−2L(0 ,m)t‖y1 − y2‖2L2(Ω)

)= e−2L(0 ,m)t d

dt

∫Ω

‖y1 − y2‖2

− 2L(0 ,m)e−2L(0 ,m)t‖y1 − y2‖2L2(Ω) ≤ 0.

Finally, by integrating between 0 and t ∈ (0, τ ] and using that y1 and y2 satisfythe same initial condition we get

e−2L(0 ,m)t‖y1(t)− y2(t)‖2L2(Ω) ≤ 0,

and hence y1(t) = y2(t) ∀t ∈ [0, τ ].

4.2.4 The maximally defined solution

Since G is defined in the whole interval [0, T ] and y = u+G, then for any y0 ∈C(Ω) there is a maximal solution of problem (4.3)–(4.4), y ∈ C(I(u0);C(Ω)),where u0 := y0 −G(0)).

We notice that this maximal solution could not be a solution of the weakproblem in I(u0) if y is not bounded as τ → sup I(u0). However, it is a weaksolution in (0, sup I(u0)− ε) for all ε > 0.

Moreover, from Proposition B.5.2 we deduce that two cases are possible:

• either the maximal solution is defined in the whole interval, i.e., I(u0) =[0, T ], (i.e., it is a global solution),

• or I(u0) = [0, τ) with τ ≤ T and y(t) is unbounded as a mapping fromI(u0) to C(Ω).


4.3 Global existence of weak solution

In this section we study the existence of a global solution for the n-dimensionalconvection-diffusion-reaction system. It is necessary to prove that the localsolution does not blow up in the interval [0, T ]. In other words, we need toprove that any maximal local solution is bounded.

In this context, there are two main properties that will be exploited. Theywill be called properties (P) and (M):

(P) The non-negativity of any solution of (4.3)–(4.4) is preserved along time.

(M) There are constants αi > 0 with i = 1, ..., N such that

ϕ(t, x, r) ·α ≤ 0 ∀r ∈ (R+)N . (4.25)

In the following, we prove that these properties are satisfied in the caseunder consideration as well as some preliminary results based on them. Later,they will be used to prove that the solution exists in the whole time interval[0, T ].

In this section, the following additional assumptions are made:

(H1) The inlet Robin boundary data g has non-negative components in [0, T ]×Γ1.

(H2) The initial data y0(x) has non-negative components ∀x ∈ Ω.

Now, we prove the positivity of solution, that is, property (P) holds. Infact, we prove that if the reaction term is quasi-positive, then property (P) issatisfied.

Definition 4.3.1. The reaction term ϕ is called quasi-positive if, for alli = 1, ..., N ,

ϕi(t, x, r1, ..., ri−1, 0, ri+1, ..., rN ) ≥ 0 ∀r ∈ [0,∞)N a.e. in [0, T ]× Ω.

Let us recall that the law of mass action [31] yields the following expressionfor the velocity of the l-th reaction (see (1.3)):

δl = kl

N∏j=1

yαljj ,

so, the source term can be written as

ϕ = Aδ,

where A is the stoichiometric matrix of the system.

4.3. GLOBAL EXISTENCE OF WEAK SOLUTION 75

Lemma 4.3.1. The reaction term ϕ = Aδ is quasi-positive for reaction sys-tems governed by the law of mass action.

Proof. Firstly, we have

ϕi(t, x, r1, ..., ri−1, ri, ri+1, ..., rN ) =

L∑l=1

(λli − νli)kl(θ(t, x))

N∏j=1

rαljj , (4.26)

where kl(θ(t, x)) ≥ 0 a.e. in [0, T ]× Ω.Now, let us assume that ri = 0 and rj ≥ 0 for j 6= i. Then, for l ∈ 1, . . . , L

we have two possibilities:

• The i-th species is a reactant in the l-th reaction

In this case, λli = 0 and νli > 0. Hence,

(λli − νli)kl(t, x)

N∏j=1

rαljj = −νlikl(t, x)

N∏j=1

rαljj = 0,

because ri = 0 and so rαlii = 0.

• The i-th species is a product in the l-th reaction

In this case, λli > 0 and νli = 0 and then (λli − νli)kl(t, x)∏Nj=1 r

αljj ≥ 0.

Theorem 4.3.1 (Positivity of solution). Let us assume (H1) and (H2) hold.If the reaction term is quasi-positive, then the maximal solution of problem(4.3)–(4.4) is non-negative along the time. This means that property (P) issatisfied.

Proof. Let us recall that the positive part of z ∈ H1(Ω) is defined by

(z+)i(x) := z+i (x) = max0, zi(x), a.e. in Ω.

Similarly, the negative part is

(z−)i(x) := max0,−zi(x).

Both z+ and z− belong to H1(Ω) (see [62]). This is because the functions

s ∈ R 7→ s± ∈ R


are global contractions, i.e., globally Lipschitz with Lipschitz constant equal to1 . Then the mapping,

ϕ : [0, T ]× Ω× RN 7→ RN

(t, x, z) 7→ ϕ(t, x, z+)

has a similar property to (4.21) and therefore, the problem

〈∂z

∂t,w〉+

∫Ω

∇zv ·w +

∫Ω

d∇z : ∇w −∫

Γ1

(v · ν)z ·w

=

∫Ω

ϕ(t, x, z) ·w +

∫Γ1

g ·w ∀w ∈ H1(Ω),

z(0) = y0,

has a unique local solution in the same way as problem (4.3)-(4.4). Selectingw = z− as test function we get

〈∂z

∂t, z−〉+

∫Ω

∇zv · z− +

∫Ω

d∇z : ∇z− −∫

Γ1

(v · ν)z · z−

=

∫Ω

ϕ(t, x, z+) · z− +

∫Γ1

g · z−.

By using that z = z+ − z−, z+ : z− = 0, ∇z+ : ∇z− = 0 and also

∇zv · z− = v · ∇zT z− = −v · (∇z−)T z− = −v · 1

2∇(‖z−‖2),

we deduce

− 1

2

d

dt

∫Ω

‖z−‖2 −∫

Ω

v · 1

2∇(‖z−‖2)−

∫Ω

d‖∇z−‖2 +

∫Γ1

(v · ν)‖z−‖2

=

∫Ω

ϕ(t, x, z+) · z− +

∫Γ1

g · z−.

Then, using a Green’s formula and that divv = 0, we obtain

− 1

2

d

dt

∫Ω

‖z−‖2 − 1

2

∫∂Ω

(v · ν)‖z−‖2 −∫

Ω

d‖∇z−‖2 +

∫Γ1

(v · ν)‖z−‖2

=

∫Ω

ϕ(t, x, z+) · z− +

∫Γ1

g · z−.

By rearranging terms, and since v · ν = 0 in Γ3, we get


− 1

2

d

dt

∫Ω

‖z−‖2 − 1

2

∫Γ2

(v · ν)‖z−‖2 −∫

Ω

d‖∇z−‖2 +1

2

∫Γ1

(v · ν)‖z−‖2

−∫

Γ1

g · z− =

∫Ω

ϕ(t, x, z+) · z−.

(4.27)Now, we use the fact that ∫

Ω

ϕ(t, x, z+) · z− ≥ 0, (4.28)

because ϕ is quasi-positive. Indeed, for all (t, x) and i ∈ 1, . . . , N we candistinguish two cases, depending on the sign of zi(t, x):

1) If zi(t, x) ≥ 0, then zi(t, x)− = 0 and ϕi(t, x, z+(t, x))zi(t, x)− = 0.

2) If zi(t, x) < 0, then zi(t, x)− = −zi(t, x) > 0. Besides, zi(t, x)+ = 0 andfrom the quasi-positiveness of ϕ, we conclude that ϕi(t, x, z

+(x, t)) ≥ 0and hence ϕi(t, x, z

+(x, t))zi(t, x)− ≥ 0.

Therefore, (4.28) holds. Moreover, since g ≥ 0 , v · ν ≥ 0 in Γ2 and v · ν ≤ 0in Γ1, we have∫

Γ1

g · z− ≥ 0,

∫Γ2

(v · ν)‖z−‖2 ≥ 0 and

∫Γ1

(v · ν)‖z−‖2 ≤ 0.

Now, by using these inequalities in (4.27) we get

d

dt

∫Ω

‖z−‖2 ≤ 0. (4.29)

Integrating this inequality in (0, t) for any t ∈ [0, τ) and using the positivity ofthe initial condition (which implies z−(0) = 0), we obtain

‖z−(t)‖2L2(Ω) ≤ 0. (4.30)

Hence, z−(t) = 0 which implies that z(t) is non-negative in [0, τ). Therefore, zis also a solution of problem (4.3)–(4.4), but from uniqueness of local solutionz must be equal to y so y is non-negative.

Now, we prove property (M). We work in two steps. Firstly, we prove anequality which is valid for any real constants. After that, we will be able toprove (M) in the case of the law of mass action by substituting these constantsby the molecular masses of the species in the mixture.


Lemma 4.3.2. Let y be the maximal solution of problem (4.3)–(4.4) and a ∈RN . Then, the following equality holds for all t ∈ (0, τ):

d

dt

∫Ω

y · a +

∫Γ2

(v · ν)y · a =

∫Ω

ϕ(t, x,y) · a +

∫Γ1

g · a. (4.31)

Proof.By selecting the test function w = a in (4.3) we obtain the following equal-

ity:

d

dt

∫Ω

y · a +

∫Ω

∇yv · a−∫

Γ1

(v · ν)y · a =

∫Ω

ϕ(t, x,y) · a +

∫Γ1

g · a. (4.32)

Now, by using a Green’s formula in the convection term we deduce∫Ω

∇y v · a =

∫Ω

v · ∇yTa =

∫Ω

v · ∇(y · a)

=

∫∂Ω

y · a v · ν −∫

Ω

divv y · a =

∫∂Ω

y · a v · ν.

Then, equation (4.32) can be written as

d

dt

∫Ω

y · a +

∫∂Ω

y · a v · ν −∫

Γ1

v · ν y · a =

∫Ω

ϕ(t, x,y) · a +

∫Γ1

g · a.

Finally, rearranging the terms and taking into account that v · ν = 0 on Γ3

we obtain equation (4.31).

Lemma 4.3.3. If the source term corresponds to the law of mass action (1.3),then property (M) holds. More precisely, we have

ϕ(t, x, r) ·M = 0 ∀r ∈ RN , (4.33)

where M is the vector of molecular masses.

Proof. Let us choose α = M in (4.25). Then,

ϕ(t, x, r) ·M = Aδ(t, x, r) ·M = δ(t, x, r) ·AtM = 0,

because the mass conservation in each reaction implies that

(AtM)l =

N∑i=1

(λli − νli)Mi = 0 ∀l = 1, . . . , L.


As a consequence of property (M), the total mass is bounded by a constantwhich depends on the initial condition y0 and boundary function g, as it isproved in Corollary 4.3.1 below (notice that the total mass is not conservedbecause the “reactor” is not closed). However, although this property suggeststhat no blow-up may occur in finite time, the L∞(Ω)-norm of the solutionmay blow up in finite time for polynomial two-dimensional systems satisfyingproperties (P) and (M), as shown in [56].

Firstly, we notice that, since y ≥ 0 , the total mass in Ω at time t is givenby

m(t,Ω) :=

∫Ω

y(t, x) ·M = ‖y(t) ·M‖L1(Ω), a.e. in [0, τ).

Corollary 4.3.1 (Boundedness of the total mass). Let y be the maximalsolution of problem (4.3)–(4.4). Then we have that

y ·M ∈ L∞(0, τ ; L1(Ω)).

Proof. From Lemma 4.3.2 for a = M and using property (M) in Lemma4.3.3 we deduce

d

dt

∫Ω

y ·M +

∫Γ2

(v · ν)y ·M =

∫Γ1

g ·M.

Now, we apply Theorem 4.3.1 regarding the positivity of y and use theassumption v · ν ≥ 0 on Γ2, to get

d

dt

∫Ω

(M · y) ≤∫

Γ1

g ·M ≤ ‖M‖∫

Γ1

‖g‖. (4.34)

Finally, by integrating in the interval (0, t) with 0 < t < τ and taking intoaccount that the initial condition belongs to L1(Ω), we deduce

‖M · y(t)‖L1(Ω) ≤ C <∞, (4.35)

where

C := ‖M · y0‖L1(Ω) + ‖M‖∫ T

0

‖g(s)‖L1(Γ1)ds.

A similar equality to (4.31) can be proved replacing y with u.


Lemma 4.3.4. Let u be the maximal local solution of problem (PAu) andψ ∈ H1(Ω). Then, the following equality holds for all t ∈ (0, τ):∫

Ω

∂(M · u)

∂tψ +

∫Ω

v · ∇(M · u)ψ +

∫Ω

d∇(M · u) · ∇ψ

−∫

Γ1

(v · ν)(M · u)ψ = −∫

Ω

∂G

∂t· Mψ (4.36)

Proof. Firstly, we consider the weak formulation of problem (Pu), de-scribed in equation (4.24) and select as test function w = Mψ, where ψ ∈H1(Ω). By taking into account Lemma 4.3.3 related to (M), we obtain∫

Ω

∂(M · u)

∂tψ +

∫Ω

∇uv ·Mψ +

∫Ω

d∇u : ∇(Mψ)

−∫

Γ1

(v · ν)u ·Mψ = −∫

Ω

∂G

∂t·Mψ, (4.37)

where we have used the following equality:

M · ϕ(t, x,u) = M ·ϕ(x, t,u + G)−M · ∂G

∂t(t, x) = −M · ∂G

∂t(t, x),

because of (4.33). Now, the second and the third terms in (4.37) can be trans-formed as follows∫

Ω

∇uv ·Mψ =

∫Ω

v · ∇uTMψ =

∫Ω

v · ∇(M · u)ψ

and∫Ω

d∇u : ∇(Mψ) =

∫Ω

d∇u : (ψ∇M + M⊗∇ψ) =

∫Ω

d∇u : (M⊗∇ψ)

=

∫Ω

d∇(M · u) · ∇ψ,

because, in general,S : (a⊗ b) = Sta · b

for any N×n matrix S, and vectors a ∈ RN and b ∈ Rn. By replacing in (4.37)we get the result.

Now, we study the global existence of solution using the above properties.

Proposition 4.3.1. If properties (P) and (M) hold, as well as hypotheses(H1) and (H2), then there exists a positive constant C such that the maximalsolution satisfies

‖y‖C([0,τ ]×Ω) ≤ C.


Proof. It is similar to the one showing the positivity of solution in Theo-rem 4.3.1.

Let M1 and M2 be any positive real numbers. From (4.36) it is easy todeduce that∫

Ω

∂(M · u−M1 −M2t)

∂tψ +

∫Ω

v · ∇(M · u−M1 −M2t)ψ

+

∫Ω

d∇(M · u−M1 −M2t) · ∇ψ −∫

Γ1

(v · ν)(M · u−M1 −M2t)ψ

= −∫

Ω

∂G

∂t·Mψ +

∫Γ1

(v · ν)(M1 +M2t)ψ −M2

∫Ω

ψ.

Now, let us choose ψ = (M · u−M1 −M2t)+. We get

1

2

d

dt

∫Ω

((M · u−M1 −M2t)

+)2

+

∫Ω

v · ∇(M · u−M1 −M2t)(M · u−M1 −M2t)+

+

∫Ω

d‖∇(M · u−M1 −M2t)+‖2 −

∫Γ1

(v · ν)‖(M · u−M1 −M2t)+‖2

= −∫

Ω

(∂G

∂t·M +M2

)(M · u−M1 −M2t)

+

+ (M1 +M2t)

∫Γ1

(v · ν)(M · u−M1 −M2t)+

and then

1

2

d

dt‖(M · u(t)−M1 −M2t)

+‖2L2(Ω) +

∫Ω

d‖∇(M · u−M1 −M2t)+‖2

+1

2

∫Γ2

v · ν((M · u−M1 −M2t)

+)2 − 1

2

∫Γ1

v · ν((M · u−M1 −M2t)

+)2

= −∫

Ω

(∂G

∂t·M +M2

)(M · u−M1 −M2t)

+

+ (M1 +M2t)

∫Γ1

(v · ν)(M · u−M1 −M2t)+.

Taking into account the sign of the different terms we deduce

1

2

d

dt‖(M · u(t)−M1 −M2t)

+‖2L2(Ω)

≤ −∫

Ω

(∂G

∂t·M +M2

)(M · u−M1 −M2t)

+ (4.38)


and integrating (4.38) in time from 0 to t we get

‖(M · u(t)−M1 −M2t)+‖2L2(Ω) ≤ ‖(M · u0 −M1)+‖2L2(Ω)

− 2

∫ t

0

∫Ω

(∂G

∂t·M +M2

)(M · u−M1 −M2t)

+.(4.39)

Now, let us take M1 and M2 such that

M · u0(x) ≤M1,∣∣∣∂G

∂t(t, x) ·M

∣∣∣ ≤M2 ∀t ∈ [0, T ], ∀x ∈ Ω.

Then, (4.39) yields

‖(M · u(t)−M1 −M2t)+‖2 ≤ 0

and finally,M · u(t, x) ≤M1 +M2T ∀t ∈ [0, T ], ∀x ∈ Ω,

from which it follows that

M · y(t, x) = M · (u(t, x) + G(t, x)) ≤M := M1 +M2T + ‖M ·G‖C([0,T ]×Ω).

From Theorem 4.3.1 the components of y are non-negative, so this inequal-ity implies that

yi(x, t) ≤M

Mi, i = 1, · · · , N,

which concludes the proof for

C = maxi=1,...,N

M

Mi

.

Finally, we prove that the maximal solution is defined in the whole timeinterval [0, T ]. That is, the solution can be prolonged to the space C([0, T ]×Ω).In the previous section, we have obtained the maximal interval for the localsolution. From there, we are able to demonstrate that the solution is boundedin that interval, I(y0) ⊆ [0, T ]. We argue by contradiction that the solutiondoes not blow-up when it tends to T on the right side.

Theorem 4.3.2. Under the assumptions (D1), (A1), (A2), (H1), (H2) and

• g ∈ W2,r(0, τ ; Lp(Γ1)) with p = n − 1 + ε if n > 2 and p = 2 if n = 2and r > 1,

• temperature θ ∈ C1([0, T ])× Ω) and θ(t, x) 6= 0 ∀(t, x) ∈ [0, T ]× Ω,


there exists a global solution of problem (4.3)–(4.4) (defined in [0, T ]) such thaty ∈ C([0, T ]× Ω).

Proof. It is a straightforward consequence of Proposition 4.3.1 and Propo-sition B.5.2 (see also Section 4.2.4).

4.3.1 Uniqueness of solution

As it is pointed out in [55] the question of uniqueness in general diffusion-reaction problems is certainly delicate since it is known that there is not unique-ness of weak solutions even for simple equations as, for instance,

∂u

∂t−∆u = u3, u(0) = u0 ≥ 0, u = 0 on ∂Ω,

and even for C∞ initial data. Further details can be found in [4] and [32].However, working with uniformly bounded solutions is satisfactory because

it allows us to prove uniqueness and so the problem is well posed in this class,as it is insured in [55]. Thus, we can conclude this chapter with the main result:

Theorem 4.3.3. Under the assumptions in Theorem 4.3.2, there exists aunique bounded global solution in the interval [0, T ] of problem (4.3)–(4.4).

Proof. Let as suppose that y1,y2 ∈ C([0, T ] × Ω) are solutions of (4.3)–(4.4). Then, for i = 1, 2, we have

〈∂yi∂t

(t),w〉+ a(yi(t),w) =

∫Ω

ϕ(t, .,yi(t, .)) ·w + l(t)(w)

∀w ∈ H1(Ω),

yi(0) = y0.

Now, if we subtract these formulations we obtain

〈∂(y1 − y2)

∂t(t),w〉+ a(y1(t)− y2(t),w)

=

∫Ω

(ϕ(t, .,y1(t, .))−ϕ(t, .,y2(t, .))) ·w

∀w ∈ H1(Ω),

y1(0)− y2(0) = 0.

By choosing w = y1(t)− y2(t) we get

1

2

d

dt‖y1 − y2‖2L2(Ω)(t) + a(y1(t)− y2(t),y1(t)− y2(t))

=

∫Ω

(ϕ(t, .,y1(t, .)−ϕ(t, .,y2(t, .))) ·(y1(t)− y2(t)

).


Now, due to coerciveness of the bilinear form proved in Proposition 4.2.1 wecan write

1

2

d

dt‖y1 − y2‖2L2(Ω) + β‖y1(t)− y2(t)‖2H1(Ω)

≤∫

Ω

(ϕ(t, .,y1(t, .))−ϕ(t, .,y2(t, .))) ·(y1(t)− y2(t)

).

Moreover, function ϕ is locally Lipschitz and hence it is Lipschitz in the unionset of the ranges of the two solutions. Therefore, there exists, K > 0 such that∫

Ω

(ϕ(t, .,y1(t, .))−ϕ(t, .,y2(t, .))) ·(y1(t)− y2(t)

)≤ K‖y1(t)− y2(t)‖2L2

and then we can write

1

2

d

dt‖y1 − y2‖2L2(Ω)(t) + β‖y1(t)− y2(t)‖2H1(Ω) ≤ K‖y1(t)− y2(t)‖2L2(Ω).

Finally, we integrate in (0, t) to obtain the following inequality

‖y1(t)− y2(t)‖2L2(Ω) ≤ 2K

∫ t

0

‖y1(s)− y2(s)‖2L2(Ω)

and by using the Gronwall’s lemma, ‖y1(t)−y2(t)‖L2(Ω) ≤ 0 ∀t ∈ [0, T ], so wecan conclude that y1 = y2 and hence the uniqueness.

Chapter 5

Numerical analysis of theconvection-diffusion-reaction n−dimensionalmodel

Convection-diffusion-reaction models are used in many applications in scienceand engineering. In general, these models are described by nonlinear equationsfor which an exact analytical solution is difficult to obtain, and from here theneed of numerical methods in order to solve such non-linear models.

This chapter is devoted to the numerical solution of the problem introducedin the previous chapter, namely,

(P )

∂y

∂t(t, x) +∇y(t, x)v − d∆y(t, x) = ϕ(t, x,y(t, x)), (t, x) ∈ (0, T )× Ω,

y(0, x) = y0(x), x ∈ Ω,

d∂y


d∂y

∂ν(t, x) = 0, (t, x) ∈ (0, T )× (Γ2 ∪ Γ3),

where Ω is a bounded domain in Rn with smooth boundary ∂Ω.Let us assume the hypotheses considered in Chapter 4 which are

• d > 0,

• v ∈ C1(Ω), divv = 0, v · ν ≤ 0 on Γ1, v · ν ≥ 0 on Γ2, v · ν = 0 on Γ3,

85

86 CHAPTER 5. NUMERICAL ANALYSIS

and there exist S ⊂ Γ2 with non-null surface measure and α > 0 suchthat v · ν ≥ α on S,

• y0 ∈ C(Ω) with non-negative components ∀x ∈ Ω,

• g ∈W2,r(0, T ; Lp(Γ1)) with p = n−1+ε for n > 2 or g ∈W2,r(0, T ; L2(Γ1))for n = 2, r > 1 and with non-negative components a.e. in [0, T ]× Γ1,

• temperature θ ∈ C1([0, T ])× Ω) and θ(t, x) 6= 0 ∀(t, x) ∈ [0, T ]× Ω.

Remark 5.0.1. In next paragraphs, we assume that ϕ ∈ C([0, T ]; Hl+1(Ω)). Inorder to obtain the required regularity we must assume θ ∈ C([0, T ]); Cl+1(Ω)).For the Law of Mass Action with integer coefficients the required regularity isinsured.

We recall that the non-negativity assumption of g and y0 has been describedas (H1) and (H2).

Let us recall that the weak formulation of this problem consists in findinga function y ∈ L2((0, T ); H1(Ω)) satisfying

d

dt

∫Ω

y(t, x) ·w(x) + a(y(t),w) =

∫Ω

ϕ(t, x,y(t, x)) ·w(x) + l(t)(w)

∀w ∈ H1(Ω) (5.1)

y(0) = y0, (5.2)

where

a(y(t),w) :=

∫Ω

∇y(t, x)v(x)·w(x)+

∫Ω

d∇y(t, x) : ∇w(x)−∫

Γ1

(v·ν)y(t, x)·w(x)

and

l(t)(w) :=

∫Γ1

g(t, x) ·w(x).

5.1 The semidiscrete problem

This section is devoted first to build a semidiscrete problem obtained by re-placing the Sobolev space H1(Ω) by a finite-dimensional approximation spaceand second to study the existence and uniqueness of a solution.

Firstly, we consider a linearly independent finite subset of the space H1(Ω)∩C(Ω): ψ1, · · · , ψndof . Then, we introduce an associated approximation of y(t)in the space spanned by these functions, namely,

5.1. THE SEMIDISCRETE PROBLEM 87

y(t) =

ndof∑i=1

αi(t)ψi ∈ H1(Ω) ∩ C(Ω),

where αi(t), i = 1, . . . , ndof are some vector functions of time,

αi : [0, T ]→ RN ,

to be determined.Let us define the space spanned by ψ1, · · · , ψndof :

W∧

= 〈ψ1, · · · , ψndof 〉

and let W∧

= W∧N

. In order to define the approximation y, we go back to theweak formulation (5.1) and introduce the following problem:

Find a function y : [0, T ]→W∧

satisfying,

d

dt

∫Ω

y(t) ·w + a(y(t),w) =

∫Ω

ϕ(t, ., y(t)) ·w + l(t)(w), ∀w ∈W∧

(5.3)

y(0) = y0, (5.4)

where y0 is a given approximation of y0 in W∧

.

5.1.1 A finite element method

Here we introduce a particular but very important example of internal ap-proximation space Vh: let Th be a collection of quasi-uniform elements (seeDefinition C.3.1) that partitions the domain Ω ⊂ Rn. These elements are tri-angles if n = 2 or tetrahedra if n = 3. Parameter h is the maximal diameter ofthe elements. Let us denote by V = H1(Ω) and by Vh the space of piecewisecontinuous functions on Ω that reduce to polynomials of degree ≤ m on eachelement of Th.

Remark 5.1.1. In the previous chapter we have supposed that Ω has a C2

boundary, for existence of a solution to the continuous problem (see 4.2.2).We notice that this assumption is not compatible with the fact that Ω has apartition in triangles or tetrahedra. Thus, Vh 6⊂ V and hence we are leadto commit a so-called variational crime in the Strang terminology. The erroranalysis arising from this variational crime is not trivial and it is beyond thescope of the present work.

We consider the canonical basis of Vh, ψh,1, · · · , ψh,ndof , where ψh,i hasvalue 1 at the i-th node and 0 at the rest of the nodes. Then, the Lagrangeinterpolation operator,

Ih : C(Ω)→ Vh,


is defined as follows: Ih(f) ∈ Vh and

Ih(f)(xj) = f(xj),

being x1, . . . , xndof the nodes associated to the finite element space (recallthat in each element the set of nodes is the n-dimensional simplex of type m(see, for instance, [22]). Therefore,

Ih(f) =

ndof∑i=1

f(xi)ψh,i

and its extension to vector functions is straightforward:

Ih(f) = (Ih(f1), . . . , Ih(fN )).

Thus, we have

Ih(f) =

ndof∑i=1

f(xi)ψh,i

and, in particular,

Ih(ϕ(t, x,y(t, x))

)=

ndof∑i=1

ϕ(t, xi,y(t, xi))ψh,i(x). (5.5)

Then, the finite element method with interpolated nonlinear term consistsin

Finding

yh(t) =

ndof∑i=1

αh,i(t)ψh,i ∈ Vh

such that

d

dt

∫Ωh

yh(t) ·wh + a(yh(t),wh)

=

∫Ωh

Ih(ϕ(t, .,yh(t))

)·wh + l(t)(wh)∀wh ∈ Vh, (5.6)

yh(0) = y0,h, (5.7)

whereVh = (Vh)N

and y0,h is an approximation of y0 in Vh.Now, we notice that the dimension of space Vh is N × ndof and introduce

a suitable basis for computer implementation:

zh,nk = ψh,ken, k = 1, ..., ndof and n = 1, ..., N,


where e1, · · · , eN is the canonical basis in RN .

Let us take wh = zh,nk. Then the nonlinear term can be written as

∫Ωh

Ih(ϕ(t, .,yh(t)) · zh,nk =

∫Ωh

Ih(ϕ(t, .,yh(t)) · enψh,k

=

∫Ωh

Ih(ϕn(t, .,yh(t))

)ψh,k =

ndof∑i=1

∫Ωh

ϕn(t, xi,αh,i(t)

)ψh,iψh,k

=

ndof∑i=1

ϕn(t, xi,αh,i(t)

) ∫Ωh

ψh,iψh,k.

We notice that the obtained expression is the product of the mass matrixMh defined by

Mh,ik =

∫Ωh

ψh,iψh,k, i, k = 1, . . . , ndof ,

times the vector

ϕnh(t) =

ϕn(t, x1,αh,1(t)

)...

ϕn(t, xndof ,αh,ndof (t)

) .

Now, we rewrite the above problem as a nonlinear numerical system. Let

αnh(t) =

αnh,1(t)...

αnh,ndof (t)

∈ Rndof

and

αh(t) =

α1h(t)...

αNh (t)

∈ RNndof .

Then, the Cauchy problem to be solved is

Mhα′h(t) +Ahαh(t) =Mhϕh(t,αh(t)) + ch(t), (5.8)

αh(0) = αh,0, (5.9)


where Mh and Ah are block-diagonal matrices

Mh =

Mh 0 · · · 0

0 Mh. . .

......

. . .. . . 0

0 · · · 0 Mh

,

Ah =

Ah 0 · · · 0

0 Ah. . .

......

. . .. . . 0

0 · · · 0 Ah

,

because the corresponding terms in the original problem do not couple thecomponents of y. Each matrix block Ah is the stiffness matrix

Ah,kj =

∫Ωh

ψh,k(∇ψh,j · v) + d

∫Ωh

∇ψh,j · ∇ψh,k −∫

Γ1

(v · ν)ψh,jψh,k,

with k, j = 1, · · · , ndof . Moreover,

cnh(t) =

cnh,1(t)...

cnh,ndof (t)

∈ Rndof ,

and

ch(t) =

c1h(t)...

cNh (t)

∈ RNndof ,

where

cnh,k(t) =

∫Ωh

ψh,k(x)gn(t, x), k = 1, · · · , ndof and n = 1, · · · , N

and

ϕh(t,αh(t)) =

ϕ1h(t,αh(t))

...ϕNh (t,αh(t))

,

where

ϕnh(t,αh(t)) =

ϕnh,1(t,αh,1(t))...

ϕnh,ndof (t,αh,ndof (t))

,


with ϕnh,k(t,αh,k(t)) = ϕn (t, xk,αh,k(t)) , k = 1, . . . ndof , n = 1, . . . , N.Finally, the initial condition is given by

αh,0 =

α1h,0...

αNh,0,

with

yh,0 =

N∑i=1

αnh,0ψh,i.

5.1.2 Local existence and uniqueness of solution to thesemidiscrete problem

In this section, we study the existence and uniqueness of solution to the semidis-crete problem (5.6)–(5.7). This problem has been rewritten in the previous sec-tion as a nonlinear ordinary differential system in (5.8). The proof of existenceof a local solution is based on the classical Picard-Lipschitz-Lindelof theorem(see, for instance [23]). Let us first recall this theorem and then we will proveour local existence result.

Theorem 5.1.1 (Picard-Lipschitz-Lindelof theorem). Let f : A ⊆ R ×Rn −→ Rn be a continuous function, locally Lipschitz with respect to x , whereA is an open set.

Then, for any given (t0, x0) ∈ A, there exists a closed interval Iα = [t0 −α, t0 + α] ⊂ R where the Cauchy problem:

x′ = f(t, x),

x(t0) = x0,

has a unique solution satisfying (t, x(t)) ∈ A ∀t ∈ Iα.

Now, we state the local existence of our ODEs system as follows:

Theorem 5.1.2. There exists δ > 0 such that the Cauchy problem (5.6)–(5.7)has a unique solution in the interval (0, δ).

Proof.Since the mass matrix Mh is invertible, the Cauchy problem (5.8)–(5.9)

can be written as

α′h(t) = ϕh(t,αh(t)) +M−1ch(t)−M−1Aαh(t),

αh(0) = αh,0.


Moreover, the mapping

(t,αh)→ ϕh(t, .,αh) +M−1h ch(t)−M−1

h Ahαh ∈ RNndof

is continuous and locally Lipschitz with respect to αh from Lemma 4.2.1.Therefore, by applying the Picard-Lipschitz-Lindelof theorem, forαh,0 given,

this problem has a unique solution in (0, δh) for some δh ∈ (0, T ].

Our next goal is to prove the existence of a global solution for the semidis-crete problem, i.e, a solution defined in the whole interval [0, T ]. Firstly, wenotice that the techniques used in the previous chapter to prove the globalexistence of solution to the continuous problem cannot be applied, essentiallybecause for wh ∈ Vh its positive or negative parts do not belong, in general,to Vh. This is why we will use a different method following some ideas from[63]. It consists in proving first an error estimate for the local solution in theL∞-norm. Thus, since the solution of the continuous problem is bounded in[0, T ]×Ω, the same will be true for the local semidiscrete solution in the timeinterval of existence, for h small enough. This fact will allow us to prolong itto the whole interval [0, T ]: let us assume that the maximal local solution forgiven h and y0,h is defined in a time interval Ih(y0,h) starting at 0. Then, wewill prove that, for h0 small enough,

supt∈Ih(y0,h)

‖yh(t)‖ <∞ ∀h < h0

and, therefore, well known results for ordinary differential equations will allowus to conclude that a global solution exists in [0, T ].

5.2 Error estimates for the semidiscrete solu-tion

In this section we follow the lines of Thomee’s book [63] in order to estimate theerror obtained when the exact solution is replaced with the maximal numericalsolution yh : Ih(y0,h)→ Vh of the semidiscrete problem

(yh,t,χh) + a (yh,χh) = (Ih(ϕ(yh)

),χh) +

∫Γ1

g · χh ∀χh ∈ Vh,

where

a(yh,χh) : =

∫Ω

∇yhv · χh +

∫Ω

d∇yh : ∇χh −∫

Γ1

(v · ν)y · χh

= (∇yhv,χh) + d (∇yh,∇χh)−∫

Γ1

(v · ν)yh · χh

5.2. ERROR ESTIMATES FOR THE SEMIDISCRETE SOLUTION 93

and yh(0) = y0h, with y0h ∈ Vh being an approximation of y0 in Vh.For the sake of simplicity, in this section we denote the derivative of a

function ϕ with respect to t by ϕ,t. Moreover, C will denote a constant maybe different at each occurence.

For a given function z ∈ H1(Ω), we recall that its elliptic projection ontoVh is the unique solution of the elliptic problem,

Find zh ∈ Vh such that

a(z− zh,χh) = 0 ∀χh ∈ Vh. (5.10)

Then, we have the following result:

Lemma 5.2.1. Let zh ∈ Vh be the elliptic projection defined in (5.10) andassume z ∈ Hl+1(Ω) with l ≤ m. Then,

‖z− zh‖L2(Ω) ≤ Chl+1‖z‖Hl+1(Ω). (5.11)

Proof. Firstly, from the classical error estimates for elliptic problems basedon Cea’s Lemma, we have

‖z− zh‖H1(Ω) ≤ Chl‖z‖Hl+1(Ω). (5.12)

Then, in order to get (5.11) we use the Aubin-Nitsche technique (see, for in-stance, [37, Th. 3.37]). Let us call e ∈ H1(Ω) the unique solution of the adjointproblem

a(w, e) = (z− zh,w)L2(Ω) ∀w ∈ H1(Ω). (5.13)

From regularity results (see [30, Th. 2.4.2.7] or [37, Th. 3.18]) we have e ∈H2(Ω) and

‖e‖H2(Ω) ≤ C‖z− zh‖L2(Ω).

Let us take w = z− zh as test function in (5.13). We get,

‖z− zh‖2L2(Ω) = a(z− zh, e) = a(z− zh, e− Ih(e))

≤ C‖z− zh‖H1(Ω)‖e− Ih(e)‖H1(Ω) ≤ C‖z− zh‖H1(Ω)h‖e‖H2(Ω)

≤ C‖z− zh‖H1(Ω)h‖z− zh‖L2(Ω) (5.14)

from which it follows that

‖z− zh‖L2(Ω) ≤ Ch‖z− zh‖H1(Ω) ≤ Chl+1‖z‖Hl+1(Ω)

by using (5.12).

Lemma 5.2.2. Let zh ∈ Vh be the elliptic projection defined in (5.10) andassume z ∈ Hl+1(Ω) with n/2 < l + 1 and l ≤ m. Then,

‖z− zh‖L∞(Ω) ≤ C. (5.15)


Proof. Firstly, we notice that z ∈ L∞(Ω) because n/2 < l + 1. Moreover,from the triangular inequality we have

‖z− zh‖L∞(Ω) ≤ ‖z− Ih(z)‖L∞(Ω) + ‖Ih(z)− zh‖L∞(Ω).

If we use Lemma C.3.2 in the first term and the inverse inequality (C.2) in thesecond one we deduce

‖z− zh‖L∞(Ω) ≤ Chl+1−n2 ‖z‖Hl+1(Ω) + Ch−n2 ‖Ih(z)− zh‖L2(Ω) (5.16)

and applying again the triangular inequality in the last term, and using Lemma C.3.2and inequality (5.11) we can write

h−n2 ‖Ih(z)− zh‖L2(Ω) ≤ h−

n2 ‖Ih(z)− z‖L2(Ω) + h−

n2 ‖z− zh‖L2(Ω)

≤ Chl+1−n2 ‖z‖Hl+1(Ω).(5.17)

Finally, we derive the following L∞−boundedness by substituting (5.17) ininequality (5.16):

‖z− zh‖L∞(Ω) ≤ Chl+1−n2 ‖z‖Hl+1(Ω),

which implies (5.15) as n/2 < l + 1.

Now we are in a position to prove the following error estimate for thesemidiscrete solution.

Theorem 5.2.1. Let us assume that the initial value y0 ∈ Hl+1(Ω) and theglobal continuous solution has the following regularity properties:

y ∈ L1((0, T ); Hl+1(Ω)) y,t ∈ L1((0, T ); Hl+1(Ω))

with l + 1 > n/2 and l ≤ m. Let [0, τh] be an interval where the local solutionof the discrete problem is defined. Then,

‖y − yh‖L∞((0,τh);L2(Ω)) ≤ C‖y0 − y0h‖L2(Ω) + Chl+1‖y0‖Hl+1(Ω)

+ Chl+1(1 + ‖y‖L1((0,T );Hl+1(Ω)) + ‖y,t‖L1(0,T );Hl+1(Ω))

). (5.18)

Proof. Firstly, we split the error into two terms involving the ellipticprojection of y, yh:

y − yh = ρh + θh,

with ρh = y − yh and θh = yh − yh.


Then, we compare the elliptic projection, yh ∈ Vh, to the solution of thesemidiscrete problem, i.e., we estimate θh. For this purpose, we notice thatfrom our definitions,(

θh,t,χh)

+ a (θh,χh) = (yh,t,χh) + a (yh,χh)− (yh,t,χh)− a (yh,χh)

= (yh,t,χh) + a (yh,χh)− (Ih(ϕ(yh)

),χh)−

∫Γ1

g · χh

= 〈yh,t − y,t + y,t,χh〉+ a (y,χh)− (Ih(ϕ(yh)

),χh)−

∫Γ1

g · χh

= 〈yh,t − y,t,χh〉+ (ϕ(y),χh)− (Ih(ϕ(yh)

),χh)

= −(ρh,t,χh) + (ϕ(y)− Ih(ϕ(yh)

),χh),

where (·, ·) represents the L2(Ω) inner product and we have assumed that y,t ∈L1(0, T ; L2(Ω)). Therefore,(

θh,t,χh)

+ a (θh,χh) = −(ρh,t,χh) + (ϕ(y)− Ih(ϕ(yh)

),χh). (5.19)

By choosing χh = θh, we rewrite equation (5.19) as

1

2

d

dt‖θh‖2L2(Ω) + a (θh,θh) = −〈ρh,t,θh〉+ (ϕ(y)− Ih

(ϕ(yh)

),θh).

Now, due to coerciveness of the bilinear form proved in Proposition 4.2.1we can write

1

2

d

dt‖θh‖2L2(Ω) +β‖θh‖2H1(Ω) ≤ ‖ρh,t‖L2(Ω)‖θh‖L2(Ω) +(ϕ(y)−Ih

(ϕ(yh)

),θh)

and, by using that ‖θh‖2L2(Ω) ≤ ‖θh‖2H1(Ω), we get that

1

2

d

dt‖θh‖2L2(Ω) +β‖θh‖2L2(Ω) ≤ ‖ρh,t‖L2(Ω)‖θh‖L2(Ω) +(ϕ(y)−Ih

(ϕ(yh)

),θh).

(5.20)The crucial issue in the error analysis is the interpolation error of the non-

linear term, that can be decomposed into three parts:

(ϕ(y)− Ih(ϕ(yh)

),θh) = (ϕ(y)− Ih

(ϕ(y)

),θh)

+(Ih(ϕ(y)

)− Ih

(ϕ(yh)

),θh) + (Ih

(ϕ(yh)

)− Ih

(ϕ(yh)

),θh).

By Cauchy–Schwartz inequality, we obtain

(ϕ(y)− Ih(ϕ(yh)

),θh) ≤

(‖ϕ(y)− Ih

(ϕ(y)

)‖L2(Ω)

+‖Ih(ϕ(y)

)− Ih

(ϕ(yh)

)‖L2(Ω) + ‖Ih

(ϕ(yh)

)− Ih

(ϕ(yh)

)‖L2(Ω)

)‖θh‖L2(Ω).


For the first term, which is the interpolation error of ϕ(y), assuming thatϕ(y) ∈ C([0, T ]; Hl+1(Ω)) and 1 ≤ l ≤ m, we have (applying Remark C.3.2 inAppendix C)

‖ϕ(y)− Ih(ϕ(y)

)‖L2(Ω) ≤ hl+1‖ϕ(y)‖Hl+1(Ω) ≤ Chl+1.

Now, for the second term, by using the locally-Lipschitz property of ϕ (withLemma 5.2.2) and the inequalities from Lemma C.4.2, we have

‖Ih(ϕ(y)

)− Ih

((ϕ(yh)

)‖L2(Ω) ≤ C2h

n2 ‖Ih

(ϕ(y)

)− Ih

(ϕ(yh)

)‖h

= C2hn2 ‖ϕ(y)−ϕ(yh)‖h ≤ C3h

n2 L‖y − yh‖h = C3h

n2 L‖Ih

(y)− yh‖h

≤ C4L‖Ih(y)− yh‖L2(Ω) ≤ C4L

(‖Ih

(y)− y‖L2(Ω) + ‖y − yh‖L2(Ω)

)= C4L

(‖Ih

(y)− y‖L2(Ω) + ‖ρh‖L2(Ω)

)≤ C5

(hl+1 + ‖ρh‖L2(Ω)

).

For the third term, we apply the inequalities from Lemma C.4.2, equality(5.5) and the locally-Lipschitz property of ϕ to obtain

‖Ihϕ(yh)− Ihϕ(yh)‖L2(Ω)

≤ C2hn2 ‖Ihϕ(yh)− Ihϕ(yh)‖h = C2h

n2 ‖ϕ(yh)−ϕ(yh)‖h

≤ C2hn2 L‖yh − yh‖h ≤ C3‖yh − yh‖L2(Ω) = C3‖θh‖L2(Ω).

In the previous estimates we have used that ‖yh(t)‖L∞(Ω) is bounded in thetime interval where we are working, i.e., in [0, τh].

Now, by replacing in inequality (5.20) we get

1

2

d

dt‖θh‖2L2(Ω) + β‖θh‖2L2(Ω) ≤ ‖ρh,t‖L2(Ω)‖θh‖L2(Ω)

+(K1h

l+1 +K2‖ρh‖L2(Ω) +K3‖θh‖L2(Ω)

)‖θh‖L2(Ω),

(5.21)

for some constants K1,K2,K3.Now, we have two possibilities:

• β ≥ K3: in this case we use the non-negativity of ‖θh‖L2(Ω) and defineλ2 := K3;

• β < K3: in this case we define λ2 := K3 − β > 0.

In any case, we can write (5.21) as

1

2

d

dt‖θh‖2L2(Ω) ≤ ‖ρh,t‖L2(Ω)‖θh‖L2(Ω)

+(K1h

l+1 +K2‖ρh‖L2(Ω) + λ2‖θh‖L2(Ω)

)‖θh‖L2(Ω),


or equivalently,

1

2

d

dt‖θh‖2L2(Ω) − λ2‖θh‖2L2(Ω)

≤(‖ρh,t‖L2(Ω) +K1h

l+1 +K2‖ρh‖L2(Ω)

)‖θh‖L2(Ω).

(5.22)

Now we compute the following derivative

1

2

d

dt

(e−2λ2t‖θh‖2L2(Ω)

)= e−2λ2t

(1

2

d

dt‖θh‖2L2(Ω) − λ2‖θh‖2L2(Ω)

).

Then multiplying inequality (5.22) by e−2λ2t and using the above equality weobtain

1

2

d

dt

(e−λ2t‖θh‖L2(Ω)

)2≤ e−λ2t

(‖ρh,t‖L2(Ω) +K1h

l+1 +K2‖ρh‖L2(Ω)

)e−λ2t‖θh‖L2(Ω).

By integrating in (0, t) for t < τh we have

1

2

(e−λ2t‖θh(t)‖L2(Ω)

)2 − 1

2‖θh(0)‖2L2(Ω)

≤∫ t

0

e−λ2s(‖ρh,t(s)‖L2(Ω) +K1h

l+1 +K2‖ρh(s)‖L2(Ω)

)e−λ2s‖θh(s)‖L2(Ω)ds

and by applying [26, Lemma A.5] we obtain,

‖θh(t)‖L2(Ω) ≤ eλ2t‖θh(0)‖L2(Ω)

+ eλ2t

∫ t

0

e−λ2s(‖ρh,t(s)‖L2(Ω) +K1h

l+1 +K2‖ρh(s)‖L2(Ω)

)ds.

(5.23)

In order to estimate ‖ρh‖L2(Ω) and ‖ρh,t‖L2(Ω) we use Lemma 5.2.1 forz = y(t) and the fact that the time derivative of the elliptic projection is theelliptic projection of the time derivative (see, for instance, [58]). In other words,the following equality holds:

a

(dyhdt− dy

dt, z

)= 0 ∀z ∈ H1(Ω).

Thus, we obtain

‖ρh‖L2(Ω) ≤ Chl+1‖y‖Hl+1(Ω) ≤ Chl+1

(‖y0‖Hl+1(Ω) +

∫ t

0

‖y,t‖Hl+1(Ω)

),

‖ρh,t‖L2(Ω) ≤ Chl+1‖y,t‖Hl+1(Ω).(5.24)


For ‖θh(0)‖L2(Ω) we have the following estimate by using Lemma 5.2.1

‖θh(0)‖L2(Ω) ≤‖y0h − y0‖L2(Ω) + ‖y0 − y0h‖L2(Ω)

≤ Chl+1‖y0‖Hl+1(Ω) + ‖y0 − y0h‖L2(Ω).(5.25)

Now, using (5.24) and (5.25) in (5.23) we get

‖θh(t)‖L2(Ω) ≤ eλ2t‖y0 − y0h‖L2(Ω) + Chl+1eλ2t‖y0‖Hl+1(Ω)

+ eλ2t

∫ t

0

e−λ2s(K1h

l+1 +K4hl+1‖y(s)‖Hl+1(Ω) +K5h

l+1‖y,t(s)‖Hl+1(Ω)

).

To conclude, the error estimate is given by the following inequality:

‖y(t)− yh(t)‖L2(Ω) ≤ eλ2t‖y0 − y0h‖L2(Ω) +K6hl+1eλ2t‖y0‖Hl+1(Ω)

+K7hl+1‖y(t)‖Hl+1(Ω) +K8h

l+1eλ2t

∫ t

0

e−λ2s(1 + ‖y(s)‖Hl+1(Ω) + ‖y,t(s)‖Hl+1(Ω)

).

5.3 Existence and uniqueness of global solutionto the semidiscrete problem

In this section, by using the above error estimate, we prove the existence anduniqueness of a global solution to the semidiscrete problem (5.6)–(5.7). Werecall that n denotes the spatial dimension, i.e., Ω ⊂ Rn.

Proposition 5.3.1. Let us assume that

• y ∈ L1((0, T ); Hl+1(Ω)), y,t ∈ L1((0, T ); Hl+1(Ω)).

• l + 1 > n/2 and l ≤ m.

• y0h is an approximation of y0 of order O(hn/2+ε) in the norm of L2(Ω),for some ε > 0.

Then the semidiscrete problem (5.6)–(5.7) has a unique global solution yh ∈C1([0, T ]; Vh).

Proof. Since the global solution of the continuous problem, y, is continuousin [0, T ]× Ω, the following real numbers,

mi := min yi(t, x) : (t, x) ∈ [0, T ]× Ω, (5.26)

Mi := max yi(t, x) : (t, x) ∈ [0, T ]× Ω (5.27)

5.3. GLOBAL SOLUTION TO THE SEMIDISCRETE PROBLEM 99

are well-defined for i = 1, . . . , N .Since we are assuming that the discrete initial condition satisfies

‖y0 − y0h‖L2(Ω) = O(hn/2+ε), (5.28)

by using Theorem 5.2.1 we obtain

‖y − yh‖L∞((0,τh(y0,h));L2(Ω)) ≤ O(hn/2+ε) +O(hl+1).

Besides, from Lemma C.3.2 we get

‖Ihy − y‖L∞((0,τh(y0,h));L2(Ω)) = O(hl+1)

and‖Ihy − y‖L∞((0,τh(y0,h));C(Ω)) = O(hl+1−n/2).

Moreover, for all t ∈ [0, τh(y0,h))

‖y(t)− yh(t)‖C(Ω) ≤ ‖yh(t)− Ihy(t)‖C(Ω) + ‖Ihy(t)− y(t)‖C(Ω)

≤ C1h−n/2‖yh(t)− Ihy(t)‖L2(Ω) + ‖Ihy(t)− y(t)‖C(Ω)

≤ C1h−n/2(‖y(t)− yh(t)‖L2(Ω) + ‖y(t)− Ihy(t)‖L2(Ω)

)+‖Ihy(t)− y(t)‖C(Ω) ≤ C1h

−n/2(C2hn/2+ε + C3h

l+1)

+ C4hl+1−n/2

= O(hε) +O(hl+1−n/2),

where we have used the inverse inequality (C.2).Therefore, for any δ > 0 there exists h0 > 0 such that for h < h0 we have

mi − δ ≤ yh,i(t, xj) ≤Mi + δ ∀t ∈ [0, τh(y0h)), j = 1, . . . , ndof

and then

‖yh(t)‖C([0,τh(y0h))×Ω) ≤ maxi=1,...N

max|mi|, |Mi|+ δ.

This estimate and the classical theory of continuation of solutions of ordi-nary differential equations (see for example [23]) allow us to conclude that forh < h0 there exists a global solution to the discrete problem belonging to thespace C([0, T ]; Vh). Now, since ϕh is continuous we deduce from (5.3) thatdyhdt

is also continuous.

Finally, the uniqueness can be proved as for the continuous problem inTheorem 4.3.3


Chapter 6

Numerical solution of thePFR model

In Chapter 2 we have described the general model for a plug-flow reactor involv-ing both species and temperature equations. These systems are usually stiff,and we should take this fact into account when choosing a numerical method.Stiff problems are characterized by large variations of the solution in a smalltime interval, hence most of the numerical methods must take small stepsto obtain satisfactory results. This means that non-stiff methods can solvestiff problems, but they are time consuming. In this work we have selecteda backward differentiation formula which is a simple and standard choice forsolving stiff ordinary differential equations. More recent numerical discretiza-tions based on Runge-Kutta schemes can be found, for instance in [33], wherea method preserving the positivity of solution is included.

This chapter will be devoted to solve numerically the PFR model in bothtransient and steady state. Let us recall these problems:

(PFR)

∂y

∂t+ v

∂y

∂z−D∂

2y

∂z2= Aδ(θ,y),

(w′(θ) · y)(∂θ∂t

+ v∂θ

∂z

)− k∂

2θ

∂z2= −∆H(θ) · δ(θ,y) +

2h

R(θext − θ),

y(z, 0) = y0(z), θ(z, 0) = θ0(z),

− d∂y

∂z(0, t) + vy(0, t) = yin(t) and θ(0, t) given,

∂y

∂z(L, t) = 0,

∂θ

∂z(L, t) = 0,

(6.1)

101

102 CHAPTER 6. NUMERICAL SOLUTION OF THE PFR MODEL

and

(PFR@SS)

v∂y

∂z−D∂

2y

∂z2= Aδ(θ,y),

(w′(θ) · y)(v∂θ

∂z

)− k∂

2θ

∂z2= −∆H(θ) · δ(θ,y) +

2h

R(θext − θ),

− d∂y

∂z(0) + vy(0) = yin and θ(0) given,

dy

dz(L) = 0,

dθ

dz(L) = 0.

(6.2)

6.1 Time and spatial discretizations of the prob-lem

For numerical solution of the transient PFR model we propose the use of finitedifference approximations for both space (z) and time (t) variables. Let usconsider NT time discretization steps and NL spatial steps. We define timeand spatial steps by

∆t =T

NTand ∆z =

L

NL,

and time and spatial meshes by

tn = n∆t, n = 1, · · · , NT and zj = (j − 0.5)∆z, j = 0, · · · , NL + 1.

We notice that first and last points in the spatial mesh are out of the interval[0, L]. More precisely,

z0 = (−0.5)L

NL= −∆z

2

and

zNL+1 = (NL + 1− 0.5)L

NL= L+

∆z

2.

Consequently, we use a second order approximation of the boundary conditionon z = L by taking a centred approximation of the first spatial derivative atthis point.

Space discretization of the first order derivative in the convection term isthe most delicate one in the system. This is because in a convection dominatedproblem (as it is the case for PFRs), a backward difference scheme to approxi-mate this derivative is needed for the sake of stability. Otherwise, the time step

6.1. TIME AND SPATIAL DISCRETIZATIONS OF THE PROBLEM 103

must be very small. This is why we have implemented two kind of schemes:centred (order 2) and backward (of orders 1 and 2), as described below:

∂ϕ

∂z(zj , t) ≈

ϕ(zj , t)− ϕ(zj−1, t)

∆z(backward formula of order 1)

∂ϕ

∂z(zj , t) ≈

1.5ϕ(zj , t)− 2ϕ(zj−1, t) + 0.5ϕ(zj−2, t)

∆z, j = 2, . . . , NL

(backward formula of order 2)

∂ϕ

∂z(zj , t) ≈

ϕ(zj+1, t)− ϕ(zj−1, t)

2∆z, j = 1, . . . , NL − 1

(centred formula of order 2).

The second order spatial derivative is approximated by the second ordercentred difference formula:

∂2ϕ

∂z2(zj , t) ≈

ϕ(zj+1, t)− 2ϕ(zj , t) + ϕ(zj−1, t)

∆z2, j = 1, . . . , NL − 1.

The integration in time is done step by step from n = 1 to n = NT . The firststep is computed using the implicit Euler scheme (which is also the first orderBackward Differentiation Formula, BDF1) and the subsequent ones with theBDF2 (second order Backward Differentiation Formula) which was presentedabove for the spatial first derivative. Moreover, the boundary condition at thereactor outlet is replaced by the centred scheme.

In what follows we write the whole discretized problem with only the back-ward scheme of order 2 for the convection term.


1. First step, n = 1 (for n = 0 all values are known from the initial condi-tions).

y11 − y0

1

∆t+ v

y11 − y1

0

∆z−Dy1

2 − 2y11 + y1

0

∆z2= Aδ(θ1

1,y11),

y1j − y0

j

∆t+ v

1.5y1j − 2y1

j−1 + 0.5y1j−2

∆z−D

y1j+1 − 2y1

j + y1j−1

∆z2= Aδ(θ1

j ,y1j ),

j = 2, · · · , NL,

(w′(θ1

1) · y11

)(θ11 − θ0

1

∆t+ v

θ11 − θ1

0

∆z

)− k θ

12 − 2θ1

1 + θ10

∆z

=2hextR

(θext(t1)− θ1

1

)−w(θ1

1) ·Aδ(θ11,y

11),

(w′(θ1

j ) · y1j

)(θ1j − θ0

j

∆t+ v

1.5θ1j − 2θ1

j−1 + 0.5θ1j−2

∆z

)− k

θ1j+1 − 2θ1

j + θ1j−1

∆z

=2hextR

(θext(t1)− θ1

j

)−w(θ1

j ) ·Aδ(θ1j ,y

1j ),

j = 2, · · · , NL,

−dy11 − y1

0

∆z+ v

y10 + y1

1

2= vyin(t1),

θ10 + θ1

1

2= θin(t1),

y1NL+1 − y1

NL

∆z= 0,

θ1NL+1 − θ1

NL

∆z= 0.

6.1. TIME AND SPATIAL DISCRETIZATIONS OF THE PROBLEM 105

2. Step n ≥ 2 (all previous fields are known).

1.5yn1 − 2yn−11 + 0.5yn−2

1

∆t+ v

yn1 − yn0∆z

−Dyn2 − 2yn1 + yn0∆z

= Aδ(θn1 ,yn1 ),

1.5yn1 − 2yn−11 + 0.5yn−2

1

∆t+ v

1.5ynj − 2ynj−1 + 0.5ynj−2

∆z

−Dynj+1 − 2ynj + ynj−1

∆z= Aδ(θnj ,y

nj ),

j = 2, · · · , NL,

(w′(θn1 ) · yn1 )

(1.5θnj − 2θn−1

j + 0.5θn−2j

∆t+ v

θn1 − θn0∆z

)− k θ

n2 − 2θn1 + θn0

∆z

=2hextR

(θext(tn)− θn1

)−w(θn1 ) ·Aδ(θn1 ,y

n1 ),

(w′(θnj ) · ynj

)(1.5θnj − 2θn−1j + 0.5θn−2

j

∆t+ v

θnj − θnj−1

∆z

)

−kθnj+1 − 2θnj + θnj−1

∆z=

2hextR

(θext(tn)− θnj

)−w(θnj ) ·Aδ(θnj ,y

nj ),

j = 2, · · · , NL,

−dyn1 − yn0∆z

+ vyn0 + yn1

2= vyin(tn),

θn0 + θn12

= θin(tn),

ynNL+1 − ynNL∆z

= 0,

θnNL+1 − θnNL∆z

= 0.

Let us notice that a nonlinear system of equations has to be solved ateach time step. The number of equations of this system equals the number of


unknowns, which is (N + 1)× (NL + 2).

6.2 Academic tests

Test at steady state

We use a simple model to construct this academic test. This is becausewe are only checking if the error reaches the expected order for the nonlinearfunctions we propose as solutions of the reaction system. Of course, this modelinvolves equations of both species and temperature.

Then, let us consider a simple reaction system of three species S1, S2, S3involved in one chemical reaction:

S1 + S2 → S3, (6.3)

and in terms of the stoichiometric matrix the reaction is represented as

A =

−1−1

1

. (6.4)

The molecular mass of each species in kg/kmol is given by the vectorM = (42, 138, 180)

tand the specific heat for all species is constant and equal,

cvi = 2.0 103 J/kgK, i = 1, 2, 3. We also assume identical diffusion term forall species equal to d = 10−3 kg/kmol. The effective coefficient of thermal con-ductivity is k = 10−4 kg/kmol and the heat transfer coefficient between thereactor and its surroundings is null.

The law of mass action and the Arrhenius law are applied for modelling thekinetics. More precisely, the reaction term is given by

δ1 = B1 exp

(Ea1

Rθ

)y1y2, (6.5)

where B1 = 3.0 107 and Ea1 = 5.0 104.The heat of the reactions at temperature 298.15 K is 4.0 103 J/K. The

volume of the reactor is 0.0029 m3 and its length 1.5 m. The mixture velocityis constant and equal to 1.0 m/s There are no catalysts.

The exact solution of this problem is given by

y1(z) = e0.5z + cos2 z,

y2(z) = e0.5z + sin2 z,

y3(z) = ez,

θ(z) = 298 + sin2 z.

6.2. ACADEMIC TESTS 107

The boundary conditions are non-homogeneous and they have the followingexpressions:

yin1 = −0.5d+ 2v,

yin2 = −0.5d+ v,

yin3 = −d+ v,

θin = 298.

yout1 = d(0.5e0.75 − 2 cos(1.5) sin(1.5)),

yout2 = d(0.5e0.75 + 2 cos(1.5) sin(1.5)),

yout3 = de1.5,

θout = 2k sin(1.5) cos(1.5).

With the complete information we can write the model we want to solve

v∂y

∂z− d∂

2y

∂z2= Aδ(θ,y) + f in [0, L],

(n∑i=1

Micviyi

)v∂θ

∂z− k∂

2θ

∂z2= −∆H(θ) · δ(θ,y) +

2h

R(θout − θ) + g in [0, L],

− d∂y

∂z(0) + vy(0) = yin and

∂y

∂z(L) = yout,

θ(0) = θin and∂θ

∂z(L) = θout.

We adjust f and g in order to obtain the solution of the system describedabove, namely

f1(z) = v(0.5e0.5z − 2 sin z cos z)− d(0.25e0.5z + 2 sin2 z − 2 cos2 z)

+B1 exp

(Ea1

R (298 + sin2 z)

)(e0.5z + cos2 z)(e0.5z + sin2 z),

f2(z) = v(0.5e0.5z + 2 sin z cos z)− d(0.25e0.5z − 2 sin2 z + 2 cos2 z)

+B1 exp

(Ea1

R (298 + sin2 z)

)(e0.5z + cos2 z)(e0.5z + sin2 z),

f3(z) = vez − dez −B1 exp

(Ea1

R (298 + sin2 z)

)(e0.5z + cos2 z)(e0.5z + sin2 z),


g(z) =

(n∑i=1

Micviyi

)v2 sin z cos z + 2k(sin2 z − cos2 z)

+∆H(θ)B1 exp

(Ea1

R (298 + sin2 z)

)(e0.5z + cos2 z)(e0.5z + sin2 z).

We have solved the problem in the space interval [0, 1.5] for different step-sizes. The results are displayed in Table 6.1.

Nz Discrete L2 error for species Disscrete L2 error for temperature4 0.085818602 0.0419018078 0.022819928 0.01100778816 0.005975571 0.00288163932 0.001532978 7.391990E-0464 3.884191E-04 1.87E-04128 9.78E-05 4.71E-05256 2.45E-05 1.18E-05512 6.14E-06 2.96E-06

Table 6.1: Table of errors in discrete L2 norm (trapezoidal rule)

This test was done using the centred scheme. We did not observe incon-sistencies in the backward schemes. The error curves in logarithm scale arerepresented in Figure 6.1 and Figure 6.2 where it can be seen that the order ofthe expected error is 2 in both cases.

101 102 103

Nz: number of space steps

10-6

10-5

10-4

10-3

10-2

10-1

Err

or

L2 error curve-Species

error in log scale

y=C/N2

Figure 6.1: Species error for the steady state test


101 102 103

Nz: number of space steps

10-6

10-5

10-4

10-3

10-2

10-1

Err

or

L2 error curve-Temperature

error in log scale

y=C/N2

Figure 6.2: Temperature error for the steady state test

Test at transient state

Firstly, we test the time discretization scheme by solving a problem whosesolution depends only on time variable. The selected test is the same as thatused for steady state academic test, but replacing z with t. Thus, the exactsolution is

y1(z, t) = e0.5t + cos2 t,

y2(z, t) = e0.5t + sin2 t,

y3(z, t) = et,

θ(z, t) = 298 + sin2 t.

The boundary conditions have the following expressions:

yin1(t) = v(e0.5t + cos2 t),

yin2(t) = v(e0.5t + sin2 t),

yin3(t) = vet,

θin(t) = 298 + sin2 t.

yout1(t) = 0,

yout2(t) = 0,

yout3(t) = 0,

θout(t) = 0.


Summarizing, we solve the problem

∂y

∂t+ v

∂y

∂z− d∂

2y

∂z2= Aδ(θ,y) + f in [0, T ]× [0, L],

n∑i=1

Micviyi

(∂θ

∂t+ v

∂θ

∂z

)− k∂

2θ

∂z2= −∆H(θ) · δ(θ,y)

+2h

R(θout − θ) + g in [0, T ]× [0, L],

−d∂y

∂z(0) + vy(0) = yin and

∂y

∂z(L) = yout,

θ(0) = θin and∂θ

∂z(L) = θout.

In order to obtain a solution, we adjust f and g as following

f1(t) = (0.5e0.5t − 2 sin(t) cos t)

+B1 exp

(Ea1

R (298 + sin(t)2)

)(e0.5t + cos2 t)(e0.5t + sin2 t),

f2(t) = (0.5e0.tz + 2 sin t cos t)

+B1 exp

(Ea1

R (298 + sin2 t)

)(e0.5t + cos2 t)(e0.5t + sin2 t),

f3(t) = et −B1 exp

(Ea1

R (298 + sin2 t)

)(e0.5t + cos2 t)(e0.5t + sin2 t),

g(t) =

(n∑i=1

Micviyi

)2 sin t cos t

+∆H(θ)B1 exp

(Ea1

R (298 + sin2 t)

)(e0.5t + cos2 t)(e0.5t + sin2 t).

We solve the problem in the time interval [0, 1] and in the space interval[0, 1.5] for different step-sizes. We recall that the size of the spatial mesh doesnot modify the error because the exact solution is independent of the variablez. The results are displayed in Table 6.2.


Nt L2 error for species L2 error for temperature4 0.027652771 0.0026683118 0.005040992 0.00075013916 0.000903504 0.00015457232 0.000160727 2.90715E-0564 2.8494E-05 5.29229E-06128 5.04384E-06 9.49283E-07256 8.92214E-07 1.69052E-07512 1.57773E-07 2.9993E-08

Table 6.2: Table of errors in L2 norm

In this case the error order we obtain is greater than 2 (2.38 approximately),as we can observe in the curves of the error in logarithm scale.

101 102 103

Nt: number of time steps

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Err

or

L2 error curve-Species

error in log scale

y=C/N2

Figure 6.3: Species error in the transient state test


101 102 103

Nt: number of time steps

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Err

or

L2 error curve-Temperature

error in log scale

y=C/N2

Figure 6.4: Temperature error in the transient state test

Now, we build a test whose solution depends on both variables (z, t). Thetest parameters are the same as in the previous one.

The exact solution is

y1(z, t) = e−t(e0.5z + cos2 z),

y2(z, t) = e−t(e0.5z + sin2 z),

y3(z, t) = etez,

θ(z, t) = 298 + e−t sin2 z.

The boundary conditions have the following expressions:

yin1= e−t(−0.5d+ 2v),

yin2= e−t(−0.5d+ v),

yin3= et(−d+ v),

θin = 298.

yout1 = e−t((0.5e0.75 − 2 cos(1.5) sin(1.5)),

yout2 = e−t((0.5e0.75 + 2 cos(1.5) sin(1.5)),

yout3 = ete1.5,

θout = 2e−t sin(1.5) cos(1.5).

In this case the auxiliary functions f and g are given by


f1(z, t) = −e−t(e0.5z + cos2 z) + ve−t(0.5e0.5z − 2 sin z cos z)

−de−t(0.25e0.5z + 2 sin2 z − 2 cos2 z)

+B1 exp

(Ea1

R (298 + e−t sin2 z)

)e−2t(e0.5z + cos2 z)(e0.5z + sin2 z),

f2(z, t) = −e−t(e0.5z + sin2 z) + ve−t(0.5e0.5z + 2 sin z cos z)

−de−t(0.25e0.5z − 2 sin2 z + 2 cos2 z)

+B1 exp

(Ea1

R (298 + e−t sin2 z)

)e−2t(e0.5z + cos2 z)(e0.5z + sin2 z),

f3(z, t) = etez + vetez − detez

−B1 exp

(Ea1

R (298 + e−t sin2 z)

)e−2t(e0.5z + cos2 z)(e0.5z + sin2 z),

g(z, t) =

(n∑i=1

Micviyi

)(−e−t sin2(z)

)+ 2k(sin2 z − cos2 z)

+∆H(θ)B1 exp

(Ea1

R (298 + sin2 z)

)(e0.5z + cos2 z)(e0.5z + sin2 z).

We solve the problem in the time interval [0, 1] and the space interval [0, 1.5]for different step-sizes. The results are displayed in Table 6.3, for fixed spatialstep corresponding to Nz = 64, and in Table 6.4, for fixed time step corre-sponding to Nt = 64. Notice that the error decreases in both cases and theorder is at least two in time and in space.

Nt L2 error for species L2 error for temperature4 0.0522535116589 0.0012043348 0.0090397291651 0.00031402416 0.0015456067776 5.78928E-0532 0.0002422882547 8.10041E-0664 0.0000307108686 9.6545066E-06



Nz L2 error for species L2 error for temperature4 0.0097958014917 0.00291640068388 0.0090397291651 0.00031402416 0.00059491811150000 0.0001866641684032 0.000123880413800 0.00004525060080064 0.0000307108686 0.0000096545066


Chapter 7

Numerical solution of theFBR model

In Chapter 3, we have described the general two-phase model for a heteroge-neous fixed-bed reactor (FBR), including both species and temperature equa-tions. These systems are more complex than PFR because one has to takeinto consideration the coupling between the fluid (macroscopic) and the solid(microscopic) phase models. Thus, the implementation of the FEM discretiza-tion and the coupling between phases is not easy to handle. For this rea-son, the implementation was done using the FeniCS library of finite elementshttps://fenicsproject.org/ through a Python program. The FeniCS li-brary allows us to write weak formulations of partial differential equations inan easy and direct way.

7.1 Weak formulation

In this section we build a weak formulation of the FBR model that will be usedto define the numerical solution by means of finite element methods. Cylindricalcoordinates are considered for the fluid phase and spherical coordinates for thesolid phase. The main technical difficulty is the information transfer betweenthe variables of the two phases, as they live in domains with different dimension(two for the fluid and three for the solid).

7.1.1 Macroscale: fluid bulk

Firstly, we consider the fluid bulk. Let us make the scalar product of equation(3.1) by a test function vector u(r, z) defined in Ω := (0, R)×(0, L). Integratingin the whole reactor domain and using the cylindrical symmetry we get,

115

https://fenicsproject.org/

116 CHAPTER 7. NUMERICAL SOLUTION OF THE FBR MODEL

2π

∫Ω

∂

∂t(εfyf ) · ur dr dz + 2π

∫Ω

∂

∂z(εfvyf ) · ur dr dz

− 2π

∫Ω

1

r

∂

∂r

(Dfr r

∂

∂r(εfyf )

)· ur dr dz − 2π

∫Ω

∂

∂z

(Dfz r

∂

∂z(εfyf )

)· u dr dz

= 2π

∫Ω

Afδf (θf ,yf ) · ur dr dz + 2π

∫Ω

g · ur dr dz.

Now we integrate by parts in the second, third and fourth terms on theleft-hand side and we obtain,

2π

∫Ω

∂

∂t(εfyf ) · ur dr dz − 2π

∫Ω

εfvyf · ∂u

∂zr dr dz

− 2π

∫ R

0

εf (r, 0)vyf (r, 0) · u(r, 0)r dr + 2π

∫ R

0

εf (r, L)vyf (r, L) · u(r, L)r dr

+ 2π

∫Ω

Dfr

∂

∂r(εfyf ) · ∂u

∂rr dr dz + 2π

∫Ω

Dfz

∂

∂z(εfyf ) · ∂u

∂zr dr dz

− 2π

∫ L

0

Dfr

∂

∂r(εfyf )(R, z) · u(R, z)Rdz + 2π

∫ R

0

Dfz

∂

∂z(εfyf )(r, 0) · u(r, 0)r dr

− 2π

∫ R

0

Dfz

∂

∂z(εfyf )(r, L) · u(r, L)r dr

= 2π

∫Ω


∫Ω

g · ur dr dz.

By using the boundary conditions, this equality becomes,

2π

∫Ω

∂

∂t(εfyf ) · ur dr dz − 2π

∫Ω

εfvyf · ∂u

∂zr dr dz

+ 2π

∫ R

0

εf (r, L)vyf (r, L) · u(r, L)r dr

+ 2π

∫Ω

Dfr

∂

∂r(εfyf ) · ∂u

∂rr dr dz + 2π

∫Ω

Dfz

∂

∂z(εfyf ) · ∂u

∂zr dr dz

= 2π

∫Ω


∫Ω

g · ur dr dz + 2π

∫ R

0

vεfyfin(t) · ur dr.

(7.1)

Now let us consider the energy equation. We multiply (3.8) by a scalartest function u(r, z) defined in Ω. Integrating in the whole reactor domain and

7.1. WEAK FORMULATION 117

using the cylindrical symmetry we get,

2π

∫Ω

εfρf cfv (θf )∂θf

∂tur dr dz + 2π

∫Ω

εfρf cfv (θf )v∂θf

∂zur dr dz

− 2π

∫Ω

1

r

∂

∂r

(kfr r

∂θf

∂r

)ur dr dz − 2π

∫Ω

∂

∂z

(kfz∂θf

∂z

)ur dr dz

= 2π

∫Ω

fur dr dz − 2π

∫Ω

w(θf ) ·(Afδf (θf ,yf ) + g

)ur dr dz.

Integrating by parts the third and fourth terms on the left-hand side, we deduce

2π

∫Ω


∂tur dr dz + 2π

∫Ω


∂zur dr dz

+ 2π

∫Ω

kfr∂θf

∂r

∂u

∂rr dr dz + 2π

∫Ω

kfz∂θf

∂z

∂u

∂zr dr dz + 2π

∫ R

0

kfz∂θf

∂z(r, 0)u(r, 0)r dr

− 2π

∫ R

0

kfz∂θf

∂z(r, L)u(r, L)r dr − 2π

∫ L

0

kfr∂θf

∂r(R, z)u(R, z)Rdz

= 2π

∫Ω

fur dr dz − 2π

∫Ω


)ur dr dz.

Since we have a Dirichlet boundary condition at the reactor inlet (temper-ature is given at z = 0), we take the test function u null there. Using theboundary conditions at the reactor outlet and on the reactor wall the aboveequality becomes

2π

∫Ω


∂tur dr dz + 2π

∫Ω


∂zur dr dz

+ 2π

∫Ω

kfr∂θf

∂r

∂u

∂rr dr dz + 2π

∫Ω

kfz∂θf

∂z

∂u

∂zr dr dz

+ 2π

∫ L

0

hextθf (R, z, t)u(R, z)Rdz = 2π

∫Ω

fur dr dz

− 2π

∫Ω


)ur dr dz + 2π

∫ L

0

hextθext(z, t)u(R, z)Rdz.

7.1.2 Micro-scale: spherical solid particles

Let us multiply equation (3.19) by a test function vector u(rs, r, z), with(rs, r, z) ∈ (0, Rs) × Ω. Integrating in the whole sphere and in the wholecylinder, and using spherical symmetry and cylindrical symmetry, respectively,


we get,

8π2

∫(0,Rs)×Ω

∂εsys

∂t· ur2

sr drs dr dz

− 8π2

∫(0,Rs)×Ω

∂

∂rs

(Dsr2

s

∂εsys

∂rs

)· ur drs dr dz

= 8π2

∫(0,Rs)×Ω

Asδs(θs,ys) · ur2sr drs dr dz.

We integrate by parts the the second term of the left-hand side and we deducethat

8π2

∫(0,Rs)×Ω

∂εsys

∂t· ur2

sr drs dr dz + 8π2

∫(0,Rs)×Ω

Ds ∂εsys

∂rs· ∂u

∂rsr2sr drs dr dz−

8π2

∫Ω

Ds ∂εsys

∂rs· uR2

sr dr dz = 8π2

∫(0,Rs)×Ω

Asδs(θs,ys) · ur2sr drs dr dz

and using the boundary conditions we get

8π2

∫(0,Rs)×Ω

∂εsys

∂t· ur2

s drsr dr dz

+ 8π2

∫(0,Rs)×Ω

Ds ∂εsys

∂rs· ∂u

∂rsr2sr drs dr dz

+ 8π2

∫Ω

ηfs(r, z, t)ys(Rs, r, z, t) · u(Rs, r, z)R2

sr dr dz

= 8π2

∫(0,Rs)×Ω

Asδs(θs,ys) · ur2sr drs dr dz

+ 8π2

∫Ω

ηfs(r, z, t)yf (r, z, t) · u(Rs, r, z)R2

sr dr dz.

(7.2)

Finally, if we make a change of variable defined by xs =rsRs

, the weak formu-

lation can be written as follows:∫(0,1)×Ω

∂εsys

∂t(xs, r, z, t) · u(xs, r, z)R3

sx2s dxsr dr dz

+

∫(0,1)×Ω

Ds ∂εsys

∂xs(xs, r, z, t) · ∂u

∂xs(xs, r, z)Rsx

2sr dxs dr dz

+

∫Ω

ηfs(r, z, t)ys(1, r, z, t) · u(1, r, z)R2

sr dr dz

=

∫(0,1)×Ω

Asδs(θs(xs, r, z, t),ys(xs, r, z, t)) · u(xs, r, z)R3sx

2sr dxs dr dz

+

∫Ω

ηfs(r, z, t)yf (r, z, t) · u(1, r, z)R2

sr dr dz.

(7.3)


Now let us consider the energy equation. We multiply equation (3.22) by atest function u(rs, r, z), with (rs, r, z) ∈ (0, Rs) × Ω. Integrating in the wholesphere and in the whole cylinder, and using spherical symmetry and cylindricalsymmetry, respectively, we get

8π2

∫(0,Rs)×Ω


∂tur2sr drs dr dz

− 8π2

∫(0,Rs)×Ω

1

r2s

∂

∂rs

(ksr2

s

∂θs

∂rs

)ur2sr drs dr dz

= −8π2

∫(0,Rs)×Ω

w(θs) ·Asδs(θs,ys)ur2sr drs dr dz.

Integrating by parts in the second term of the left-hand side we deduce that

8π2

∫(0,Rs)×Ω


∂tur2sr drs dr dz

+ 8π2

∫(0,Rs)×Ω

ks∂θs

∂rs

∂u

∂rsr2sr drs dr dz

− 8π2

∫Ω

ks∂θs

∂rs(Rs, r, z, t)u(Rs, r, z)R

2sr dr dz

= −8π2

∫(0,Rs)×Ω

w(θs) ·Asδs(θs,ys)ur2sr drs dr dz,

and using the boundary conditions, we obtain the weak formulation for theenergy equation in the micro-scale model:

8π2

∫(0,Rs)×Ω


∂tur2sr drs dr dz

+ 8π2

∫(0,Rs)×Ω

ksr2s

∂θs

∂rs

∂u

∂rsr drs dr dz

+ 8π2

∫Ω

hfs(r, z, t)θs(Rs, r, z, t)u(Rs, r, z)R

2sr dr dz

= −8π2

∫(0,Rs)×Ω

w(θs) ·Asδs(θs,ys)ur2sr drs dr dz

+ 8π2

∫Ω

hfs(r, z, t)θf (r, z, t)u(Rs, r, z)R

2sr dr dz.


Finally, we introduce the change of variable xs =rsRs

:

∫(0,1)×Ω

εsρscsv(θs(xs, r, z, t))

∂θs

∂t(xs, r, z, t)u(xs, r, z)x2

sR3sr dxs dr dz

+

∫(0,1)×Ω

ksRs∂θs

∂xs(xs, r, z, t)

∂u

∂xs(xs, r, z)rx2

s dxs dr dz

+

∫Ω

hfs(r, z, t)θs(1, r, z, t)u(1, r, z)R2

sr dr dz

= −∫

(0,1)×Ω

w(θs(xs, r, z, t)) ·Asδs(θs(xs, r, z, t),ys(xs, r, z, t))u(xs, r, z)x2sR

3sr dxs dr dz

+

∫Ω

hfs(r, z, t)θf (r, z, t)u(1, r, z)R2

sr dr dz.

7.1.3 Mass conservation at steady-state

In this paragraph we obtain a mass conservation equation for the whole multi-scale model at steady-state. Hence, all fields are time independent, thus theaccumulation terms are null in all equations.

Let us start by using the micro-scale model and take the test function

u(r, z) =a(r, z)

R2s(r, z)

M

in (7.2), where a is given by (3.31). As the partial derivative of this functionwith respect to rs is null, we obtain

8π2

∫Ω

a

R2s

ηfs(r, z)ys(Rs, r, z) ·MR2

sr dr dz

= 8π2

∫(0,Rs)×Ω

a

R2s

Asδs(θs,ys) ·Mr2sr drs dr dz

+ 8π2

∫Ω

a

R2s

ηfs(r, z)yf (r, z) ·MR2

sr dr dz

= 8π2

∫(0,Rs)×Ω

a

R2s

δs(θs,ys) · (As)tMr2sr drs dr dz

+ 8π2

∫Ω

a

R2s


sr dr dz

= 8π2

∫Ω

a

R2s


sr dr dz,


because (As)tM = 0 (see (1.2)). Therefore,

8π2

∫Ω

a(r, z)ηfs(r, z)ys(Rs(r, z), r, z) ·Mr dr dz

= 8π2

∫Ω

a(r, z)ηfs(r, z)yf (r, z) ·Mr dr dz.

(7.4)

Now, let us consider the bulk fluid. We take the test function u = M in theweak formulation (7.1) and we get

2π

∫ R

0

vεf (r, L)yf (r, L) ·Mr dr = 2π

∫Ω

Afδf (θf ,yf ) ·Mr dr dz

+ 2π

∫Ω

g ·Mr dr dz + 2π

∫ R

0

vεfyfin ·Mr dr (7.5)

= 2π

∫Ω

δf (θf ,yf ) · (Af )tMr dr dz + 2π

∫Ω

g ·Mr dr dz

+ 2π

∫ R

0

vεf (r, 0)yfin ·Mr dr = 2π

∫Ω

g ·Mr dr dz + 2π

∫ R

0

vεf (r, 0)yfin ·Mr dr,

again, because (Af )tM = 0 (see (1.2)).

In the following we prove that

∫Ω

g ·Mr dr dz = 0. Indeed, from the

definition of g (see (3.33)) and by using (7.4) we have,∫Ω

g ·Mr dr dz =

∫Ω

a(r, z)ηfs(r, z)(ys(Rs(r, z), r, z

)− yf (r, z)

)·Mr dr dz = 0.

Then (7.5) yields

2π

∫ R

0

vεf (r, L)yf (r, L) ·Mr dr = 2π

∫ R

0

vεf (r, 0)yfin ·Mr dr.

The left-hand side is the total mass flow rate (kg/s) leaving the reactor, whilethe right-hand side is the total mass flow rate (kg/s) entering the reactor.Therefore, we have proved that the model conserves the total mass.


7.2 An academic test in steady state

In order to validate the finite element implementation we consider an academicexample for a steady FBR with resistance. We recall that using the notationfrom Chapter 3 the full model is the following:

∂

∂z(εfyfv)− 1

r

∂

∂r

(Dfr r

∂

∂r(εfyf )

)− ∂

∂z

(Dfz∂

∂z(εfyf )

)= Afδf (θf ,yf ) + aηfs

(ys(Rs)− yf

),

εfρf cfv(θf )v

∂θf

∂z− 1

r

∂

∂r

(kfr r

∂θf

∂r

)− ∂

∂z

(kfz∂θf

∂z

)= ahfs

(θs(Rs)− θf

)− w(θf ) ·Afδf (θf ,yf ),

− 1

r2s

∂

∂rs

(Dsr2s

∂εsys

∂rs

)= Asδs(θs,ys),

− 1

r2s

∂

∂rs

(ksr2s

∂θs

∂rs

)= −w(θs) ·Asδs(θs,ys),

−Dfz∂

∂z(εfyf )(r, 0) + vεfyf (r, 0) = vεfyfin,

θf (r, 0) = θfin,

∂εfyf

∂z(r, L) = 0,

∂θf

∂z(r, L) = 0,

∂εfyf

∂r(R, z) = 0,

kfr∂θf

∂r(R, z) = hext

(θext − θf (R, z)

),

∂εfyf

∂r(0, z) = 0,

∂θf

∂r(0, z) = 0,

∂(εsys)

∂rs(0, r, z) = 0,

∂θs

∂rs(0, r, z) = 0,

Ds ∂εsys

∂rs(Rs(r, z), r, z) = ηfs(r, z)

(yf (r, z)− ys(Rs(r, z), r, z)

),

ks∂θs

∂rs(Rs(r, z), r, z) = hfs(r, z)

(θf (r, z)− θs(Rs(r, z), r, z)

).

7.2. AN ACADEMIC TEST IN STEADY STATE 123

We define a stationary problem with known analytical solution. The reac-tion system is described by 4 species involved in 2 reactions in the fluid phasewhose kinetic is governed by law of mass action:

Sf1 + Sf2 → Sf3 −→ δf1 (θf ,yf ) = B1e(−Ea1

Rθf)yf1 y

f2

Sf1 + Sf3 → Sf4 δf2 (θf ,yf ) = B2e(−Ea2

Rθf)yf1 y

f3 ,

with coefficients B1 = 1.0 1010, Ea1 = 7.5 104, B2 = 1.0 108 and Ea2 = 9.0 104.

We assume that the reaction system is the same at the microscale level,that is, inside the solid particles.

Now we define the solution:

ys1(xs, r, z, t) = cos2 z + r2 + x2s,

ys2(xs, r, z, t) = sin2 z + r3 + x3s,

ys3(xs, r, z, t) = sin2 z + cos2 r + x4s,

ys4(xs, r, z, t) = cos2 z + sin2 r + x5s,

θs(xs, r, z, t) = 2z2 + r2 + xs + 280,

(7.6)

yf1 (r, z, t) = 2εsDs

1

ηfsRs+ cos2 z + r2 + 1,


2

ηfsRs+ sin2 z + r3 + 1,


3

ηfsRs+ sin2 z + cos2 r + 1,


4

ηfsRs+ cos2 z + sin2 r + 1,

θf (r, z, t) =εsks

hfsRs+ 2z2 + r2 + 281.

(7.7)

These functions are a solution of the above system of equations by addingan auxiliary source function at each equation, namely,


ffauxi =∂

∂z(εf yfi v)− 1

r

∂

∂r

(Dfr r

∂

∂r(εf yfi )

)− ∂

∂z

(Dfz

∂

∂z(εf yfi )

)−

L∑j=1

Afij δf

j − aηfs(ysi (1)− yfi

),

gfaux = εfρf cfv (θf )v∂θf

∂z− 1

r

∂

∂r

(kfr r

∂θf

∂r

)− ∂

∂z

(kfz∂θf

∂z

)+ w(θf ) ·Af δ

f− ahfs

(θs(1)− θf

),

fsauxi = − 1

r2s

∂

∂rs

(Dsr2

s

∂εsysi∂rs

)−

L∑j=1

Asij δs

j ,

gsaux = − 1

r2s

∂

∂rs

(ksr2

s

∂θs

∂rs

)+ w(θs) ·Asδ

s, i = 1, ..., N,

where δs

= δf (θf , yf ) and δs

= δf (θs, ys).

We also need to modify the homogeneous Neumann boundary conditionsat the reactor inlet and replace them by Robin boundary conditions.

The coefficients appearing in the model are summarized in the followingtables:

Species coefficients S1 S2 S3 S4

Molecular weigths (M) 58 138 196 254Specific heat (cfv ) 2000 2000 2000 2000Radial diffusions in macro scale (Df

r ) 1.0e-1 1.0e-1 1.0e-1 1.0e-1Axial diffusions in macro scale (Df

z ) 1.0e+1 1.0e+1 1.0e+1 1.0e+1Mass diffusion in micro scale (Ds) 1.0e-10 1.0e-10 1.0e-10 1.0e-10

Table 7.1: Coefficients of chemical species

Reaction measurements T a formation Reac 1 Reac 2

Heat of form. in macro scale (∆Hf (θ∗)) 298.15 1.0e+08 8.0e+07

Heat of form. in micro scale (∆Hs(θ∗)) 298.15 1.0e+08 8.0e+07

Table 7.2: Heats of formation in macro and micro scales


Effective, transfer and porosity coefficientsEnergy coefficient in macro scale: radial and axial (kfr and kfz ) 1.0e-07Energy coefficient in micro scale (ks) 1.0e-07Transfer fluid-solid coefficient for species (ηfs) 3.38e-7Transfer fluid-solid coefficient for temperature (hfs) 1.0e-5Overall heat transfer coefficient (hext) 30.0Bulk porosity (εf ) 0.45Solid porosity (εs) 0.4

Table 7.3: Transference, effectiveness and porosity coefficients

In this example the radius of the reactor is 0.01 m, its length is 1.0 m andthe expression of the particles radius is Rs(r, z, t) = 0.004 m.

The academic test we present has been computed using Batea which is aninterface over FEniCS that resolves the full model with unknowns that live indifferent domains. Numerical versus analytical solution can be observed in thefollowing graphs for the most refined mesh:

a) Analytical solution of yf4 b) Numerical solution of yf4

1.5

1.65

1.8

1.95

2.1

1.440e+00

2.148e+00yf_4

1.5

1.65

1.8

1.95

2.1

1.439e+00

2.147e+00yf_4

Figure 7.1: Analytical vs Numerical concentrations in the macro-scale


a) Analytical solution of ys2 b) Numerical solution of ys2

0.35

0.7

1.05

1.4

0.000e+00

1.708e+00ys_2

0.35

0.7

1.05

1.4

-1.250e-07

1.708e+00ys_2

Figure 7.2: Analytical vs Numerical concentrations

Notice that the profile of numerical and analytical solutions is the same.We observe a little difference in lower values of solution near zero.

To perform the error and order of accuracy analysis we employ three uni-form meshes (with number of steps nr, nz and nxs in [0, R], [0, L] and [0, 1]respectively), both for micro and macro scales, described in Table 7.4 and Table7.5.

Mesh nr nz Elements-triangles

Mf1 15 5 112

Mf2 30 10 2412

Mf3 60 20 10412

Table 7.4: Macro scale mesh features

Mesh nr nz nxs Elements-tetrahedraMs

1 15 5 5 1344Ms

2 30 10 10 14094Ms

3 60 20 20 127794

Table 7.5: Micro scale mesh features

Finally, we present the error table for the different meshes when using con-tinuous piecewise linear finite elements.


Error in L2(Ω) norm Order of the method

Mf1 Mf

2 Mf3 OMf

1 /Mf2

OMf2 /M

f3

yf 5.31040e-03 1.04863E-03 2.37881e-04 2.34030924 2.14020096

θf 2.63086e-03 4.99771e-04 9.34071e-05 2.39619717 2.41966297Ms

1 Ms2 Ms

3 OMs1/M

s2

OMs2/M

s3

ys 5.77722e-03 1.16707e-03 2.62987e-04 2.30748936 2.14982345θs 2.28600e-03 4.55834e-04 1.12632e-04 2.32624447 2.01688713

Table 7.6: Observed errors and convergence orders

The order has been computed as

OMi/Mj=

log(eMi/eMj

)

log(hMi/hMj )

and

eMi=

(∫Ω

||yf − yf ||2dx) 1

2

represents the error in the L2(Ω) norm in Mfi and Ms

i meshes for the numberof elements specified in Table 7.4 and Table 7.5, respectively.


Conclusions

Through this Part II we have proved a global existence theorem for convection-diffusion-reaction systems. The property of boundedness of total mass neededfor the global existence cannot be repeated for the energy equation. Then,we have proved the theorem only for the species system. We have followedthe variational approach combined with semigroup theory. The proof of thistheorem is based on techniques presented in details in [55]. Thus, we exploreproperties (P) and (M) that were verified because of the form of our particularreaction term (the law of mass action), and because the variables in our problemrepresent chemical species and so the positivity of their concentrations is anatural property.

We have proposed a space semidiscretization for which an existence theoremis proved and error estimates are also given.

Finally, we obtained the numerical solution of the models that interest usfrom the practical point of view: PFR and FBR models. For the first one wehave used a finite difference scheme, while for the FBR model a finite elementmethod was proposed. The implementation was validated using an academiccase of FBR in steady state with resistance.

129

130 Conclusions

Part III

Identification in reactionsystems

131

Introduction

Reaction systems are widely used for controlling, monitoring and optimizingindustrial processes. Their study makes extensive employment of mathemati-cal modelling in terms of differential equations expressing the conservation ofmass and energy in order to describe concentrations, volume or temperature.Building these models needs the identification of the reactions taking place andtheir corresponding kinetics. One of the most challenging task is the identifica-tion of the kinetic laws: the identification of the best kinetic model from a listof proposed functional forms and also finding the optimal values of their cor-responding parameters. The main difficulties appear in the a priori statementof the shape of the kinetics and in the amount of degrees of freedom in theoptimization problem. The first one requires the help and the experience of anexpert in order to define the general expression of the functions with the pa-rameters to be identified. The second one is related to overfitting. This can beavoided using adequate optimization techniques and including as parameterssusceptible to optimize those that the expert considers necessary.

In this part we present a methodology for solving the inverse problem de-scribed above, also called model identification problem. We are interested inidentifying kinetic models and their corresponding parameters, using a set ofexperimental data and the reactions taking place. The identification can bedone in one step via an integral approach or sequential via an incrementalapproach [12]. This method decomposes the initial identification problem insub-problems in which each reaction can be determined individually [5, 18]. Inthe following sections we describe a methodology introduced in [9] that consistsin the combination of two methods: incremental and integral. This approach isillustrated with examples of stirred tank reactors described in details in Chap-ter 1.

There are situations when not all the species are measurable, or speciesinformation is missing in some time instants. In these cases, the methodologyfrom [9] for solving inverse identification problems sometimes does not producegood results. Thus, the identification problem to solve is replaced by the prob-lem of inferring parameters in chemical reactions networks where the available

133

134 Introduction

information, either in transient or steady state, has missing concentration val-ues. In some cases, the available data is enough to recover the parametersin the kinetics and thus “the system is identifiable” [17]. In other cases, themethodology provides a range of the missing concentrations of species usingextreme (highest and lowest) concentrations under incomplete data measure-ments. The model parameters associated with such extreme concentrations areobtained [13].

Another challenging problem in this field is the so called model selectionthat consists in determining the experimental initial values in the ODEs systemwhich allow us to discriminate among several models. A way of model selectionin chemical reaction network is done using global optimization method, asdescribed in detail in [14].

Identification problem regards different approaches, apart of kinetics iden-tification. See for example the work of Burnham and Willis [21] in which theyidentify chemical reaction networks assuming no a priori information aboutreaction stoichiometries or species structures, through the analysis of processdata obtained in a laboratory environment. Another approach is described in[64]. In this article the stoichiometry and kinetic model are selected using twoconsecutive optimization steps using integer linear programming. The first oneconsists onobtaining a list of all feasible stoichiometric relations is developedand the second one uses these relations to construct all plausible combinationsof the stoichiometric equations which are used to instantiate kinetic modelstructures.

Chapter 8

The identification problemin reaction systems

In order to solve our identification problem, we apply two approaches in cas-cade: incremental and integral methods [9]. The use of these two methods incascade can be replicated for some of the most typical reactors, such as stirredtank reactors (batch, semi batch or continuous) and plug flow reactors, bothextensively used in literature and industry. Such identification processes areusually studied in systems where the phenomena of interest can be observedin isolation, without other physical phenomena interference. It is the case ofreaction kinetics in liquid phase, where a stirred batch or semi batch reactor isused in the majority of cases [17]. Of course, an important aspect to be takeninto account is the set of measurements obtained in laboratory that will beincluded in the parameters adjustment of these kinetics.

In this chapter, we focus on the identification of kinetic models on stirredtank reactors, using a set of experimental data and the reactions taking place.A catalogue of kinetic models containing the parameters to be identified willbe provided too.

It is important to mention that the incremental method described in Section8.4.1 can be applied only to STR and PFR reactors. However, the integralmethod can be used even for FBR model.

8.1 Measurements and reactions scheme

Experimental data is required in the optimization process for the adjustmentof the parameters characterizing the kinetic model. These measurements arespecies concentration, temperature, inlet/outlet flow rates, among others. Inaddition, physico-chemical parameters of the species such as molecular weight

135

136 CHAPTER 8. THE IDENTIFICATION PROBLEM

(kg/kmol), specific heat (J/kgK) and reaction heat (J/kmol) at formationtemperature (usually 298.15 K) are also needed.

In the case of the batch reactor we consider in the following, measurementsare related to the concentration of chemical species, volume of the mixture andalso mixture temperature at some time instants. All these data are collectedin several experiments under different conditions.

8.2 Kinetic models

An important step in the identification process is a good definition of a kineticcatalogue. The specification must take into account the chemical knowledgeand the experience of an expert for defining reaction rates susceptible to beselected. It is important to consider the most relevant parameters to avoidoverfitting.

In practice it is possible to write “ad hoc” kinetics to define any functionalform in the identification, but the integral method works better with generalexpressions known a priori because it is very useful for computing the gradientof the functional cost by means of adjoint method. This is explained in detailsin Section 8.4.2.

The general expression of the reaction rates we use is given by:

δr(θ,y, z) = Brexp

(−EarRθ

) Mr∏j=1

P rj∑m=1

Grj,m

N∏n=1

yβrj,m,nn

Nc∏n=1

zβrj,m,n+Nn + brj

αrj

with 1 ≤ r ≤ L.The first part corresponds to the Arrhenius law, described in Section 1.1.3.

Two parameters, the pre-exponential (or frequency) factor (Br) and the ac-tivation energy (Era) must be adjusted. Notice that if Br tends to zero, thecorresponding reaction term can be neglected. The second part is formed bypowers of combinations of powers of concentrations. In this case, divisions byzero may appear because negative exponents are allowed.

During computational tests we have used the following common bounds forthe above mentioned parameters:

• Br ∈ [0, 1014],

• Era ∈ [0, 2.0× 105],

• Grj,m ∈ [0, 1],

• βrj,m,n ∈ [0, 2],

8.3. THE GENERAL MODEL 137

• brj ∈ [0, 100]

• αrj ∈ [−2, 2].

8.3 The general model

As we explained at the beginning of this chapter, we focus on an important fam-ily of chemical reactors: the so-called stirred tank reactors (STR). We assumethat the mixture inside these reactors is homogeneous because of stirring, thusthe physico-chemical magnitudes do not depend on position. Hence, they aremodelled as (usually stiff) coupled, non-linear ordinary differential equations.

We consider a model involving mass and heat balance equations. In addi-tion, we have an equation for volume variation, but this equation is decoupledfrom the rest. In the experimental environment, additional variables appearin the model. They are the catalysts, which help the reactions to occur or tomake the process faster, but in our model they are not considered. The modelis written in general form as

dy

dt= f(θ,y, z) in [0, T ], mass balance euqations

dθ

dt= h(θ,y, z), heat balance equation

dV

dt= f2 − f3, volume equation

y(0) = y0, θ(0) = θ0 and V (0) = V0,

(8.1)

with the source terms

f = Aδ(θ,y, z) +1

V(Ff1 − f2y),

h = −

∆H(θ) · δ(θ,y, z)− g

V(θout − θ)−w′(θ) ·

(F

P∑p=1

f1p(θsp − θ)ep

)w′(θ) · y

,

with

∆H(θ) = Atw(θ), wi(θ) =Miei and ei(θ) = e∗i +

∫ θ

θ∗ci(s)ds for the

i− th species

and

y represents the vector of species concentrations,

θ represents the temperature of mixture,


z represents the vector of catalysts,

V represents the volume of the mixture,

A is the stoichiometric matrix,

δ represents the vector of reaction velocities,

F represents the inlet composition,

f1 is the vector of inlet flow rates,

f2 is the sum of components in f1,

f3 is the outlet flow rate,

∆H is the vector of heat of reactions,

g is a heat transfer coefficient,

θout is the outside temperature,

θsp is the temperature of the p-th stream, where P is the number ofstreams,

Mi is the molecular mass of the i-th species,

ci is the specific heat of the i-th species,

ei is the internal energy of the i-th specie,

e∗i is the internal energy of formation of the i-th species at temperatureθ∗.

In general, we have continuous inlet and outlet streams. In this case, thereactor is called continuous STR. If we have only inlet streams the reactor iscalled semi-batch STR, and if no inlet or outlet streams are considered, thereactor is called batch STR.

8.4 Model selection and parameter identifica-tion

For an optimal identification process, a suitable model is desired includinginformation of stoichiometry matrix and a “good” expression for the rate ofreactions. The model can be solved by considering several techniques such asdifferential, integral or incremental methods [12], employing experimental data.

8.4. MODEL SELECTION AND PARAMETER IDENTIFICATION 139

Of course, all they can be used independently or we can combine some of themto obtain more accurate solutions.

Differential method uses cubic spline functions interpolating the data andtrying to minimize the residual of the differential equations system taking theirderivatives at time measurements. The error in these derivatives may affectthe accuracy of the results.

In this chapter we focus on incremental and integral methods. The incre-mental method works with the concept of “extent”, which provides an analyticsolution of an equivalent decoupled system. The second one needs to solvenumerically the initial ODEs model. Then, the unknown parameters are de-termined comparing experimental data with model predictions. In some cases,if the solution obtained using the incremental method is good enough we canconclude the identification process, but this is not always true and thus theintegral method is needed to improve the solution using the initial solutionprovided by the incremental method.

8.4.1 Initial approximation: The incremental method.

The incremental approach is characterized by the fact that each rate processis modeled individually, independently of the other rate processes, thus theidentification problem is decomposed into a set of subproblems, one for eachkinetic. The incremental technique is firstly introduced, although of differentialtype, as the reactions and inlet-outlet flows are obtained by differentiation ofmeasured concentrations, for Batch STR reactors in [5] and for CSTR reactorsin [18] (only for mole balance equation in both articles). These references led toa relatively recent concept, called extent. Its definition appears in [2] where alinear transformation that computes the extents of reaction from the numbersof moles in homogeneous reaction systems with inlet and outlet streams isproposed. It is extended in [10] for gas-liquid reaction systems. After that, ithas been studied for the STR model with mole and heat balance equations in[12] and for PFR model in [59].

The main features of this method are the decoupling of the reaction equa-tions using algebraic procedures and obtaining direct solution of the trans-formed equations. Thus, the kinetic models and their parameters can be iden-tified in parallel for all reactions. The parameters are obtained via local opti-mization techniques.

In this chapter we introduce an alternative method where the heat balanceequation is treated independently. Volume equation can be solved indepen-dently, but the ODE system (8.2) remains coupled. That is why we workin two stages: the concentrations system is rewritten as a decoupled extentssystem and the temperature equation is treated separately.


Then, the following systemdy

dt= f(θ,y, z,Θ) in [0, T ],

y(0) = y0

(8.2)

is replaced by its corresponding extents decoupled systemde

dt= g(θ, e, z,Θ) in [0, T ],

e(0) = 0,(8.3)

where Θ represents all the parameters to be identified and g is the sourceterm that will be described in the next paragraphs.

The goal is to minimize the following functional cost in terms of extents

Jm,l(Θml ) =

∑e∈E

∑s∈Se

|eesl − e(m)l (tes,Θ

ml )|2,∀m = 1, ...,Ml and l = 1, ..., L,

(8.4)

where Θml is the parameters vector, e

(m)l (tes,Θ

ml ) and eesl are the l-th com-

ponent of the extents model and measurements, respectively at time tes ∈ Seand experiment e ∈ E . Ml is the set of kinetics for the l-th reaction.

To construct functions Jm,l we need to define the extents as

e = Sy

for a matrix S ∈ ML×N , such that SA = Id SF(0) = 0 and Sy0 = 0. Thetheory related to the matrix S is described in detail in [11].

Initially, we have a set of measurement species yes, at time instants tes, s ∈ Seand a set of experiments e ∈ E . Then, we can compute ees = Syes (observed

extents) and also their derivativesdee

dtusing ee constructed as cubic splines of

e ∈ E .Next, we need to construct model (8.3). Then, by multiplying mole balance

system in (8.2) by S we have the extents model (8.3) and the source termdefined by

g = δ(θ,y, z,Θ) + SFf1 + f2e.

Notice that this model is still coupled. So, we solve heat balance equationin parallel using

δe

=dee

dt− SFf1 − f2ee, e ∈ E .

Then, we can solve heat balance equation independently to obtain θe, e ∈ E .


Now, the solution of this problem is given by

e(t,Θ) =

∫ t

0

exp

(−∫ t

τ

f2(s)ds

)[δ(θ(τ),y(τ), z(τ),Θ) + SF(τ)f1(τ)

]dτ,

∀t ∈ [0, T ] and it is solved using a numerical integration formula and replacing

θ, y, z, F, f1 and f2 by cubic splines of θe, ye, ze, Fe, f1e and f2e e ∈ E .

Writing temperature in terms of extents in Batch reactors

Let us recall the extents ODE system in the case of a simple batch reactor withno heat exchange with the exterior as

de

dt= δ,

e(0) = e0.(8.5)

We assume constant specific heat of species, then heat of reactions is alsoconstant. Now, multiplying at the left by ∆Ht we get,

d∆Hte

dt= ∆Htδ = −ρcdθ

dt. (8.6)

Hence,

d(∆Hte + ρcθ)

dt= 0, (8.7)

and so

∆Hte + ρcθ = constant = ∆Ht0 + ρcθ0 = ρcθ0. (8.8)

This implies that

θ(t) = θ0 −1

ρc∆Hte(t). (8.9)

Thus, we have an expression for θ in terms of the extent vector e.

Moreover, since e(t) = Sy(t) we also have

θ(t) = θ0 −1

ρc∆HtSy(t) (8.10)

an expression giving temperature in terms of the concentrations at eachtime instant.


8.4.2 Improvements in solution: The integral method.

Integral method allows to use experimental data in determining reaction rateparameters. But we do no use it in a local optimization only. An heuristicbased on the variable neighbourhood search (VNS) [47] has been implemented.This method uses as initial values of the parameters those values previouslycomputed using the incremental method. Thus, new solutions are generateddoing successive perturbations both in kinetics and in parameters. The value ofthe integral functional cost is updated with the help of the derivatives computedvia adjoint computation when a new combination of parameters and kineticmodels improve the solution.

We describe the method in the following flow chart:

iter=1

reaction=1

iter kinetic=1

Local optimization

of Jint

Improvementsin Jint?

iter kinetic<

maxiterkinetic?

Yes

iter kinetic=

iter kinetic+1

No

reaction=

reaction+1

No

reaction< L?Yes

iter<maxiter&t< Tmax

iter=iter+1 No

Yes

Yes

No

Incemental opt. params.

Jint opt.

Integral opt. params.

Update opt. params

t=0

Figure 8.1: Flow chart VNS perturbations

Notice that t is the time of execution.


The integral method

The integral method is based on a direct comparison of species measurementsand computed concentrations via theoretical model. Sometimes, it also includesthe comparison between experimental and theoretical temperatures. Due tothis, integral method is obviously more time consuming.

The main difficulties lie in identifying the huge number of parameters whichappear in the functional cost (we need to identify at the same time the param-eters of each reaction) and in computing the derivatives of the functional costwith respect to these parameters.

Then candidate kinetic laws are integrated numerically and the parametersare calculated comparing experimental data and model prediction. We usea finite differences scheme described in Chapter 6 (BDF2 initialized with aBDF1) to solve the time derivative in the reactor model at each evaluation ofthe functional cost. Simultaneously, the derivatives of the functional cost arecomputed using the adjoint-state method we describe below.

Thus, the goal here is to minimize the following functional cost:

Jint(Θ) :=∑e∈E

∑i∈S

∑s∈Se

ωeis(yei (t

es,Θ)− yesi)2, (8.11)

where Θ is the parameters vector, yesi(tes,Θ) and yesi are the i-th component

of the solution of the model varying in S = 1, · · · , N with parameters Θ andof measurements, respectively at time tes ∈ Se and experiment e ∈ E .

The adjoint method

The adjoint method is a classical technique in optimal control theory. It hasbeen successfully used for both chemical systems STR in [6] and PFR in [8].

For most local optimizers the derivatives of the cost function are needed.They are usually calculated using finite-difference formulas. The main problemis that the functional changes each time in the process and thus the computa-tional time is greater. For this reason, the derivatives can be computed muchmore efficiently and accurately by the so-called adjoint method. Of course,this method is related to the parameters appearing in chemical reaction mod-els. There are two different approaches. When the adjoint method is appliedto the continuous system (respectively, discretization scheme) it is called con-tinuous adjoint approach (respectively, discrete adjoint approach). Inthe implementation, we have used the second approach.

Throughout this section, we explain the use of the implemented computerprogram to compute the gradient of the regularized fitting function, i.e. thatincluding the difference between theoretical and empirical concentrations.

Let us consider the following optimization problem:


minuJ(u) (8.12)

s.a.dy

dt(t) = f(t,y(t),u) en [0, T ],

dy

dt(0) = y0,

being

J(u) = J(y,u) =‖Cy − y‖2ω + zB‖B‖2 + zE‖E‖2 + zG‖G‖2

+ zβ‖β‖2 + zb‖b‖2 + zα‖α‖2,(8.13)

where u = (B,E,G,β,b,α)t.In (8.13) C is the observation operator that extracts from the state y(t) their

values at the observation times to be compared with the vector of observations,denoted by y. B = (B1, ..., BL)t and E = (Ea1, ..., EaL)t denote the pre-exponential factor and the activation energy of the reactions. The rest of theparameters are:

G =

(G1

1,1, ..., G11,P1

1, ..., G1

M1,1, ..., G1

M1,P1M1

, ..., GL1,1, ..., GLML,P

LML

)t∈ RM1P

1M1

+...+MLPLML ,

β =

(β11,1,1, ..., β

LML,P

LML

,1, β1

1,1,N , ..., βLML,P

LML

,N

)t∈ RN(M1P

1M1

+...+MLPLML

),

b =(b11, ..., b

1M1

, ..., bL1 , ..., bLML

)t∈ RM1+...+ML ,

α =(α11, ..., α

1M1

, ..., αL1 , ..., αLML

)t∈ RM1+...+ML .

The source term in the ODE system represents one of the following expres-sions:

Aδ(θ,y) for a batch STR,

Aδ(θ,y) +1

VWu− 1

Vy

P∑p=1

up for a CSTR.(8.14)

Notice that we have renamed the parameters vector called Θ as u followingthe usual notation in control theory.

In addition, we have only considered species equations. That is because inmost of the cases we are interested only in the adjustment of concentrations andalso because the ranges of the species and the temperature are quite different.However, one can consider the entire system and also compute the gradient


using the adjoint method. The only thing we should take into account is theselection of good weights in the functional cost to avoid errors that can appeardue to different ranges.

In the following paragraphs we calculate the gradient for the above problem,previously discretized, using the adjoint-system method.

1. Building the state equation

We need to construct a system for solving our problem. For this purpose, weuse a finite-difference scheme called BDF2. We obtain N × (S − 1) equationsfor N × S variables (where S is the number of steps and h is the time stepand ts = sh), so we need to introduce one more equation given by the BDF1scheme. Both formulas are described in Chapter 6.

Now, we can write the state equation as follows:

F (y,u) = By − b = 0 and y = (y11, ..., yN1, ..., y1S , ..., yNS)t, (8.15)

with

B =

1

hIN×N 0N×N ... ... ... ... ... 0N×N

−2

hIN×N

3

2hIN×N 0N×N ... ... ... ... 0N×N

1

2hIN×N −

2

hIN×N

3

2hIN×N 0N×N ... ... ... 0N×N

0N×N

...

...

...

...

.

.

.

.

.

.

...

...

...

...

...

.

.

.

.

.

.

...

...

...

...

...

.

.

.

.

.

.

...

...

...

... 0N×N

0N×N ... ... ... 0N×N1

2hIN×N −

2

hIN×N

3

2hIN×N

,

b =

1

h

y01

...y0N

+Aδ(θ(t1),y1)

− 1

2h

y01

...y0N

+Aδ(θ(t2),y2)

Aδ(θ(t3),y3)

...

Aδ(θ(tS),yS)

in a batch STR,


where ys = (y1s, ..., yNs)t 1 ≤ s ≤ S, or

b =

1

h

y01

...y0N

+Aδ(θ(t1),y1)

− 1

2h

y01

...y0N

+Aδ(θ(t2),y2)

Aδ(θ(t3),y3)

...

Aδ(θ(tS),yS)

+1

V

Wu− y1

P∑p=1

up(t1)

Wu− y2

P∑p=1

up(t2)

Wu− y3

P∑p=1

up(t3)

...

Wu− yS

P∑p=1

up(tS)

in a CSTR.

2. Building the adjoint-state equation

We define p ∈ RN×S as the solution of the following lineal system:

pt∂yF (y,u) = ∂yJ(y,u). (8.16)

In the following, we calculate the above derivatives.

2.1. Building the derivatives of state equation with respect to y

In this case the derivatives also change if the reactor type changes. For abatch STR they can be written in matrix form as follows:

∂yF (y,u) = B −

AC1 0

. . .

0 ACS

,

where each Cs for 1 ≤ s ≤ S is described by (Cs)rn =∂δr(θ(ts),ys)

∂yn1 ≤ r ≤ L, 1 ≤ n ≤ N and partial derivatives can be easily computed as


∂δr(θ(ts),ys)

∂yn=c0(r)

Mr∑j=1

αrj

Prj∑m=1

Grj,mβrj,m,ny

βrj,m,n−1ns

N∏k=1k 6=n

yβrj,m,kks

Nc∏k=1

zk(ts)βrj,m,k+N

Prj∑m=1

Grj,m

N∏k=1

yβrj,m,kks

Nc∏k=1

zk(ts)βrj,m,k+N + brj

αrj−1

c1(r, j),

(8.17)with

c0(r) = Brexp

(− EarRθ(ts)

),

c1(r, j) =

Mr∏k=1k 6=j

P rk∑m=1

Grk,m

N∏n=1

yβrk,m,nns

Nc∏n=1

zβrk,m,n+Nn + brk

αrk

,

both evaluated in ts.

For a CSTR the matrix of derivatives is the following:

∂yF (y,u) = B −

AC1 0

. . .

0 ACS

− 1

V

P∑p=1

up(t1)I 0

. . .

0

P∑p=1

up(tS)I

.

2.2. Building derivatives of the fitting function with respect to y

They can be written as a vector,

∂yJ(y,u) = 2(y11 − ˆy11, · · · , yNS1 − ˆyN1, · · · , y1S − ˆy1S , · · · , yNS − ˆyNS

)t.

3. Building the gradient of the fitting function

We define the gradient of the fitting function as,

∇yJ(y,u) = ∂(u)J(y,u)− pt∂(u)F (y,u). (8.18)

In the following sections, we calculate the above derivatives.

3.1. Building the derivatives of state equation with respect to u


They can be written as a matrix,

∂(u)F (y,u) = −

AZ1

...

AZS

where each Zs, 1 ≤ s ≤ S has the following equation

Zs =

J1s

∣∣∣J2s

∣∣∣G11,1s

∣∣∣ · · · ∣∣∣GLML,PLMLs

∣∣∣β11,1,1s

∣∣∣ · · · ∣∣∣βLML,PLML,N

s

∣∣∣b11s∣∣∣ · · · ∣∣∣bLMLs∣∣∣α11s

∣∣∣ · · · ∣∣∣αLMLs

.

We describe below each of the derivatives in Zs:

J1s =

exp

(− Ea1Rθ(ts)

)c2(1) 0

. . .

0 exp

(− EaLRθ(ts)

)c2(L)

with

c2(r) =

Mr∏j=1

P rj∑m=1

Grj,m

N∏n=1

yβrj,m,nns

Nc∏n=1

zn(ts)βrj,m,n+N + brj

αrj

(evaluated in ts),

J2s =

− B1

Rθ(ts)exp

(− Ea1Rθ(ts)

)c2(1) 0

. . .

0 − BLRθ(ts)

exp

(− EaLRθ(ts)

)c2(L)

,


Grj,ms =

0

...

0

c0(r)αrj

P rj∑m=1

Grj,m

N∏n=1

yβrj,m,nns

Nc∏n=1


αrj−1

N∏n=1

yβrj,m,nns

Nc∏n=1

zn(ts)βrj,m,n+N c1(r, j)

0

...

0

,

βrj,m,ns =

0

...

0

c0(r)

Mr∑j=1

αrj

Prj∑m=1

Grj,m

N∏k=1

yβrj,m,kks

Nc∏k=1

zk(ts)βrj,m,k+N + brj

αrj−1

Prj∑m=1

Grj,m log(βrj,m,n)yβrj,m,nns

N∏k=1k 6=n

yβrj,m,kks

Nc∏k=1

zk(ts)βrj,m,k+N

c1(r, j)

0

...

0

,


brjs =

0

...

0

c0(r)αrj

Prj∑m=1

Grj,m

N∏n=1

yβrj,m,nns

Nc∏n=1


αrj−1

c1(r, j)

0

...

0

,

and

αrjs =

0

...

0

c0(r) log

P rj∑m=1

Grj,m

N∏n=1

yβrj,m,nns

Nc∏n=1


c2(r)

0

...

0

.

3.2. Building derivatives of the fitting function with respect to u

They can be written as a vector,

∂uJ(y,u) = 2(ZbB1, · · · , ZBBL, · · · , Zαα1

l , · · · , ZααLML

)t.

4. Adjoint method flow chart

Figure 8.2 shows the flow chart of the program which calculates the valueof the fitting function, and its partial derivatives with respect to the involvedparameters.

8.5. AN EXAMPLE 151

A,B,E, G, β, b,αti, y(ti), T

zB, zE, zG, zβ, zb, zα

Solve the state equation

by a BDF2 scheme

y(tj), j = 1, ..., S∗

u = (B,E, G, b,α)t, t = tS∗

Build ∂yF (y,u)

Build ∂yJ(y,u)

Obtain p as the solution of

the linear system

pt∂yF (y,u) = ∂yJ(y,u)

Build ∂yF (y,u)

Build ∂yJ(y,u)

Obtain the gradient of the

fitting function as

∇uJ(u) = −pt∂yF (y,u) + ∂uJ(y,u)

Buildingtheadjointstate

equation

Buildingthegrad

ientof

thefittingfunction

t = t1No

t = t−∆t

J(u),∇uJ(u)

Yes

Figure 8.2: Flow chart for a batch STR adjoint method

8.5 An example

We consider an academic example that represents a batch type reactor withknown temperature. The reaction system is described by 12 species, involved


in 6 reactions and 1 catalyst with constant concentration equals to 0.001 mol/l.

A+B → C +D,

B → C + 0.5E,

B → F +G,

B → H + I,

B → J +K,

D +G→ L.

(8.19)

The data related to these species and its reactions is represented in thefollowing image:

Figure 8.3: Species and reactions data

We have 10 experiments with different initial conditions and time measure-ments from 0 to 100 seconds at each 10 seconds. An example of the measure-ments in one of these experiments is represented in the following picture:

Figure 8.4: Experimental data set

The list of kinetics for each reaction is:

8.5. AN EXAMPLE 153

Reaction 1

δ(1)1 (θ,y, z) = B1e

(−Ea1Rθ

)yα11 yα2

2 zα31 ,

δ(2)1 (θ,y, z) = B1e

(−Ea1Rθ

)yα11 zα2

1 ,

δ(3)1 (θ,y, z) = B1e

(−Ea1Rθ

)yα12 zα2

1 ,

δ(4)1 (θ,y, z) = B1e

(−Ea1Rθ

)yα11 yα2

2 ,

δ(5)1 (θ,y, z) = B1e

(−Ea1Rθ

)yα11 ,

δ(6)1 (θ,y, z) = B1e

(−Ea1Rθ

)yα12 ,

δ(7)1 (θ,y, z) = B1e

(−Ea1Rθ

)zα11 ,

δ(8)1 (θ,y, z) = B1e

(−Ea1Rθ

)yαint11 y

αint22 z

αint31 .

Reaction 4

δ(1)4 (θ,y, z) = B4e

(−Ea4Rθ

)yα12 zα2

1 ,

δ(2)4 (θ,y, z) = B4e

(−Ea4Rθ

)yα12 ,

δ(3)4 (θ,y, z) = B4e

(−Ea4Rθ

)zα11 ,

δ(4)4 (θ,y, z) = B4e

(−Ea4Rθ

)yαint12 zα2

1 .

Reaction 2

δ(1)2 (θ,y, z) = B2e

(−Ea2Rθ

)yα12 zα2

1 ,

δ(2)2 (θ,y, z) = B2e

(−Ea2Rθ

)yα12 ,

δ(3)2 (θ,y, z) = B2e

(−Ea2Rθ

)zα11 ,

δ(4)2 (θ,y, z) = B2e

(−Ea2Rθ

)yαint12 zα2

1 .

Reaction 3

δ(1)3 (θ,y, z) = B3e

(−Ea3Rθ

)yα12 zα2

1 ,

δ(2)3 (θ,y, z) = B3e

(−Ea3Rθ

)yα12 ,

δ(3)3 (θ,y, z) = B3e

(−Ea3Rθ

)zα11 ,

δ(4)3 (θ,y, z) = B3e

(−Ea3Rθ

)yαint12 zα2

1 ,

δ(5)3 (θ,y, z) = B3e

(−Ea3Rθ

)yα12 z

αint21 ,

δ(6)3 (θ,y, z) = B3e

(−Ea3Rθ

)yαint12 z

αint21 .

Reaction 5

δ(1)5 (θ,y, z) = B5e

(−Ea5Rθ

)yα12 zα2

1 ,

δ(2)5 (θ,y, z) = B5e

(−Ea5Rθ

)yα12 ,

δ(3)5 (θ,y, z) = B5e

(−Ea5Rθ

)zα11 ,

δ(4)5 (θ,y, z) = B5e

(−Ea5Rθ

)yαint12 zα2

1 .

Reaction 6

δ(1)6 (θ,y, z) = B6e

(−Ea6Rθ

)yα14 yα2

7 zα31 ,

δ(2)6 (θ,y, z) = B6e

(−Ea6Rθ

)yα14 yα2

7 ,

δ(3)6 (θ,y, z) = B6e

(−Ea6Rθ

)yα17 zα2

1 ,

δ(4)6 (θ,y, z) = B6e

(−Ea6Rθ

)yα14 zα2

1 ,

δ(5)6 (θ,y, z) = B6e

(−Ea6Rθ

)yα14 ,

δ(6)6 (θ,y, z) = B6e

(−Ea6Rθ

)yα17 ,

δ(7)6 (θ,y, z) = B6e

(−Ea6Rθ

)zα11 ,

δ(8)6 (θ,y, z) = B6e

(−Ea6Rθ

)yαint14 y

αint27 zα3

1 .

R is the universal gas constant, B ∈ [0, 1014] and Ea ∈ [0, 200000] representthe pre-exponential factor and the activation energy, respectively in the Ar-rhenius law, and αi ∈ [0, 2] ∀i = 1, 2, 3. The super index int in the exponentsmeans that we do integer optimization on these parameters.

The incremental method selects the following kinetics after computing the


parameters of all the kinetics in the list in about 4732 seconds

δ(4)1 (θ,y, z) = 1.76 105e

(− 4.60 105

Rθ

)y0.921 y0.902 , with J4,1(Θ

41) = 5.48 10−2,

δ(2)2 (θ,y, z) = 1.06 104e

(− 3.80 104

Rθ

)y1.872 , with J2,2(Θ

22) = 5.48 10−2,

δ(2)3 (θ,y, z) = 201.29e

(− 3.80 104

Rθ

)y1.882 , with J2,3(Θ

23) = 5.48 10−2,

δ(2)4 (θ,y, z) = 1.16 104e

(− 1.14 105

Rθ

)y0.932 , with J2,4(Θ

24) = 5.48 10−2,

δ(1)5 (θ,y, z) = 1.62 108e

(− 1.14 105

Rθ

)y0.952 z0.961 , with J1,5(Θ

15) = 5.48 10−2,

δ(2)6 (θ,y, z) = 2.27 106e

(− 6.86 104

Rθ

)y0.984 y0.977 , with J2,6(Θ

26) = 5.48 10−2.

The objective value of the integral method for these kinetics is 0.2184.

The integral method provides a value of the functional cost of 0.2071 after11755 seconds. The selected kinetics are the following:

δ(8)1 = 1.97 108e

(− 4.60 104

Rθ

)y1y2z1,

δ(1)2 = 1.05 107e

(− 3.80 104

Rθ

)y1.872 z0.991 ,

δ(4)3 = 6.46 104e

(− 3.44 104

Rθ

)y22z

0.841 ,

δ(2)4 = 1.16 104e

(− 6.97104

Rθ

)y0.932 ,

δ(4)5 = 7.99 108e

(− 1.18 105

Rθ

)y2z

1.021 ,

δ(8)6 = 4.96 109e

(− 6.91 104

Rθ

)y4z

1.091 .

In Figures 8.5 and 8.6 we can see the comparison between the data and thenumerical solution of the model with the selected kinetics and their parametersin both incremental and integral methods in one of the experiments.

8.5. AN EXAMPLE 155

Figure 8.5: Numerical vs Experimental concentrations: Incremental method

Figure 8.6: Numerical vs Experimental concentrations: Integral method


The incremental method provides a good solution as it can be seen in Figure8.5. However, experimental measurements are not compared directly in thismethod because an algebraic transformation of the data was previously done.Then, the integral method is used to correct these possible differences betweendata and numerical solution. The largest differences appear in species A, E andF . It is not recommendable to use only the integral method by itself becauseit is computationally expensive.

In fact, these methods generate better results when used together. Theincremental method provides good results when enough measurements and/orexperiments (not affected by noise) are given. In other cases, incrementalmethod generates an initial solution for the integral method which is essentialin order to obtain a better adjustment. Moreover, this last method is computa-tionally expensive and so, to improve this, an adjoint method is considered forcomputing functional cost derivatives and a VNS heuristic is also considered inthe optimization process to select faster the best combination of kinetic modelsfrom the catalogue.

Conclusions

In conclusion, a tandem of two methods has been used. The first technique,called incremental method, is a methodology able to decouple the model forworking separately with each kinetic. Then, the identification problem is di-vided into a set of smaller subproblems, one for each kinetic. The secondtechnique, called integral method, is an heuristic based on the variable neigh-bourhood search (VNS). This method uses as initial values of the parametersthose previously computed by the incremental method. New solutions are gen-erated by optimizing the integral functional cost and updating the parametersby doing successive perturbations both in kinetics and in parameters. Thederivatives of this functional cost are computed via the adjoint method, as in-tegral method by itself is time consuming. When the value of the functionalcost decreases means that a new combination of parameters and kinetic modelsimprove the solution. The use of both incremental and integral methods, withthe help of adjoints calculation, provides good results in the identification ofkinetic models, as exemplified for batch reactor.

157

158 Conclusions

Future work

To conclude the dissertation of this thesis, we briefly introduce some of theresearch lines that we would like to develop as part of future research:

• Existence of solution of the full convection-diffusion-reaction model

Prove the existence of solution of the full model, including the equation oftemperature. The property of boundedness of total mass needs an anal-ogous one for the energy equation. Then, we need to explore a differentapproach for the complete system.

• Existence of solution of the FBR model

It is a more complex model, with the added difficulty of coupling betweenthe macro-scale and the micro-scale model. Therefore, we should studythe most appropriate techniques for treating this problem.

• Identification of parameters in the reaction term

The problem of identification of parameters for this type of reactors canbe addressed. The ideal approach is to do it in two phases. In the firstone, the identification of the stationary model can be studied to latertreat the transient one. In this reactor the most adequate method is theintegral. But it is important to take into account the computation time.Maybe the adjoint method must be used too. The residence time in thistype of reactor is large in most of the cases. Then probably only in thefirst time steps the transient model is needed.

• Deactivation of the catalyst

In many cases it is assumed that the effectiveness of the catalysts for in-creasing the speed of the reactions does not change over time. Sometimesthat is not truth, because the activity decreases as the catalyst is used.Sometimes this procedure is very fast, in other cases it is so slow thatregeneration or replacement of the catalyst is only necessary after a longperiod of time. That is an interesting phenomena to include in the modeland then study it.

159

160 Future work

Appendix A

Summary of equations ofcontinuumthermomechanics

We recall the general equations of continuum thermomechanics for reactingmixtures. Further details can be found, for instance, in [7].

A.1 Equations of continuum thermomechanics

Species mass

∂(ρYi)

∂t+ div(ρYiv)−div(ρDigradYi) =Mi

R∑j=1

aijδj(θ,y), i = 1, ..., N. (A.1)

Mixture mass

∂ρ

∂t+ div(ρv) = 0. (A.2)

Momentum

∂(ρv)

∂t+ div(ρv ⊗ v) = divT + b. (A.3)

Energy

∂(ρe)

∂t+ div(ρev) + divq = T ·D + f. (A.4)

161

162 APPENDIX A. SUMMARY CONTINUUM THERMOMECHANICS

Constitutive equations (ideal solution)

We assume that the mixture density can change due to change in compositionand/or temperature. More precisely, we make the following assumptions:

• The mixture is an ideal solution and then its volume is the sum of thevolumes occupied by the species if they were isolated.

• The density of each pure species only depends on temperature. For thei-th species it will be called di.

Then it is easy to see that the density is given by

ρ =

(N∑i=1

Yidi

)−1

(kg/m3), (A.5)

with di = di(θ). Indeed, the specific volume ν = 1/ρ is the volume occupiedby 1 kg of mixture. Since the mass of the i-th species in this kg is Yi kg, thenits volume is Yi/di m

3. Therefore, the total volume of 1 kg of mixture is

ν =

N∑i=1

Yidi.

Since yi = ρYi/Mi, then equation (A.5) is equivalent to

1 =

N∑i=1

Miyidi

. (A.6)

Constitutive equations (for mixture of perfect gases)

If we deal with a mixture of perfect gases, then

p = ρRθ, (A.7)

where R is the gas constant given by

R =RM

and M is the molecular mass of the mixture defined by

1

M=

N∑i=1

YiMi

.

Moreover, in both cases (ideal solution or mixture of perfect gases) we have

A.1. EQUATIONS OF CONTINUUM THERMOMECHANICS 163

e =

N∑i=1

Yiei, with ei = ei(θ), (J/kg)

q = −k grad θ (Fourier′s law), (W/m2),

ci = cvi(θ) =∂ei∂θ

(θ) (specific heat), (J/(kgK)),

T = −pI + Tv, with Tv = 2ηD (the viscous stress tensor) (N/m2).

(A.8)

Notations

ρ: density (kg/m3)

v: velocity (m/s)

p: presure (N/m2)

η, ξ: viscosity coefficients (Ns/m2)

b: density of body force (N/m3)

D = 12 (gradv + gradvt) (strain rate) (s−1)

Yi: mass fraction of the i-th species

yi: concentration of the i-th species, yi = ρYi/Mi (kmol/m3 = mol/l)

di: density of the i-th pure species (kg/m3)

Di: diffusion coefficient of the i-th species (m2/s)

δj : kinetic of the j-th reaction (kmol/(m3s))

Mi: molecular mass of the i-th species (kg/kmol)

R: gas constant (J/(kgK))

R: universal gas constant (8314.4621 J/(kmolK))

M: molecular mass of the mixture

e: specific internal energy (J/kg)

q: density of the heat flux vector (W/m2)

θ: absolute temperature (K)

A = (aij): stoichiometric matrix

164 APPENDIX A. SUMMARY CONTINUUM THERMOMECHANICS

Remark A.1.1. By replacing the constitutive equation (A.5) in the mass con-servation equation (A.2), we get a condition for the divergence of the velocityin terms of the mixture temperature and composition. Coupled with the mo-mentum equation (A.3) it can be used to determine pressure.

Remark A.1.2. Thus, in order to solve the models we could use a segregatedmethod. Firstly, for a given temperature (θ) and composition Yi we can computedensity by (A.5) and then solve the momentum equation to determine velocityand pressure. Next, by using the computed velocity we can solve the speciesconservation equations (A.1) and the energy equation (A.4) to determine massfractions and temperature, respectively.

Appendix B

Abstract semilinearinitial-value problems

This appendix has been extracted from [42] (see also [43]).

B.1 Operators. Spectrum and resolvent

Throughout this section X 6= 0 is a real or complex Banach space and L(X)denote the Banach space of the bounded linear operators form X into itself.Even in the case where X is a real vector space, we need to deal with complexspectrum and resolvent: so we introduce the complexification of X, defined as

X = x+ iy : x, y ∈ X; ‖x+ iy‖X = sup0≤θ≤2π

‖x cos θ + y sin θ‖.

If A: D(A) ⊂ X 7→ X is a linear operator, the complexification of A is definedby

D(A) = x+ iy : x, y ∈ D(A), A(x+ iy) = Ax+ iAy.

In the sequel if no confusion will arise we shall drop out all the tildes, and byspectrum and resolvent of A we shall mean spectrum and resolvent of A.

Definition B.1.1. Let A : D(A) ⊂ X 7→ X be a linear operator. The resolventset ρ(A) and the spectrum σ(A) of A are defined by

ρ(A) = λ ∈ C : ∃(λI −A)−1 ∈ L(X), σ(A) = C\ρ(A).

If λ ∈ ρ(A) , we set

R(λ,A) := (λI −A)−1

and R(λ,A) is called resolvent operator or simply resolvent.

165

166 APPENDIX B. ABSTRACT SEMILINEAR PROBLEMS

B.2 Sectorial operators

Definition B.2.1. We say that a linear operator A: D(A) ⊂ X → X issectorial if there are constants ω ∈ R, θ ∈ (π/2, π) , M > 0 such that

ρ(A) ⊃ Sθ,ω := λ ∈ C : λ 6= ω, | arg(λ− ω)| < θ, (B.1)

‖R(λ, A)‖L(X) ≤M

|λ− ω|, λ ∈ Sθ,ω, (B.2)

where the set Sθ,ω is represented in the following graph:

Figure B.1: Sector image

Proposition B.2.1. Let A : D(A) ⊂ X → X be a linear operator such thatρ(A) contains a halfplane λ ∈ C : Reλ ≥ ω, and

‖λR(λ,A)‖L(X) ≤M, Reλ ≥ ω,

with ω ≥ 0,M ≥ 1. Then A is sectorial.

For every t > 0, Definition B.2.1 above allows us to define, for any sectorialoperator A, a bounded linear operator etA on X, through an integral formula.For r > 0, η ∈ (π/2, θ), in the curve γr,η defined by

λ ∈ C : | arg λ| = η, |λ| ≥ r ∪ λ ∈ C : | arg λ| ≤ η, |λ| = r,

B.3. SECOND ORDER DIFFERENTIAL OPERATORS 167

oriented counterclockwise, as in Figure B.2.

ω ω + γ

σ(A) η

γr,η + ω

Figure B.2: Curve γr,η + ω

Definition B.2.2. Let A be a sectorial operator. The function

t ∈ [0,∞)→ etA ∈ L(X)

defined by means of the Dunford integral

etA =1

2πi

∫γr,η+ω

etλR(λ, A)dλ, t > 0

is called the analytic semigroup generated by A in X and e0A = I.

B.3 Second order differential operators

Let us consider general second order elliptic operators in an open set Ω ⊂ Rn.The set Ω can be either the whole Rn or a bounded open set with uniformlyC2 boundary ∂Ω. Let us denote by n(x) the outer unit vector normal to ∂Ωat x.

Let A be the differential operator

A(x,D) :=

n∑i,j=1

aij(x)Dij +

n∑i=1

di(x)Di + c(x)I, (B.3)

where

Di =∂

∂xi, Dij =

∂2

∂xi∂xj,


I is the identity operator, and aij , di, c are real, bounded and continuous coef-ficients defined on Ω. We assume that for every x ∈ Ω the matrix aij(x), i, j =1, . . . , n is symmetric and uniformly strictly positive definite, i.e., there existsa positive constant α such that

N∑i,j=1

aij(x)ξiξj ≥ α|ξ|2, x ∈ Ω, ξ ∈ Rn. (B.4)

Moreover, if Ω = Rn we need the leading coefficients aij to be uniformlycontinuous.

We also consider a first order differential operator acting on the boundary:

Bu(x) := b0(x)u(x) +

n∑i=1

bi(x)Diu(x), (B.5)

where the coefficients bi, i = 0, 1, . . . , n are in C1(Ω) and the following transver-sality condition holds:

n∑i=1

bi(x)ni(x) 6= 0, x ∈ ∂Ω.

These hypotheses regarding operator B are needed in the following theorem

Theorem B.3.1. Let Ω ⊂ Rn be a bounded open set with uniformly C2 bound-ary ∂Ω, and X = C(Ω). We define the operator

D(A) = u ∈⋂

1≤p<+∞

W 2,p(Ω) : Bu|∂Ω = 0, Au ∈ C(Ω),

Au = Au, u ∈ D(A).

Then A is sectorial in X and D(A) is dense in X.

B.4 Local existence results

Let us consider the initial-value problem

du

dt= Au(t) + F (t, u(t)) , t > 0, (B.6)

u(0) = u0, (B.7)

where A : D(A) ⊂ X → X is a sectorial operator and F : [0, T ] ×X → X.We shall assume that F is continuous, and that for every R > 0 there is L > 0such that

B.4. LOCAL EXISTENCE RESULTS 169

‖F (t, x)− F (t, y)‖ ≤ L‖x− y‖, t ∈ [0, T ], x, y ∈ B(0, R),

where B(0, R) denotes the open ball with center 0 ∈ X and radius R.This means that F is Lipschitz continuous with respect to x on any boundedsubset of X, with Lipschitz constant independent of t.

Definition B.4.1. Let I be defined by I = [0, τ) or I = [0, τ ], with τ ≤ T ,

• We say that a function u : I → X is a strict solution of problem (B.6)–(B.7) in I if it is continuous with values in D(A) and differentiable withvalues in X in the interval I, and it satisfies (B.6)–(B.7).

• We say that it is a classical solution if

– it is continuous with values in D(A) and differentiable with valuesin X in the interval I\0,

– it is continuous in I with values in X, and

– it satisfies (B.6)–(B.7).

• We say that it is a mild solution if it is continuous with values in X, inI\0, and it satisfies

u(t) = etAu0 +

∫ t

0

e(t−s)AF (s, u(s))ds, t ∈ I. (B.8)

One can prove that every strict or classical solution satisfies (B.8). More-over, we notice that, in general, if u is a mild solution it may be discontinuousat t = 0 because, in general, limt→0+ etAu0, needs not to be u0. However, onecan prove that

limt→0+

etAu0 = u0 ∀u0 ∈ D(A).

Theorem B.4.1. Let us denote by Cb((0, a];X) the Banach space of mappingsfrom (0, a] in X that are continuous and bounded. The following statementshold:

a) If u, v ∈ Cb((0, a];X) are mild solutions for some a ∈ (0, T ], then u ≡ v.

b) For every u ∈ X there exist r, δ > 0,K > 0 such that for ‖u0 − u‖ ≤ rproblem (B.6)–(B.7) has a mild solution u = u(.;u0) ∈ Cb((0, δ] ;X).Function u belongs to C([0, δ];X) if and only if u0 ∈ D(A).

Moreover for every u0, u1 ∈ B(u, r) we have

‖u(t;u0)− u(t;u1)‖ ≤ K‖u0 − u1‖, 0 ≤ t ≤ δ.


B.5 The maximally defined solution

We can construct a maximally defined solution as follows.

Proposition B.5.1. Set

τ(u0) = supa > 0 : problem (B.6), (B.7) has a bounded mild solution ua in [0, a].

Then the mapping given by u(t) := ua(t) , if t ≤ a, is well defined in theinterval

I(u0) := ∪[0, a] : problem (B.6), (B.7) has a mild solution ua in [0, a].

Besides, τ(u0) = sup I(u0) and u is the maximally defined solution correspond-ing to initial condition u0.

Now, we state some results concerning regularity of the maximally definedsolutions.

Theorem B.5.1. Assume that there is α ∈ (0, 1) such that for every R > 0we have

‖F (t, x)− F (s, x)‖ ≤ C(R)(t− s)α, 0 ≤ s ≤ t ≤ T, ‖x‖ ≤ R.

Then, for every u0 ∈ X,u ∈ C0,α([ε, τ(u0) − ε];D(A)) ∩ C1,α([ε, τ(u0) −ε];X), ∀ε ∈ (0, τ(u0)/2). Moreover the following statements hold:

(i) If u0 ∈ D(A) then, u is a classical solution of (B.6), (B.7).

(ii) If u0 ∈ D(A) and Au0 + F (0, u0) ∈ D(A), then u is a strict solution of(B.6)–(B.7).

In order to prove the next proposition, we introduce an auxiliary result.

Lemma B.5.1. Let f ∈ Cb((0, T );X). Then the function v defined by

v(t) :=

∫ t

0

e(t−s)Af(s)ds, 0 ≤ t ≤ T,

belongs to Cα(0, T ];X) for every α ∈ (0, 1), and there is C = C(α, T ) suchthat

‖v‖C0,α([0,T ];X) ≤ C sup0<s<T

‖f(s)‖.

Proposition B.5.2. Let u0 be such that I(u0) 6= [0, T ]. Then t ∈ I(u0) →u(t) ∈ X is unbounded in I(u0).

B.5. THE MAXIMALLY DEFINED SOLUTION 171

Proof. Assume by contradiction that u is bounded in I(u0) and set τ =τ(u0). Then t 7→ F (t, u(t;u0)) is bounded and continuous with values in X inthe interval (0, τ). Since u satisfies the variation of constants formula (B.8),it may be continuously extended to t = τ , in such a way that the extension isHolder continuous in every interval [ε, τ ], with 0 < ε < τ . Indeed, t 7→ etAu0 iswell defined and analytic in the whole halfline (0, +∞), and u− etAu0 belongsto C0,α([0, τ ];X) for each α ∈ (0, 1) by Lemma B.5.1.

By Theorem B.4.1, the problem

dv

dt(t) = Av(t) + F (t, v(t)) , t ≥ τ, v(τ) = u(τ),

has a unique mild solution v ∈ C([τ, τ + δ];X) for some δ > 0. Note that v iscontinuous up to t = τ because u(τ) ∈ D(A).

The function w defined by w(t) = u(t) for 0 ≤ t < τ , and w(t) = v(t)for τ ≤ t ≤ τ + δ, is a mild solution of (B.6)–(B.7) in [0, τ + δ]. This is incontradiction with the definition of τ . Therefore, u cannot be bounded.

From the above proof we can deduce the following result:

Corollary B.5.1. If I(u0) 6= [0, T ], then τ(u0) = sup I(u0) 6∈ I(u0).


Appendix C

Interpolation ErrorEstimates

This appendix includes some background on Lagrange finite element methodsneeded for the numerical analysis of the problem in Section 5.2: Error esti-mates for the semidiscrete solution. It has been mainly extracted from Ernand Guermond [27]. Other classical references on the mathematical analysisof the finite element method are Ciarlet [22] and Brenner and Scott [19]. Wealso include a discrete norm and some inverse-like inequalities following Wang’sarticle [66].

The following definition of finite element has been given by Ciarlet [22].

Definition C.0.1. A finite element consists of a triplet K, P, Σ where:(i) K is a compact, connected, Lipschitz subset of Rd with non-empty inte-

rior.(ii) P is a vector space of functions p : K → Rm for some positive integer

m.(iii) Σ is a set of nsh linear forms σ1, . . . , σnsh

acting on the elementsof P , such that the linear mapping

P 3 p→ (σ1(p), . . . , σnsh(p)) ∈ Rnsh ,

is bijective, i.e., Σ is a basis for L(P ;R). The linear forms σ1, . . . , σnsh are

called local degrees of freedom.

C.1 Local interpolation operator

The first step is to define the local interpolation operator associated to a fam-ily of finite elements. It will be used in the next section to build the globalinterpolation operator that is the one we are interested in.

173

174 APPENDIX C. INTERPOLATION ERROR ESTIMATES

Let Ω be a domain in Rn and Th = Kl1≤l≤nel be a mesh of Ω, where nel is

the number of elements in the mesh and Ωh =⋃k∈Th

K (Ωh is called geometric

interpolation of Ω). Let K, P , Σ be the reference finite element.

Denote by σ1, ...σndof the local degrees of freedom and by θ1, ...θndof the local shape functions. Let V (K) be the domain of the local interpolationoperator IK associated with K, P , Σ, i.e.,

IK : v ∈ V (K) ndof∑i=1

σi(v)θi ∈ P .

For all K ∈ Th, one must first define the counterpart of V (K), i.e., a Banachspace V (K) and a linear bijective mapping

ΦK : V (K)→ V (K).

Then, a set of Th-based finite elements can be defined as follows:

Proposition C.1.1. For K ∈ Th, the triplet K,PK ,ΣK defined byK = TK(K)

PK = Φ−1K (p) : p ∈ P

ΣK = σK,i1≤i≤ndof : σK,i(p) = σi(ΦK(p)), p ∈ PK(C.1)

is a finite element. The local shape functions are θK,i = Φ−1K (θi) 1 ≤ i ≤ ndof ,

and the associated local interpolation operator is

IK : v ∈ V (K) ndof∑i=1

σK,i(v)θK,i ∈ PK .

Remark C.1.1. Let K, P , Σ be a Lagrange finite element. Then, one maychoose V (K) = C(K). Defining V (K) in a similar way, the mapping

ΦK : v ∈ V (K)→ ΦK(v) = v TK ∈ V (K)

is linear and bijective. Then, for all K ∈ Th, the finite element K,PK ,ΣKconstructed in Proposition C.1.1 is a Lagrange finite element. Moreover, since

σK,i(v) = σi(ΦK(v)) = ΦK(v)(ai) = v TK(ai),

setting aK,i = TK(ai) for 1 ≤ i ≤ ndof , we infer that aK,i1≤i≤ndof are thenodes of K,PK ,ΣK.

Now, we are in a position to define the global interpolation operator.

C.2. GLOBAL INTERPOLATION OPERATOR 175

C.2 Global interpolation operator

Using the Th-based family of finite elements K,PK ,ΣKK∈Th generated inProposition C.1.1, a global interpolation operator Ih can be constructed asfollows: firstly, choose its domain to be

D(Ih) = v ∈ L1(Ωh) : ∀K ∈ Th, v|K ∈ V (K).For a function v ∈ D(Ih), the quantities σK,i(v|K) are meaningful on all the

mesh elements and for all 1 ≤ i ≤ ndof · Then, the global interpolant Ihv canbe specified elementwise using the local interpolation operators defined above,i.e.,

∀K ∈ Th, (Ihv)|K = IK(v|K) =

ndof∑i=1

σK,i(v|K)θK,i.

The global interpolation operator is defined as follows:

Ih : v ∈ D(Ih)→∑K∈Th

ndof∑i=1

σK,i(v|K)θK,i ∈Wh,

where Wh, the codomain of Ih, is

Wh = vh ∈ L1(Ωh) : ∀K ∈ Th, v|K ∈ PK.The space Wh is called an approximation space. Notice that, we abuse

of notation by implicitly extending θK,i by zero outside K.

Definition C.2.1. Let a1, . . . , andof be the nodes associated to the finiteelement space and ψh,1, · · · , ψh,ndof the global shape functions. The globalLagrange interpolation operator is defined as follows:

Ih : v ∈ C(Ωh)→ndof∑i=1

v(ai)ψh,i ∈ Vh.

Notice that the domain of Ih can also be taken to be Hs(Ωh) for s >n

2.

C.3 Some bounds for the interpolation operator

Let us start with a bound for the local intepolation operator which will begeneralized later for the global one.

Lemma C.3.1. Let K, P , Σ be a finite element with associated normed vec-tor space V (K). Let 1 ≤ p ≤ ∞, and assume that there exists k an integersuch that

Pk ⊂ P ⊂W k+1,p(Ω) ⊂ V (K).


Let TK : K → K be an affine bijective mapping and IK be the local inter-polation operator on K. Let l be such that 0 ≤ l ≤ k and W l+1,p(K) ⊂ V (K)

with continuous embedding. Then, setting σK =hKρk

(ρK is the diameter

of the largest ball that can be inscribed in K and hK is the diameter of KhK = max

x,y∈K‖x− y‖), there exists c > 0 such that, for all m ∈ 0, ..., l + 1,

∀K ∈ Th,∀v ∈W l+1,p(K) |v − IKv|m,p,K ≤ chl+1−mσmK |v|l+1,p,K .

Remark C.3.1. For a Lagrange finite element of degree k, V (K) = C(K).

Hence, the condition on l in the above lemma isn

p− 1 < l ≤ k.

Definition C.3.1. A family of meshes is said to be shape-regular if there

exists σ0 such that ∀h, ∀K, σK =hKρK≤ σ0.

It is said quasi-uniform if and only if it is shape-regular and there exista constant C such that, for all h and K, hK ≥ Ch.

Lemma C.3.2. Let p, k, and l satisfying the assumptions of Lemma C.3.1.Let Ω be a polyhedron and Thh>0 be a shape-regular family of affine meshes

of Ω. Denote by Vh the approximation space based on Th and K, P , Σ. LetIh be the corresponding global interpolation operator. Then, there exists c suchthat, for all h,

• for p <∞, if v ∈W l+1,p(Ω),

‖v − Ihv‖Lp(Ω) +

l+1∑m=1

hm

( ∑K∈Th

|v − Ihv|pm,p,K

) 1p

≤ chl+1|v|l+1,p,Ω;

• for p =∞, if v ∈W l+1,∞(Ω),

‖v − Ihv‖L∞(Ω) +

l+1∑m=1

hm(

maxK∈Th

|v − Ihv|m,∞,K)≤ chl+1|v|l+1,∞,Ω;

• for p = 2 and l > n2 − 1, if v ∈ H l+1(Ω),

‖v−Ihv‖L∞(Ω) +

l+1∑m=1

hm

( ∑K∈Th

|v − Ihv|2m,2,K

) 12

≤ chl+1−n/2|v|l+1,2,Ω.

C.4. SOME INVERSE INEQUALITIES 177

Remark C.3.2. Consider a Lagrange finite element of degree k. Take p = 2and assume n ≤ 3. Then, from Remark C.3.1 one can take 1 ≤ l ≤ k andhence the previous inequality yields, for all v ∈ H l+1(Ω),

‖v − Ihv‖L2(Ω) + h|v − Ihv|1,Ω ≤ chl+1|v|l+1,Ω.

This estimate is optimal if v is smooth enough, i.e., v ∈ Hk+1(Ω). However,if v is in Hs(Ω) and not in Hs+1(Ω) for some s ≥ 2, increasing the degree ofthe finite element beyond s− 1 does not improve the interpolation error.

C.4 Some inverse inequalities

Lemma C.4.1 (Global inverse inequalities). Let us assume the hypothesesof Lemma C.3.1. Let us also assume that the family of meshes Thh>0 isquasi-uniform. Let l be such that P ⊂ W l,∞(K). Set Wh = vh : ∀K ∈Th, vh TK ∈ P. Then, using the usual convention if p =∞ or q =∞, thereis c, independent of h, such that, for all vh ∈Wh and 0 ≤ m ≤ l,( ∑

K∈Th

‖vh‖pl,p,K

) 1p

≤ chm−l+min(0,np−nq )

( ∑K∈Th

‖vh‖qm,q,K

) 1q

.

Remark C.4.1. In the particular case p = ∞ and q = 2 the above inequalityyields

‖vh‖L∞(Ω) ≤ Ch−n/2‖vh‖L2(Ω) ∀vh ∈Wh. (C.2)

To estimate the interpolation of nonlinear terms in the error estimates weuse Lemma 4.3 in [66] that we state below.

Let us introduce an auxiliary semi-norm in C(Ω) by

‖χ‖h :=

(N∑l=1

|χ(xi)|2)1/2

.

Since P is finite-dimensional, we have the equivalence between norms on thereference element. Then, by using a straightforward homogeneity argument,the following lemma can be proved:

Lemma C.4.2. There exist two strictly positive constants C1 and C2 indepen-dent of h such that

C1hn/2‖χh‖h ≤ ‖χh‖L2(Ω) ≤ C2h

n/2‖χh‖hfor all χh ∈ Vh.


Appendix D

Resumen

En terminos generales, un reactor quımico puede entenderse como un recipienteutilizado para transformar ciertas especies quımicas en los productos finales de-seados. Estos recipientes pueden ser simplemente reactores ideales de tanqueagitado en el caso mas simple o reactores mas complejos como pueden ser losreactores de lecho fijo. En cualquier caso, es importante que el tiempo deresidencia dentro del reactor sea el suficiente como para que se produzcan lasreacciones quımicas esperadas.

En el ambito de la ingenierıa quımica, el diseno de los reactores abarca almenos tres campos: termodinamica, cinetica y transferencia de calor. Ası, sipor ejemplo se produce una reaccion en un reactor batch STR, una preguntarazonable serıa cual es la conversion maxima esperada. Esta es una cuestionde termodinamica. Si quisieramos saber en cuanto tiempo deberıa transcurrirla reaccion para lograr una conversion en los productos deseados, estarıamoshaciendonos una pregunta sobre la cinetica (deberemos conocer no solo la es-tequiometrıa, sino tambien las tasas de la reaccion). Finalmente, si queremossaber cuanto calor debe transferirse al reactor o desde el para mantener lacondicion isotermica, estamos tratando un problema de transferencia de calorcombinado con un problema termodinamico (deberemos saber si la reaccion esendotermica o exotermica).

Despues de la reaccion quımica generalmente debe realizarse un tratamientofısico para purificar el producto y reciclar, si es necesario, el material que noha reaccionado. La cantidad de material a producir es un factor clave paradeterminar que tipo de reactor se debe usar. Para cantidades pequenas sue-len utilizarse en la industria normalmente reactores batch STR. Para grandesvolumenes, como en la industria petrolera, los reactores de flujo en piston soncomunes.

179

180 APPENDIX D. RESUMEN

Para describir mas detalladamente un reactor es necesario distinguir entrelos diferentes tipos. Hay muchas clasificaciones en la literatura. Cada una de el-las realizada de acuerdo con alguna caracterıstica. Describiremos los esquemasmas habituales de acuerdo con el artıculo de Foutch [29]:

Tipo de operacion: Esta clasificacion se realiza de acuerdo con la con-figuracion del reactor. Esta es la clasificacion que usamos principalmenteen la tesis (batch STR, semi-batch, CSTR, PFR o FBR son algunos delos diferentes nombres de reactor segun su configuracion).

1. Batch STR: los reactivos se introducen en el reactor solo en el mo-mento inicial. No hay flujos de entrada ni de salida a lo largo delproceso.

2. Semi-batch STR: algunos de los reactivos se introducen en el reactoren el momento inicial; otros se introducen continuamente a lo largodel proceso.

3. Reactor de tanque agitado (de flujo) continuo en estado transitorio(CSTR): los reactivos se introducen continuamente a lo largo deltiempo. Tambien hay un flujo de salida a lo largo del proceso.

4. Reactores de flujo en piston: es un reactor tubular donde se asumeflujo en piston. Es decir, la velocidad es constante en cualquierseccion transversal del reactor.

5. Reactores de lecho fijo: el reactor de lecho fijo es un reactor cilındricocon extremos convexos y un lecho relleno de partıculas catalıticas detamano uniforme, que se inmovilizan o se fijan dentro del tubo.

Numero de fases: Los reactores tambien se pueden clasificar por elnumero de fases presentes en el reactor en cualquier momento. Se de-nominan reactores homogeneos y heterogeneos. Los primeros representanlos reactores con una sola fase (los STR son reactores homogeneos). Elsegundo contiene mas de una fase. Varios tipos de reactores heterogeneosestan disponibles debido a varias combinaciones de fases (como PFR oFBR).

Tipos de reaccion: Esta clasificacion se realiza teniendo en cuenta eltipo de reacciones que se estan produciendo. Algunos de los mas impor-tantes son:

1. Catalıticas: reacciones que requieren la presencia de un catalizadorpara obtener por ejemplo las condiciones de velocidad necesariaspara ese diseno de reactor en particular. Un ejemplo de este reactores el FBR.

181

2. No-catalıticas: reacciones que no incluyen un catalizador homogeneoo heterogeneo. Son las opuestas a las anteriores.

3. Autocatalıticas: en estas reacciones, uno de los productos aumentala velocidad de las reacciones.

4. Biologicas: reacciones que involucran celulas vivas (enzimas, bacte-rias, etc.).

5. Polimerizaciones: reacciones que involucran la formacion de cadenasde polımeros moleculares.

Finalmente, dependiendo del destino final en la industria, consideramosuna clasificacion de acuerdo con dos motivaciones diferentes.

1. Reactores industriales: simulacion de su funcionamiento con el ob-jetivo final de optimizarlo economicamente modificando las condi-ciones de funcionamiento (condiciones iniciales, temperatura, ...).

2. Reactores de laboratorio / planta piloto: simulacion (para la opti-mizacion del diseno del reactor: determinacion de las condicionesgeometricas y de operacion optimas para ese futuro reactor). El ob-jetivo del diseno es determinar las caracterısticas del reactor, comotuberıas, valvulas o mezcladores. Por ejemplo, el reactor debe tenerun volumen suficiente para permitir que la reaccion alcance un nivelde conversion o permitir el intercambio de calor necesario.

El diseno del reactor requiere conocer, en primer lugar, el tipo de reaccionesque tendan lugar en el y sus dimensiones y tambien el metodo de operacion deacuerdo con el proceso quımico deseado, como se explica en [24]. Pero en eseproceso es importante conocer tambien las reacciones quımicas, en cualquierade los reactores descritos, y las expresiones de velocidad de reaccion (cinetica)que deben a traves de una expresion matematica. Para predecir el tamano delreactor necesario para la obtencion de la conversion deseada de reactantes a losproductos finales, se requiere informacion sobre la composicion y los cambiosde temperatura, ası como la velocidad de reaccion, obtenidos de las ecuacionesde balance de moles y energıa.

Suponiendo conocidos los datos disponibles de los experimentos y la este-quiometrıa (las reacciones), debemos buscar una metodologıa de identificacion.De hecho, hay varias tecnicas disponibles en la literatura, como los metodosdiferencial, integral e incremental [12]. El metodo diferencial compara el ladoderecho del modelo con las derivadas de los datos. El metodo incrementaltrabaja con el concepto “extent”, que nos proporciona una solucion analıtica deun nuevo sistema desacoplado. El metodo integral resuelve numericamenteel modelo y lo compara con los datos. En cualquier caso, se formula un prob-lema de optimizacion, en funcion de los parametros cineticos. Ademas, para


la obtencion de resultados mas precisos es posible combinar algunos de estosmetodos.

Previo al diseno del reactor, es importante estudiar estos modelos rigurosa-mente. El analisis matematico de los modelos mencionados ha despertadocuriosidad desde hace muchos anos hasta nuestros dıas. En particular, lasecuaciones generales de conveccion-difusion-reaccion tienen un notorio interescientıfico. De hecho, se han estudiado desde diferentes enfoques y utilizandouna variedad de metodos. Por ejemplo, bifurcaciones y estabilidad, teorıade semigrupos, perturbaciones singulares o siguiendo un enfoque variacional.Seguido de este analisis, naturalmente resulta necesario simular estos modelos.La resolucion numerica de los mismos puede basarse en un esquema de difer-encias finitas en la mayorıa de los casos y en algunos otros en el metodo deelementos finitos.

Esta tesis se ha dividido en tres partes en las que pretende hacer un estudioexhaustivo de reactores resolviendo algunas cuestiones que hemos introducidoya, tanto teoricas como practicas sobre ellos, ademas de dos apendices en losque se describen herramientas y resultados utiles a lo largo del documento enaras de obtener un texto auto-contenido. A continuacion, describiremos endetalle el contenido de cada una de las partes.

I Modelando reactores quımicos

Una parte importante en el modelo es la velocidad de la reaccion quea priori puede entenderse como un parametro independientemente de laforma y la longitud del reactor. Sin embargo, induce variaciones en latemperatura y la composicion y viceversa, estas magnitudes influyen enlas velocidades de reaccion. Por ello, el comienzo de capıtulo 1 estadedicado a recordar algunos conceptos basicos sobre las especies y lasreacciones quımicas. Tambien a las velocidades de reaccion, recordandolas mas importantes a traves de la literatura (como las que se describenen [46], [34] y [31]); y a la definicion de la tasa de reaccion segun la leyde Arrhenius.

Ahora bien, dado que muchos procesos se han modelado tradicionalmenteen la industria como reactores ideales: reactores de tanque agitado o deflujo en piston, no podemos olvidarnos de ellos. De hecho, los primerosseran los protagonistas en la parte III. En los capıtulos 1 y 2 derivamos elmodelado matematico de reactores de tanque agitado (batch, semi-batchy CSTR) y reactores de flujo en piston (PFR). Consideramos tanto elcaso transitorio como el estacionario y asumimos que los reactores notienen porque ser adiabaticos ni isotermicos, por lo que la temperatura ylas concentraciones de las especies deben calcularse mediante los modelos

183

que se obtienen a partir de las ecuaciones de conservacion de la masa yde la energıa, respectivamente. Suponemos densidad constante.

En el capıtulo 2, se describe el modelo general de conveccion-difusion-reaccion que es el que despues se reduce a dimension 1 para presentarun caso particular, el PFR. El objetivo de describir este modelo en elCapıtulo 2 es estudiar la existencia y unicidad de la solucion, ademasde analizar el comportamiento del error en la solucion. Todo esto paraobtener finalmente la solucion numerica del modelo.

En el capıtulo 3 derivamos el modelo para reactores de lecho fijo, tambienllamados reactores de lecho empacado (PBR) o reactores catalıticos delecho empacado. Nos centramos en los modelos continuos que se utilizancon frecuencia en algunos procesos industriales. Algunos de ellos son laoxidacion del etileno y la oxidacion del metanol al formaldehıdo. A pesarde la existencia de un tipo mas nuevo de reactores, como los reactores delecho fluidizado, los reactores de lecho fijo se utilizan ampliamente parael procesamiento a gran escala tanto en la industria del petroleo y comoen la industria quımica basica.

De hecho, en la industria se suelen considerar un conjunto de tubos llenosde catalizador, generalmente dispuestos dentro de una gran carcasa delreactor. En este sentido, se supone que la temperatura en el tubo per-manece constante y que las condiciones son iguales en cada tubo (hay unfluido alrededor del exterior de los tubos para mantener una temperaturaadecuada). Aunque estas suposiciones no son ciertas en la practica.

En nuestro caso, el termino ”reactor de lecho empacado” se relaciona conuna unica carcasa cilındrica con cabezas convexas con un lecho fijo departıculas catalıticas de tamano uniforme, que se inmovilizan o se fijandentro del tubo. En ese tubo se introduce una mezcla fluida de reactivosen la entrada del reactor que se mueve a lo largo del reactor e interactuacon las partıculas activas catalıticas. Las reacciones generalmente pro-ducen intercambios de calor. Ası que, si es necesario, la temperatura seregula a traves de la pared del tubo.

Consideramos los FBR como sistemas de reaccion heterogeneos. Se suponeun flujo en piston, es decir, v = vez donde z es la direccion axial.

El modelo para estos reactores se basa en las leyes de conservacion para lamasa, la energıa y el momento y todas ellas nos conducen a un sistema deecuaciones en derivadas parciales. Sin embargo, debido a la complejidaddel sistema, la descripcion de estos reactores debe simplificarse. Por estarazon, existen diferentes modelos de reactores de lecho empacado validos.De hecho, cada problema debe analizarse para hacer las simplificacionesadecuadas. En algunos casos, el reactor puede considerarse como unmodelo pseudo-homogeneo. Si las diferencias entre las fases fluidas y


solidas son significativas, se deben considerar modelos heterogeneos comoes nuestro caso. Ademas, tenemos en cuenta la resistencia intraparticular.

Consideramos el modelo multi-escala. El lecho se modela a nivel de micro-escala como un continuo de pequenas partıculas de material solido quecontienen el catalizador y que interactuan con el fluido (suponemos queestas partıculas son esfericas, pero otras geometrıas como el cilindro sepueden considerar mediante modificaciones directas en el modelo). Elfluido se modela a nivel de macro-escala y a traves de un medio poroso.Para el modelo de macro-escala, el efecto de la micro-escala se representamediante terminos fuente tanto en las ecuaciones de concentracion de es-pecies como en la ecuacion de energıa. A su vez, el modelo de micro-escalase acopla a las magnitudes de macro-escala a traves de sus condicionesde contorno.

II Analisis matematico y solucion numerica

El objetivo de esta parte es hacer un estudio exhaustivo del modelo sobreanalisis matematico y solucion numerica. Nos centramos en los reactoresdescritos en el Capıtulo 2. Como ya hemos mencionado, las ecuacionesen los sistemas de reaccion-difusion se han estudiado desde diferentesenfoques utilizando una variedad de metodos diferentes.

A traves de esta parte, el objetivo es demostrar la existencia global desoluciones para sistemas de reaccion de conveccion-difusion. La prueba deeste teorema se basa en las tecnicas de [55]. En este artıculo, la existencialocal de los sistemas de reaccion-difusion se proporciona a traves de lateorıa de los semigrupos al considerar el problema parabolico semilineal.Nosotros utilizamos esta teorıa para nuestro sistema parabolico de PDEde primer orden. Ademas nos seran necesarias ciertas hipotesis, comola condicion de Holder y la propiedad local de Lipschitz en el terminode reaccion. Esta teorıa nos proporciona una solucion continua unica.Por supuesto, esta solucion se entiende en el sentido debil. Volviendo ala solucion global, las propiedades (P) y (M) que se verificaran debidoa la forma de nuestro termino de reaccion particular (la ley de accionde masas) juegan un papel importante en la prueba de existencia. Lasvariables en nuestro problema representan la concentracion de especies,por lo que la positividad de este es una propiedad natural. Construiremosuna contraccion y el teorema de punto fijo se aplicara para la solucionlocal. En primer lugar, estudiaremos la existencia en L2, pero el objetivoes el lımite de L∞.

Los problemas de control y optimizacion en ingenierıa quımica y sus apli-caciones a menudo requieren muchas simulaciones numericas de sistemasdinamicos a gran escala con diferentes condiciones. Si se desea una es-trategia de control rapida o en tiempo real, una simulacion numerica

185

directa de fuerza bruta no funciona bien. Es importante saber como secomporta el error. En el Capıtulo 5 se obtiene una estimacion de errorsiguiendo las tecnicas en [63]. El enfoque propuesto se aproxima a lafuncion no lineal por su interpolante de Lagrange. Para asegurarnos deque podemos hacer estas estimaciones, necesitamos probar previamentela existencia de la solucion del problema semidiscreto que utilizamos enlas estimaciones. Una vez realizado este estudio, es necesario obtener unasolucion numerica de los modelos que nos interesan desde el punto de vistapractico. Nos centramos en los modelos PFR y FBR. Para el primero us-amos un esquema de diferencias finitas y el metodo de elementos finitospara el modelo FBR.

III Identificacion en sistemas de reaccion

Esta ultima parte esta relacionada con la identificacion del modelo. Masprecisamente, en la identificacion del mejor modelo cinetico de una lista deformas funcionales propuestas y tambien en los valores de sus parametroscorrespondientes en el proceso de optimizacion. Para resolver el prob-lema, utilizamos una combinacion de un metodo incremental y un metodointegral. Tales procesos de identificacion generalmente se estudian en sis-temas donde los fenomenos de interes se pueden observar de forma ais-lada, sin otros fenomenos fısicos. Ese es el caso para la identificacion dela cinetica de una reaccion en fase lıquida, en la que en la mayorıa de loscasos se usa un reactor batch STR o un reactor semi-batch. Eso se explicaen [17]. Por este motivo, nos centramos en los reactores de tanque agi-tado, utilizando un conjunto de datos experimentales y las reacciones quetienen lugar. Tambien proporcionamos un catalogo de modelos cineticosque contienen los parametros a identificar. Describimos tambien la ex-presion general de esta cinetica.

Como decıamos, utilizamos una combinacion de un metodo incrementaly un metodo integral, tambien conocido como metodo de identificacionsimultanea. El uso de estos dos metodos en cascada se puede replicarpara algunos de los reactores mas tıpicos como el reactor de tanque agi-tado (batch, semi-batch o CSTR) y el reactor de flujo en piston; ambosampliamente utilizados en la literatura y la industria. De hecho, una vezque las reacciones son fijas, podemos obligarlas a ocurrir en uno u otroreactor y esto sera lo que hara que nuestra cinetica identificada sea una uotra. Por supuesto, una herramienta importante para tener en cuenta esun conjunto de mediciones en el laboratorio que se incluiran en el ajustede parametros de estas cineticas.

Hay que tener en cuenta que el primer metodo que describiremos se puedeaplicar solo en los tipos de reactor STR y PFR; sin embargo, el segundopodrıa aplicarse incluso a nuestro modelo FBR.


El metodo incremental funciona con el concepto de ”extent”, que nosproporciona una solucion analıtica de un nuevo sistema desacoplado. Enalgunos casos, la solucion dada por el metodo incremental es suficientey podemos concluir el proceso de identificacion, pero no siempre es ası.Mas en detalle, la idea de este metodo es descomponer la tarea de identi-ficacion en un conjunto de subproblemas, uno para cada modelo cinetico.Por tanto, este metodo se caracteriza por el desacoplamiento de las ecua-ciones de reaccion mediante procedimientos algebraicos y la solucion di-recta de las ecuaciones transformadas. Ası, los modelos cineticos y susparametros se pueden identificar en paralelo para todas las reacciones.Los parametros se obtienen a traves de una optimizacion local.

Nosotros presentamos una alternativa al metodo en el que la ecuacionde balance de energıa se trata de forma independiente. La ecuacion devolumen del modelo se puede resolver al principio e independientemente.El resto de EDOs del sistema estan acopladas. Por eso trabajamos endos etapas: el sistema de concentraciones se reescribe como un sistemade extents desacoplado y la ecuacion de temperatura se resuelve de man-era independiente utlizando las medidas experimentales de las concentra-ciones.

Si es necesario mejorar la solucion del metodo incremental, disponemosde una heurıstica basada en la busqueda por entornos (VNS) [47]. Estemetodo utiliza como valores iniciales de los parametros aquellos previ-amente calculados por el metodo incremental. Se generan nuevas solu-ciones en el momento de la ejecucion al hacer perturbaciones sucesivastanto en la cinetica como en los parametros.

El valor del funcional coste del metodo integral se actualiza con la ayudade las derivadas calculadas a traves del metodo del adjunto cuando unanueva combinacion de parametros y modelos cineticos mejoran la solucion.

El metodo integral se basa en una comparacion directa de las medicionesde concentraciones de especies calculadas a traves del modelo teorico.A veces, tambien incluye la comparacion entre temperaturas experimen-tales y teoricas. Debido a esto, este metodo es obviamente mas costosocomputacionalmente.

Las principales dificultades se encuentran en la gran cantidad de parame-tros que aparecen en el funcional coste porque necesitamos identificar almismo tiempo los parametros de cada reaccion; y en el calculo de lasderivadas con respecto a estos parametros en el funcional coste.

Finalmente, necesitamos resolver numericamente el modelo utilizandoun esquema de diferencias finitas (BDF2 inicializado con un BDF1) entiempo en cada evaluacion del funcional coste. Simultaneamente, calcu-lamos las derivadas del funcional utilizando el metodo del adjunto [6].

187

Apendices

El Apendice A resume las ecuaciones generales de la mecanica de medioscontinuos para mezclas de reactantes que pueden verse con mas detalleen el libro [7].

El Apendice C se basa en los libros de Ciarlet [22] y de Ern y Guer-mond [27], donde se presentan el operador de interpolacion de Lagrangeglobal y algunas acotaciones para este operador; tambien se ha utilizado elartıculo de Wang en [66] donde se incluye una equivalencia entre la normaen L2 y una norma euclıdea. En los parrafos siguientes, enunciamos lasdesigualdades mencionadas que son necesarias para las estimaciones deerror del problema semidiscreto en la Seccion 5.2.


189


Bibliography

[1] H. Amann. Existence and regularity for semilinear parabolic evolutionequations. Annali della Scuola Normale Superiore di Pisa - Classe diScienze, 11(4):593–676, 1984.

[2] M. Amrhein, N. Bhatt, B. Srinivasan, and D. Bonvin. Extents of reactionand flow for homogeneous reaction systems with inlet and outlet streams.AIChE Journal, 56:2873–2886, 11 2010.

[3] S. Banach. Sur les operations dans les ensembles abstraits et leur ap-plication auxequations integrales. Fundamenta Mathematicae, 3:133–181,1922.

[4] P. Baras. Non-unicite des solutions d’une equation d’evolution non lineaire.In Annales de la Faculte des Sciences de Toulouse, volume 5, pages 287–302. Universite Paul Sabatier, 1983.

[5] A. Bardow and W. Marquardt. Incremental and simultaneous identifica-tion of reaction kinetics: methods and comparison. Chemical EngineeringScience, 59(13):2673 – 2684, 2004.

[6] M. Benıtez, A. Bermudez, and J. Rodrıguez-Calo. Adjoint method forparameter identification problems in models of stirred tank chemical reac-tors. Chemical Engineering Research and Design, 123:214 – 229, 2017.

[7] A. Bermudez. Continuum Thermomechanics, volume 37 of Progress inMathematical Physics. Birhhauser Verlag, 2005.

[8] A. Bermudez, N. Esteban, J. Ferrın, J. Rodrıguez-Calo, and M. Sillero-Denamiel. Identification problem in plug-flow chemical reactors using theadjoint method. Computers & Chemical Engineering, 98:80 – 88, 2017.

[9] A. Bermudez, E. Carrizosa, O. Crego, N. Esteban, and J. F. Rodrıguez-Calo. A two-step model identification for stirred tank reactors: Incremen-tal and integral methods. In M. Mateos and P. Alonso, editors, Computa-tional Mathematics, Numerical Analysis and Applications: Lecture Notes

191

192 BIBLIOGRAPHY

of the XVII ’Jacques-Louis Lions’ Spanish-French School, pages 213–220,Cham, 2017. Springer International Publishing.

[10] N. Bhatt, M. Amrhein, and D. Bonvin. Incremental identification of reac-tion and mass–transfer kinetics using the concept of extents. Industrial &Engineering Chemistry Research, 50(23):12960–12974, 08 2011.

[11] N. Bhatt, M. Amrhein, B. Srinivasan, P. Mullhaupt, and D. Bonvin. Min-imal state representation for open fluid-fluid reaction systems. In Proceed-ings of the American Control Conference, pages 3496–3502, 06 2012.

[12] N. Bhatt, N. Kerimoglu, M. Amrhein, W. Marquardt, and D. Bonvin. In-cremental identification of reaction systems—a comparison between rate-based and extent-based approaches. Chemical Engineering Science, 83:24– 38, 2012. Mathematics in Chemical Kinetics and Engineering Interna-tional Workshop 2011.

[13] R. Blanquero, E. Carrizosa, O. Chis, N. Esteban, A. Jimenez-Cordero,J. F. Rodrıguez, and M. R. Sillero-Denamiel. On extreme concentrationsin chemical reaction networks with incomplete measurements. Industrial& Engineering Chemistry Research, 55(44):11417–11430, 2016.

[14] R. Blanquero, E. Carrizosa, A. Jimenez-Cordero, and J. Fran-cisco Rodrıguez. A global optimization method for model selection inchemical reactions networks. Computers & Chemical Engineering, 93:52–62, 2016.

[15] N. Boudiba. Existence Globale pour des Systemes de Reaction-Diffusionavec Controle de Masse. PhD thesis, Universite de Rennes 1, France, 1999.

[16] N. Boudiba and M. Pierre. Global existence for coupled reaction–diffusionsystems. Journal of Mathematical Analysis and Applications, 250(1):1–12,2000.

[17] M. Brendel. Incremental identification of complex reaction systems. PhDthesis, RWTH Aachen, 2005.

[18] M. Brendel, D. Bonvin, and W. Marquard. Incremental identification ofkinetic models for homogeneous reaction systems. Chemical EngineeringScience, 61(16):5404 – 5420, 2006.

[19] S. Brenner and L. Scott. The mathematical theory of finite element meth-ods, volume 15 of Texts in applied mathematics. Springer-Verlag New York,2008.

BIBLIOGRAPHY 193

[20] H. Brezis. Operateurs Maximaux Monotones et Semi-Groupes de Contrac-tions dans les Espaces de Hilbert, volume 5 of North-Holland MathematicsStudies. North Holland, 1973.

[21] S. C. Burnham and M. J. Willis. Determining reaction networks. InR. M. de Brito Alves, C. A. O. do Nascimento, and E. C. Biscaia, editors,10th International Symposium on Process Systems Engineering: Part A,volume 27 of Computer Aided Chemical Engineering, pages 561 – 566.Elsevier, 2009.

[22] P. Ciarlet. The finite Element Method for Elliptic Problems. Classics inApplied Mathematics. SIAM, 2002.

[23] E. Coddington and L. Levinson. Theory of Ordinary Differential Equa-tions. McGraw-Hill, 1987.

[24] A. K. Coker. Modeling of Chemical Kinetics and Reactor Design. GulfProfessional Publishing, 2001.

[25] L. Desvillettes, K. Fellner, M. Pierre, and J. Vovelle. About global ex-istence for quadratic systems of reaction-diffusion. Advanced NonlinearStudies., 7(3):491–511, 2007.

[26] H. B. (Eds.). Operateurs Maximaux Monotones: Et Semi-Groupes DeContractions Dans Les Espaces De Hilbert. North-Holland MathematicsStudies 5. North Holland, 1st edition, 1973.

[27] A. Ern and J.-L. Guermond. Theory and Practice of Finite Elements.Applied Mathematical Sciences 159. Springer-Verlag New York, 2004.

[28] W. B. Fitzgibbon, S. L. Hollis, and J. J. Morgan. Stability and lyapunovfunctions for reaction-diffusion systems. SIAM Journal on MathematicalAnalysis, 28(3):595–610, 1997.

[29] G. L. Foutch and A. H. Johannes. Reactors in process engineering. Ency-clopedia of Physical Science and Technology (Third Edition), pages 23–43,12 2003.

[30] P. Grisvard. Elliptic Problems in Nonsmooth Domains. Pitman, London,1983.

[31] C. M. Guldberg and P. Waage. Studies concerning affinity. Journal ofchemical education, 63(12):1044, 1986.

[32] A. Haraux and F. B. Weissler. Non-uniqueness for a semilinear initialvalue problem. Indiana University Mathematics Journal, 31(2):167–189,1982.

194 BIBLIOGRAPHY

[33] I. Higueras. Positivity properties for the classical fourth order runge-kuttamethod. Monografıas de la Real Academia de Ciencias de Zaragoza, 33:125– 139, 2010.

[34] A. V. Hill. The possible effects of the aggregation of the molecules ofhæmoglobin on its dissociation curves. The Journal of Physiology, 40:i–vii, January 1910.

[35] S. L. Hollis, R. H. Martin, and M. Pierre. Global existence and bounded-ness in reaction-diffusion systems. SIAM Journal on Mathematical Anal-ysis, 18(3):744–761, 1987.

[36] V. I. Istratescu. Fixed Point Theory, An Introduction. Springer Nether-lands, 1981.

[37] P. Knabner and L. Angermann. Numerical Methods for Elliptic andParabolic Partial Differential Equations. Springer-Verlag New York, 2003.

[38] S. Kouachi. Existence of global solutions to reaction-diffusion systemsvia a lyapunov functional. Electronic Journal of Differential Equations,2001(68):1–10, 2001.

[39] E. H. Laamri and M. Pierre. Global existence for reaction–diffusion sys-tems with nonlinear diffusion and control of mass. Annales de l’InstitutHenri Poincare (C) Non Linear Analysis, 34(3):571–591, 2017.

[40] O. A. Ladyzenskaya, V. A. Solonnikov, and N. N. Ural’ceva. Linear andQuasi-linear Equations of Parabolic Type. American Mathematical So-ciety, translations of mathematical monographs. American MathematicalSociety, 1988.

[41] J. Lions. Quelques methodes de resolution des problemes aux limites nonlineaires. Etudes mathematiques. Dunod, 1969.

[42] L. Lorenzi, A. Lunardi, G. Metafune, and D. Pallara. Analytic semigroupsand reaction-diffusion problems.

[43] A. Lunardi. Analytic Semigroups and Optimal Regularity in ParabolicProblems. Springer Basel, 1995.

[44] E. Lund. Guldberg and waage and the law of mass action. Journal ofChemical Education, 42(10):548, 1965.

[45] R. H. Martin and M. Pierre. Nonlinear reaction-diffusion systems. InW. Ames and C. Rogers, editors, Nonlinear Equations in the Applied Sci-ences, volume 185 of Mathematics in Science and Engineering, pages 363– 398. Elsevier, 1992.

BIBLIOGRAPHY 195

[46] L. Michaelis and M. L. Menten. Die kinetik der invertinwirkung. Biochem,49:333 – 369, 1913.

[47] N. Mladenovic and P. Hansen. Variable neighborhood search. Computers& Operations Research, 24(11):1097 – 1100, 1997.

[48] J. Morgan. Global existence for semilinear parabolic systems via lya-punov type methods. In T. L. Gill and W. W. Zachary, editors, NonlinearSemigroups, Partial Differential Equations and Attractors, pages 117–121,Berlin, Heidelberg, 1989. Springer.

[49] J. Morgan. Boundedness and decay results for reaction-diffusion systems.SIAM Journal on Mathematical Analysis, 21(5):1172–1189, 1990.

[50] J. Morgan. Global existence for semilinear parabolic systems on one–dimensional bounded domains. Rocky Mountain J. Math., 21(2):715–725,1991.

[51] J. Necas. Direct methods in the theory of elliptic equations. SpringerScience & Business Media, 2011.

[52] R. Nittka. Regularity of solutions of linear second order elliptic andparabolic boundary value problems on lipschitz domains. Journal of Dif-ferential Equations, 251(4):860 – 880, 2011.

[53] C. V. Pao. Nonlinear Parabolic and Elliptic Equations. Plenum, NewYork, 1992.

[54] A. Pazy. Semigroups of operators in banach spaces. In H. W. Knoblochand K. Schmitt, editors, Equadiff 82: Proceedings of the internationalconference held in Wurzburg, FRG, August 23–28, 1982, pages 508–524,Berlin, Heidelberg, 1983. Springer Berlin Heidelberg.

[55] M. Pierre. Global existence in reaction-diffusion systems with control ofmass: a survey. Milan Journal of Mathematics, 78(2):417–455, 2010.

[56] M. Pierre and D. Schmitt. Blowup in reaction-diffusion systems withdissipation of mass. SIAM Review, 42(1):93–106, 2000.

[57] H. F. Rase. Fixed-Bed Reactor Design and Diagnostics. Butterworth-Heinemann, 1990.

[58] B. Riviere. Discontinuous Galerkin Methods for Solving Elliptic andParabolic Equations: Theory and Implementation. SIAM. Philadelphia,2008.

196 BIBLIOGRAPHY

[59] D. Rodrigues, J. Billeter, and D. Bonvin. Incremental model identificationof distributed two-phase reaction systems. IFAC-PapersOnLine, 48(8):266– 271, 2015. 9th IFAC Symposium on Advanced Control of ChemicalProcesses ADCHEM 2015.

[60] F. Rothe. Global Solutions of Reaction-Diffusion Systems, volume 1072 ofLecture Notes in Mathematics. Springer, Berlin, 1984.

[61] D. Schmitt. Existence Globale ou explosion pour les Systemes de Reaction-Diffusion avec Controle de Masse. PhD thesis, Universite Henri Poincare,1995.

[62] G. Stampacchia. Le probleme de dirichlet pour les equations elliptiquesdu second ordre a coefficients discontinus. Ann. Inst. Fourier (Grenoble),15(1):198–258, 1965.

[63] V. Thomee. Galerkin Finite Element Methods for Parabolic Problems.Springer Series in Computational Mathematics. Springer Berlin Heidel-berg, 2013.

[64] J. Tsu, V. H. G. Dıaz, and M. J. Willis. Computational approaches tokinetic model selection. Computers & Chemical Engineering, 121:618 –632, 2019.

[65] A. M. Turing. The chemical basis of morphogenesis. Philosophical Trans-actions of the Royal Society of London. Series B, Biological Sciences,237(641):37 –72, 1952.

[66] Z. Wang. Nonlinear model reduction based on the finite element methodwith interpolated coefficients: Semilinear parabolic equations. NumericalMethods for Partial Differential Equations, 31(6):1713–1741, 2015.

Documents

1&'..+0) /#6*'/#6+%#.#0#.;5+5 07/'4+%#.51.76+10#0&2#4#/'6 ... · Preface In general terms, a chemical reactor can be understood as a vessel used in trans-forming the initial chemical