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University of Dortmund

Department of Bio and Chemical Engineering

Chair of Technical Chemistry-Reaction Engineering

Prof. Dr. David W. Agar

Master Thesis

Comparison between Advanced Finite Element and Finite Difference Methods for the Direct Calculation of Cyclic Fixed

Bed Processes

by Srinivas Nanam

Examiner: Prof. Dr. David W. Agar Co-Examiner: Prof. Dr. Stefan Turek Supervisor: Dipl-Ing. Frank Platte Date: Friday, 18th March 2005

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TABLE OF CONTENTS ACKNOWLEDGEMENT................................................................................................................ 7 ABSTRACT .............................................................................................................................. 8 1. INTRODUCTION................................................................................................................ 9

1.1 BACKGROUND ................................................................................................................... 9 1.2 CYCLIC FIXED BED PROCESSES ....................................................................................... 10 1.3 APPLICATION EXAMPLE ................................................................................................... 12 1.4 NUMERICAL DEMANDS FOR CATALYTIC COMBUSTION IN REVERSE FLOW OPERATION ..... 13

2. NUMERICAL TREATMENT FOR CYCLIC FIXED BED PROCESSES................. 14 2.1 MODELLING OF REVERSE FLOW REACTOR ....................................................................... 14

2.1.1 Modelling of reverse flow reactor for N2O decomposition .................................... 14 2.2 PROPERTIES OF NUMERICAL METHODS ............................................................................ 15 2.3 SOLUTION OF CYCLIC PROFILES IN A CYCLIC FIXED BED REACTOR................................... 16

2.3.1 Dynamic simulation................................................................................................. 16 2.3.2 Direct calculation method....................................................................................... 16

2.4 NUMERICAL MODELS....................................................................................................... 17 2.5 THE NEED FOR STABILIZATION......................................................................................... 18

2.5.2 Steep solution gradients .......................................................................................... 19 2.6 STABILIZATION TECHNIQUES ........................................................................................... 19

3A TEST PROBLEMS USING FEM................................................................................... 21 3.1A ABOUT FEMLAB........................................................................................................... 21 3.2A CAUCHY PROBLEM ........................................................................................................ 21

3.2.1a Solution approach ................................................................................................. 22 3.2.2a Stabilization Techniques ....................................................................................... 25 3.2.3a Comparison with analytical solution .................................................................... 27

3.3A BURGERS PROBLEM..................................................................................................... 29 3.3.1a Comparison with analytical solution .................................................................... 30

3.4A COUPLED REACTION ENGINEERING PROBLEM................................................................ 32 3.4.1a Solution approach ................................................................................................. 32

3B TEST PROBLEMS USING FINITE DIFFERENCE METHOD................................ 35 3.1B TVD SCHEMES.............................................................................................................. 35

3.1.1b TVD Methodology ................................................................................................. 35 3.1.2b Discrete upwinding ............................................................................................... 36

3.2B CAUCHY PROBLEM ........................................................................................................ 38 3.2.1b Leapfrog scheme .................................................................................................. 39

3.3B BURGERS EQUATION.................................................................................................... 47

4. DECOMPOSITION OF N2O IN REVERSE FLOW REACTOR................................. 49 4.1 INTRODUCTION................................................................................................................ 49 4.2 REACTION MECHANISM FOR N2O DECOMPOSITION.......................................................... 50

4.2.1 Kinetic modelling .................................................................................................... 51 4.3 HYBRID REACTION........................................................................................................... 52

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4.3.1 Separation of hybrid reaction ................................................................................. 52 4.4 REACTOR MODELLING ..................................................................................................... 53 4.5 COUPLING MECHANISM ................................................................................................... 54

4.5.1 TC model ................................................................................................................. 54 4.5.2 TCC model............................................................................................................... 56 4.5.3 Model with Enhancement factor ............................................................................. 56

FIG 4.7 HYBRID REACTION RATE FOR TC MODEL AND WITH ENHANCEMENT FACTOR ........... 57 4.6 SURFACE COVERAGE DEPENDANT MODEL........................................................................ 57

4.6.1 Influence of activation energy on N2O decomposition............................................ 58 4.6.2 Effect of NO on N2O decomposition........................................................................ 58 4.6.4 Production of NO .................................................................................................... 60 4.6.4 Kinetic oscillations in N2O decomposition.............................................................. 61

5. CONCLUSION................................................................................................................... 62 REFERENCES....................................................................................................................... 63 APPENDIX ............................................................................................................................. 65

APPENDIX 1A........................................................................................................................ 65 APPENDIX 1B ....................................................................................................................... 67 APPENDIX 2........................................................................................................................... 69

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List of Figures

Fig 1.1 Principles of reverse flow operation: (a) Temperature (T) and conversion(X) profiles of a moving reaction front; (b) temperature and conversion profiles at the end of the first two semi cycles.6

Fig 1.2 Scheme of reverse flow reactor: Cyclic opening and closing of thevalve pairs V1/V3 and V2/V4..........8

Fig 2.1 Pattern of different numeric methods...11

Fig 3.1 Representation of computational domain and boundary conditions16

Fig 3.2 Computational domain with unstructured mesh with 900 elements (left) and structured mesh with 1600 elements.17

Fig 3.3 Solution to Cauchy problem with triangular coarse mesh with 900 elements in 2D (left) and 3D (right)..18

Fig 3.4 Solution to Cauchy problem with quadrilateral coarse mesh with 1600 elements in 2D (left) and 3D (right) .18

Fig 3.5 Solution of Cauchy problem with Stream line artificial diffusion and with mesh refinement level 2.19

Fig 3.6 Solution Cauchy problem with Cross wind artificial diffusion and with mesh refinement level 2.20

Fig 3.7 Solution with stream line artificial diffusion using quadrilateral mesh with 14400 elements (left) and 60000 elements (right)...20

Fig 3.8 Representation of error for refinement level 1 and without any stabilization .21 techniques Fig 3.9 Error norm for different stabilization techniques using triangular elements...22 Fig 3.10 Error norm for different stabilization techniques using quadrilateral elements22

Fig 3.11 Solution to Burgers problem with triangular (left) and quadrilateral mesh (right)..24 Fig 3.12 Representation of error for Burgers problem...25

Fig 3.13 Representation of error norm with and without artificial diffusion....25

Fig 3.14 Concentration profiles for species 1(left) and 2(right)...27

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Fig 3.15 Concentration profiles for species 3(left) and 4(right)...27

Fig 3.16 concentration profiles for different species using time dependant solver..28

Fig 3.17 Schematic diagram of a space- time grid...32

Fig 3.18 Sparse matrix with 5points in space and time for leap frog scheme 33

Fig 3.19 3D plot for a leap frog scheme (left) and the final value of the step (right)..34

Fig 3.20 Representation of the error using leap frog scheme..35 Fig 3.21 Convergence behaviour for velocity = 0.5 for leap frog scheme...36

Fig 3.22 Convergence behaviour for velocity = 0.2 for leap frog scheme...36

Fig 3.23 Sparse matrix for CDS/PW37

Fig 3.24 Representation of solution in 3D (left) and the final value of the step using 38 CDS/UPW Fig 3.25 Convergence behaviour for CDS/UPW.39

Fig3.26 3D plot of the step and the corresponding final value for imp.Euler/CDS40

Fig 3.27 Convergence behaviour for Impeuler/CDS40

Fig 3.28 Numerical solution to Burgers problem using FD method...42

Fig 4.1 Schematic Diagram of reverse flow reactor.44

Fig 4.2 Scheme of reaction for a conventional catalyst (left) and for a structured catalyst (right). Course of temperature T, concentration C and reaction rates (rhyb,rhom, rhet) are shown. Grey and white areas depicts active and inert section..46

Fig 4.3 Schematic representation of the computational domain..46

Fig 4.4 Concentration and Temperature profile in N2O decomposition..48

Fig 4.5 Reaction rate for pure heterogeneous and for hybrid reaction48

Fig 4.6 Comparison between TC and TCC model with a correction term gamma=0.2..49 Fig 4.7 Hybrid reaction rate for TC model and with enhancement factor50

Fig 4.8 Schematic representation of proposed interaction...51

Fig 4.9 Concentration profile for NO for two cycles....53

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List of Tables Table 1 Physical and Chemical parameters for N2O decomposition...13

Table A1 Results for Cauchy problem with triangular elements.61

Table A2 Results for Cauchy problem with quadrilateral elements61

Table A3. Results for Burgers problem with triangular elements..62 Table A4. Results for Burgers problem with quadrilateral elements.62 TableB1 Results for the step problem with velocity=0.5 using leap frog scheme..63

Table B2. Results for the step problem with velocity=0.2 using leap frog scheme.63

Table B3. Results for CDS in time and Upwind in space for different grid points.64

Table B4. Results Implicit Euler in time and CDS in space for different grid points.64 List of Symbols

C [mol/m3] Concentration

Cp [J/mol.K] Specific Heat

[Kg/m3] Density [-] Porosity T [K, OC] Temperature

eff [J/m2..K] Effective thermal conductivity R [J/mol K] gas constant

r [mol/m3S] Reaction rate

rH [J/mol] Reaction Enthalpy cyct [s] Cycle time

k [1/s] Rate constant

p [bar] Partial Pressure

[-] Vacant active site EA [J/mol] Activation energy

NT [mol/gcat] Active site concentration

L [m] Reactor length

T [s] Time

v [m/s] velocity

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Acknowledgement I would like to express my sincere gratitude to all of those who help me to complete this

thesis. First and foremost, I have to thank my thesis supervisor, Mr. Frank Platte for the

incredible amounts of time and effort he contributed to this thesis. It was a great pleasure for

me to conduct this thesis under his supervision.

Second I would like to acknowledge Prof. Dr. David W.Agar and Prof Dr. Turek for their

valuable feedback helped me to improve the thesis in many ways. Further more I would like

to thank Mr. Shu-Ren Hysing for his support.

Last but not least, I would also like to thank my parents for supporting me through all these

years. I am grateful for their invaluable support.

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ABSTRACT Cyclic fixed bed processes have proven to be a cost effective technique for pollutant removal

in industrial application. In this thesis we investigated N2O decomposition in reverse flow

reactor. When modelling reactor usually yields PDEs, but due to numerical problems these

exhibit shocks and sharp fronts due to convection term and high reaction rates. To overcome

these numerical problems, initially different test problems were solved using finite element

method and finite difference method with different types of stabilization techniques. For the

treatment N2O decomposition different coupling mechanisms were investigated. Lastly the

coupling mechanism was investigated by introducing a new species and theoretically

investigated the effect of that new species on the N2O decomposition.

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CHAPTER 1

1. INTRODUCTION

1.1 Background

Fixed bed reactors are widely used in chemical industry. They facilitate a wide variety of

processes and catalytic reactions, this large variety of processes results in different types of

fixed bed reactors like adiabatic single bed, multi-tube surrounded with heat exchange fluid

and adiabatic multi tube with internal heat exchange. Normally catalytic reactions are of two

types i.e. catalytic liquid phase and catalytic gas phase.

Catalytic liquid phase reactions are carried out as homogeneous catalytic reactions in stirred-

tank reactors in batch, semi-batch, or continuous mode of operation with dissolved

organometal complexes as catalysts. The reactions of commercial interest are

hydroformylations, carbonylations, polymerizations, hydrogenations, and oxidations. In

addition, tubular reactors are also used for liquid-phase reactions like polymerization,

hydrogenation, hydrolysis, or dehydroclorination.

Catalytic gas-phase reactions play a major role in refinery, petrochemistry, and industrial

organic technology. Various bulk products and intermediates are manufactured in this way

using solid catalysts. The reactions take place in tubular reactors which are designed as fixed-

bed or fluidized-bed reactors, depending upon the heat of reaction and the thermodynamic

stability of the products formed. The least expensive kind to build is an adiabatic reactor with

a fixed bed of catalyst and without internals for transferring heat. They are generally more

practical for large-scale and relatively slow reactions without large heat effects. Several beds

may be used in series (multistage reactor) so that the reacting gas can be cooled between beds

or a cool reactant gas is injected as quench gas between them. Adiabatic fix-bed reactors are

also used for highly reactions which occur extremely rapidly at high temp. (e.g.,

formaldehyde synthesis). The very short reaction times needed, on the order of milliseconds,

are realized by a very thin layer of catalysts (< l cm) or by using gauzes (also called shallow-

bed reactors, metal gauze reactors). For more exothermic reactions which are temperature.-

sensitive relating to the product selectivity, multitube fixed-bed reactors with external cooling

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are used. The coarse catalyst particles (2 to 8 mm in diameter) are used to fill in several

thousands (up to 27 000) of tubes having diameters from 2 to 5 cm and lengths from 1 to 10

m. A variety of reactions such as partial oxidations (e.g., maleic anhydride from butane,

ethylene oxide), partial hydrogenations (low-temperature hydrogenation), dehydrogenation

(of ethyl benzene to styrene), as well as oxichlorination, isomerization, and cracking

reactions, take place in multitube reactors. Special types of fixed-bed reactors are build with

catalyst monoliths or microstructured wafers (monolitic honeycomb). Another type of fixed-

bed reactor is the micro channel reactor, designed with a stack of microstructured, typically

metallic, wafers.

Traditionally adiabatic fixed bed reactor can be operated with a separate heat exchanger but

more recently the concept of integrating the apparatus into the reactor has received much

attention, such type of reactors are called multi functional reactors. One of the simplest and

most common examples of multi functional reactors are auto thermal reactors for weakly to

moderately exothermic reactions, where the cold feed is heated up to the reaction temperature

by the hot reactor effluent. Auto thermal reactor can be operated in counter currently or with

periodically switching the flow direction. The reactor which is operated with periodic flow

reversal has considerable advantages and the will discussed in the following section.

1.2 Cyclic Fixed Bed Processes

When fixed bed reactor is started by heating the catalyst bed above the ignition temperature of

the exothermic reaction considered, the feed temperature can subsequently be lowered to

ambient. As a consequence a moving reaction front develops where the bed is cooled by the

cold feed and the feed is heated up by hot bed (fig 1.1a). As long as the moving front is still

inside the catalyst bed, full conversion will be obtained but if it moves out the reactor will

extinguish. The best way to reignite the reaction is to reverse the flow direction after the front

has moved to a certain position into the bed and to repeat this flow reversal periodically until

a periodic steady state has been established. Fig 1.1b shows the profiles at the end of the first

and the second semi cycle. Under periodic flow reversal the both ends of the fixed bed are

obviously used as regenerative heat exchangers. Since regenerative heat exchange is generally

considered simpler and more efficient than recuperative heat exchange, the reverse flow

reactor has found considerable industrial application primarily for the catalytic or

homogeneous combustion of organic pollutants in exhaust air.

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(a)

(b)

Fig 1.1 Principles of reverse flow operation: (a) Temperature (T) and conversion(X) profiles of a moving reaction front; (b) temperature and conversion profiles at the end of the first two

semi cycles.

Due to their regenerative nature these processes have to be operated in a cyclic mode with

usually one loading step and one or several regeneration steps. For example in case of

adsorptive air purification process the loading step is where the pollutant is adsorbed by the

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adsorbent and clean air is the product released from the outlet. The regeneration can for

example be achieved by purging the adosorber bed with the hot product. In order to obtain a

continuous process, multi bed setups are commonly used. Although it is not too difficult to

design loading and regeneration steps separately, it is much more complicated to design an

entire process cycle where the end of the loading cycle is the initial state of the regeneration

and vice versa. If, for example, in the above mentioned adsorption process the regeneration is

incomplete, the pollutant increases to accumulate in fixed bed during each cycle until the

required clean air specification can no longer be met. Generally small changes in one cycle

can accumulate during cyclic operation and strongly influence the cyclic steady state. From a

practical point of view the state where the process repeats itself after one cycle is of primary

interest. This is called Cyclic Steady State. This is the reason why the proper design and scale

up of cyclic processes often requires both, a large experimental effort and long time practical

experience. There is a great need for rigorous methods for the efficient analysis and design of

cyclic processes.

Multi functional processes are either inherently instationary or forced instationary. Their

behaviour based on properly chosen operating conditions allow additional enhancement of the

performance. Moreover, the time dependent behaviour leads to more data which can be

exploited for model evaluation. Unfortunately, due to the non-linearity and stiff ness of these

instationary fixed bed processes, experiments and simulations are rather time-consuming

projects.

1.3 Application example

1.3.1 Catalytic Combustion in reverse flow operation

One important instationary process is the reverse flow reactor which is operated in a forced

periodical way by switching the side/direction of the flow (Fig 1.2).

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V1 V2 V1 V2

V3 V4

Fig 1.2 Scheme of reverse flow reactor: Cyclic opening and closing of thevalve pairs V1/V3 and V2/V4

The classical Reverse Flow Reactor (RFR) operates with two identical half cycles, i.e. the

reaction and regeneration steps are fulfilled simultaneously. Due to an inherently low heat

loss weak exothermic processes can remain ignited without additional heat or fuel gas.

Suggested examples for industrial application are the treatment of waste gases in air,

oxidation of SO2 and many more.

It is well known that the RFR reaches the cyclic steady state after a long operation time and a

large number of flow reversals. Moreover, high reaction rates at elevated temperature levels

lead to sharp fronts in the distribution of temperature and concentration.

1.4 Numerical demands for catalytic combustion in reverse flow operation In order to overcome these sharp fronts, one should need

1. Good numerical algorithm

2. Direct calculation of cyclic steady state

3. Stabilization techniques

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CHAPTER 2

2. NUMERICAL TREATMENT FOR CYCLIC FIXED BED PROCESSES

2.1 Modelling of reverse flow reactor The mathematical modelling of catalytic reactor operation with periodic flow reversal has

received attention by Eigenberger and Nieken (1988), who have used a pseudo homogenous

packed bed model, neglecting gas- solid temperature differences. Heterogeneous models

allowing for such differences have been conducted by Matros (1988), Gawdzik and Rakowski

(1988, 1989) and Bhatia (1991). A common feature of all these studies is the large amount of

computational time required to calculate cyclic profiles attained in the reactor. One should

need a good numerical algorithm to solve the model equations and a direct calculation method

for the calculation of cyclic profiles.

2.1.1 Modelling of reverse flow reactor for N2O decomposition

In this work we are dealing with the hybrid decomposition of N2O, that is combination of

both homogeneous and heterogeneous, in reverse flow reactor. The heat and mass balance

yields two partial differential equations in order to describe the distribution of temperature

and concentration. Due to the dominating behaviour of heat balance, we regard the mass

balance as to be pseudo steady state. The system of equations can be represented in the

following expression

( )( ) ( ) ( ) ( ) ( )2 21 ,ambp eff p r hybs gT T T kc c T T H r T Ct z z r = + + (1.1) ( )2 20 ,eff hybC CD v r T Cz z

= (1.2)

where rhyb is the rate of hybrid reaction, it is the combination of both homogeneous and

heterogeneous reaction rates

( ) hom. . ( , ), ( , ) .. ( , )TC v hethyb v heta k T Cr T C k T C C

a k T C

= + + (1.3)

The physical and chemical parameters are shown in Table1. As mentioned in section 1.3 in

order to solve the above PDEs one need good numerical algorithm, which stabilizes the

convective term and also at high reactions rates it produces sharp profiles

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symbol value Unit symbol value unit

Av 1100 m2/m3 0.69 -

L 1.5 M D 0.12 M

dhyd 0.002 M C0 0-0.1 Mol/m3

Deff,g 0.00691 m2/s cp,g 1093 J/kg.k

g 0.1 W/mK varrhog 0.486 Kg/m3

s 1.26 w/mK cp,s 840 J/kgK

eff 0.85 W/mK s 1645 Kg/m3

kwv 30 w/m2K Tamb 300 K 2N O

rH -81.6 . 103 J/mol kohom 4.4 . 1011 1/s EAhom 2.5 . 105 J/mol Kohet 3.0 . 108 1/s

EAhet 1.5 . 105 J/mol R 8.3144 J/molK

91 W/m2K tcyc 18-180 S

0.18 m/s vg 0.4 m/s

Table 1 Physical and Chemical parameters for N2O decomposition

2.2 Properties of numerical methods A good numerical algorithm should fulfil the following properties

1. Consistency: The discretization of a PDE should be exact as the mesh size tends to

zero(truncation errors should vanish)

2. Stability: Numerical errors which are generated during the solution of discretized

equations should not be magnified.

3. Convergence: The numerical should solution should approach the exact solution of the

PDE and converge to it as the mesh size tends to zero.

4. Conservation: Underlying conservation laws should be respected at the discrete level

(artificial sources or sinks are to be avoided)

5. Bounded ness: Quantities like density, temperature and concentration should remain

non negative and free of spurious wiggles.

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2.3 Solution of cyclic profiles in a cyclic fixed bed reactor As mentioned in the earlier, the modelling of a reactor yields PDE, in order to calculate the

cyclic profiles, there are different methods available, which are illustrated in fig 2.1

Method of lines Global discretization

Dynamic Schiess Newton Method

Simulation Procedure

Fig 2.1 Pattern of different numeric methods

2.3.1 Dynamic simulation

Due to the inherently dynamic nature of these processes dynamic simulation is the first choice

for the theoretical approach. By using this method the analytically insolvable PDEs are

Reactor

PDE

1 2 3 ( )t zz za u a u a u f u= +

Zero Equation

R(u)

Dynamic Behavior Cyclic Stationary Behavior

DAE-System

( , ) . ( , )uu t f u tt

=

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discretised in space and the resulting ODE-system is integrating in time. This method is called

as method of lines. This method computes the cyclic steady state by calculating the full

transient from an initial state to the cyclic steady state. However the computation of cyclic

steady state by direct dynamic simulation is not always straight forward. This has been proved

in the research work by J. Unger, G. Kolios, G.Eigenberger [1]. They took a simple transport

equation with appropriate initial and boundary conditions and simulated the process using

method of lines, and then they have observed a very large number of cycles greater than 5000

is required to obtain the cyclic steady state. The above mentioned authors summarized that it

can be said that the dynamic simulation is a valuable tool for getting a detailed understanding

of the dynamics of cyclically operated processes.

It also monitors the accumulation of the small errors over a large number of cycles. However,

if the transition from the initial state to cyclic steady state is slow, dynamic simulation

requires a large computational effort since a large number of cycles has to be computed. So,

for the design of cyclically operated processes dynamic simulation is not a suitable choice.

2.3.2 Direct calculation method [2]:

For the calculation of cyclic steady states this method makes use of periodicity conditions that

are treating the differential equation system as a boundary value problem in time, with the

stationary profiles at the beginning and the end of the half cycle being mirror images of each

other. This allows direct solution of the cyclic profiles independent of the initial conditions,

and dispenses with the need of repeatedly solving the problem for an enormous number of

cycles until stationarity is attained. By this the computation can be reduced by a factor of 4-7

and can be used for any value of cycle time.

A typical mirror symmetry and symmetry boundary condition in time in for the temperature in

the case of RFR is

( ) ( ), , / 2cycT Z t T L Z t t= + ( ) ( ), , / 2cycT Z t T Z t t= +

2.4 Numerical models To illustrate the change of a property with respect to one or more independent variables, one

needs to solve these PDEs. The best and most precise method is solving these equations

analytically. Analytical solutions are typically obtained in closed form and represent exact

solutions; however, they are available only for very simple or idealized configurations.

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Obtaining the analytical solution for a given differential equation is not always easy, and in

many cases is absolutely impossible. As a result, numerical methods were introduced.

In fact, solving differential equations numerically meant coding the method into an iterative

procedure and looping over and over again to produce a perfectly usable form of the answer

in many applied problems. It has become a major tool in the engineering field, as computing

power has increased to such a large extent. Some of the popular techniques used in

engineering applications are

1. Finite Difference method (FD)

2. Finite Element method (FE)

3. Finite volume method (FV)

In this work we choose global discretization based on direct calculation method which is

discretizing both in space and in time. Alternatively, a direct calculation based on dynamical

simulation is also a good method but global discretization allows better implementation of

modern algorithms developed for 2D/3D problems.

In this work we restrict only to FD and FE methods.

2.5 The need for stabilization

Using numerical methods in a straight forward way for the approximation of PDEs may cause

severe problems. Oscillations, excessive numerical diffusion and singular matrices, may be

the result of disregarding important basic rules related with a certain concrete problem. Then,

stabilization is needed. In this section, it is described under which circumstances problems

occur and stabilization may be needed to obtain satisfactory approximations.

2.5.1 Convection-dominated problems The phenomenon of convection, typically identified by first order terms in the differential

equations of a model, divides the usability of the methods.

The approximation of a convection dominated problem using FEM shows spurious

oscillations in the solutions, worsening with growing convection domination. This does not

lead to qualitatively bad results but even violates basic physical principles like positive

boundedness. One finds that the pollution of the solution with oscillations is dependent on the

domination of the convective terms over other terms of the differential equation, like diffusion

terms. The role of convection in differential equation is defined by well known identification

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numbers such as Peclet number and Reynolds number. The higher these numbers are, the

more dominant the convection term is and the stronger the pollution with oscillations.

In FDM context it is well known that upwind differencing on the convective term does not

show oscillatory solutions, but introduces excessive numerical diffusion. A simple Taylor

series analysis proves that upwinding is only first order accurate, in contrast to the second

order accurate but oscillatory central differences. This analysis also elucidates that

upwinding can also be interpreted as central difference plus artificial diffusion. Thus the right

combination of central and upwind difference may introduce the optimal amount of artificial

diffusion which leads to accurate and oscillation free solutions. Thus the idea of including

upwind effects in FEM has given interest to many authors, and then they developed different

stabilization techniques.

2.5.2 Steep solution gradients

In the above section, it has been shown that convection-dominated problems require

stabilization such that a pollution of the overall solution with oscillations is prevented.

However, these stabilizations do not preclude over- and undershooting about sharp internal

and boundary layers [34]. These localized (in that they do not influence the whole domain)

oscillations can be suppressed by getting control over the solution gradient. The aim is to

obtain a monotone solution without any oscillations. There is a severe restriction concerning the monotonicity of a numerical scheme that is no linear higher order method can obtain

monotone solutions. Thus there are only two ways to achieve monotonicity: Using first order

accurate schemes such as upwind finite difference or using non linear schemes. As higher

order accuracy is essential in the reliable simulation of many problems, so upwinding finite

difference scheme is no real alternative, consequently non linear schemes have to be

developed.

In finite difference context this can for example be realized with the so called slope limiter

methods, a subclass of Total Variation Diminishing (TVD) schemes. Recently there has been

considerable research going on implementing monotone methods on FEM context.

2.6 Stabilization techniques 1. Streamline-Upwind/ Petrov-Galerkin (SUPG):

This method is the first successful stabilization technique to prevent oscillations in the

convection dominated problems in the FEM. The main steps are: introducing a certain amount

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of artificial diffusion in the streamline direction only, ensuring that no diffusion perpendicular

to the flow direction is introduced, which was the reason for excessive over diffusion in other

methods. The details how the SUPG introduces artificial diffusion in stream line direction and

the determination of right amount of stabilization parameter (weights the influence of the added stabilization terms), is not considered in this study.

2. Galerkin/Least squares (GLS):

This method is similar to SUPG in certain aspects, and for purely hyperbolic equations and

linear interpolation functions it becomes identical. In the GLS method, least squares forms of

the residuals are added to the Galerkin method, enhancing stability without giving up

consistency or degrading accuracy.

3. Discontinuity capturing:

As mentioned in the section 2.5.2, the over and undershoots in the solution can be prevented

by getting control in the direction of the solution gradient. This can be done by using a

Petrov/Galerkin approach.

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CHAPTER 3

3a TEST PROBLEMS USING FEM For solving PDEs using FE method, here we used a commercial software package FEMlab.

All the calculations were done using this software package.

3.1a About FEMlab FEMLAB is a powerful interactive environment for modelling and solving all kinds of

scientific and engineering problems based on partial differential equations (PDEs). When

solving the PDEs, FEMLAB uses the proven finite element method (FEM). The software runs

the finite element analysis together with adaptive meshing and error control using a variety of

numerical solvers.

In this work we used FEMlab3.1. FEMlab can generate a mesh with triangular mesh that is

unstructured mesh and also with quadrilateral mesh which is a structured mesh. In this work

the problems were solved using both types of solvers. FEMlab includes different types of

solvers like linear solver, non linear solver, parametric solver and iterative solvers and also

the resulting linear system can be solved by UMFPACK or conjugate gradient method.

FEMlab includes different stabilization techniques which are described in section 3.2.2a. It is

possible in FEMlab to refine the mesh adaptively that is minimizing the error in the quantities

of interest.

3.2a Cauchy problem A pure transport is considered. A step moves with a constant positive velocity of 0.5 in space:

0.5 0t xu u+ = ( ) ( )2, 0,1 ,x t = (3.1)

( )0,

, 0 1,0,

u z t= =

for 0.0 0.20.2 0.40.4 1.0

zzz

The numerical treatment of this problem is very hard to solve due to the discontinuities.

Therefore this problem is an adequate test for the quality of the numerical method. The

solution can then be compared with the analytical one.

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3.2.1a Solution approach:

1. The computational domain was chosen to be a unit square with x and y axis as space and

time domain varying from 0 to 1.

2. Appropriate boundary and sub domain settings were given, the boundary settings is

represented in the figure 3.1.

Convective flux

1

Time

u(z=0,t) = 0

0 Space 1

u(t=0, z)=u0

Fig 3.1 Representation of computational domain and boundary conditions 3. In FEMlab, its possible to generate the mesh in an unstructured mesh that is with

triangular elements and in a structured mesh that is with quadrilateral elements. The

structured and unstructured mesh on the computational domain is shown in fig 3.2. The

calculations have been done on both types of the mesh. The unstructured mesh generator

is based on Delaunay algorithm. Initially the problem is solved on a coarse mesh and

further the mesh is refined.

3 1 4 2

23

Fig 3.2 Computational domain with unstructured mesh with 900 elements (left) and structured mesh with 1600 elements

4. As we have chosen global discretization method, stationary nonlinear solver has been

used to solve the problem. This non linear solver uses the damped Newton method. The

solver algorithm is as follows

I. The discrete form of equations can be written as f(U)=0 where f(U) is the residual

vector and U is the solution vector

II. Start with an initial guess Uo, form linearized model using U0 as the linearization

point.

III. Solve the discretized form linearized model, f (Uo) U = -f (Uo), where U is the

Newton step.

IV. Compute U1 = Uo+.U, where is the damping factor.

V. Estimate the error for the new iterate U1 by solving f (Uo) E = -f (Uo).

VI. If E is larger than relative error of previous error, select new , recomputed step 4

and 5.

The linear system is solved with UMFPACK direct solver, which solves the system of

un symmetric multi-frontal method and the direct LU factorization of sparse matrix. [

http://www.cise.ufl.edu/research/sparse/umfpack]

5. The problem has been solved with a relative tolerance of 1E-06 and the maximum

number of iterations is 25.

24

The solution with the coarse mesh in 2D and 3D is given in fig 3.3 and fig 3.4

Fig 3.3 Solution to Cauchy problem with triangular coarse mesh with 900 elements in 2D (left) and 3D (right)

Fig 3.4 Solution to Cauchy problem with quadrilateral coarse mesh with 1600 elements in 2D

(left) and 3D (right) From the plots 3.4 and 3.5, one can see the oscillation that is instabilities, which violates the

properties of numerical methods. In order to minimize these oscillations one needs good

stabilization algorithm and the need to refine the mesh.

25

3.2.2a Stabilization Techniques

FEMlab includes different stabilization techniques in order to minimize the oscillations. The

available techniques are,

I. Stream line diffusion: It adds artificial diffusion in streamline direction and it

stabilizes oscillations and instabilities

II. Cross wind diffusion: It adds artificial diffusion in cross direction and it preferably

with shock capturing and can minimize unnatural side effects.

III. Isotropic diffusion: It stabilizes most problems of convection diffusion problems.

It is possible to control the amount of artificial diffusion being added by using a tuning

parameter.

In FEMlab its possible to refine the mesh with different level, now the problem has been

solved with different mesh refinement levels and also included the artificial stabilization

techniques.

The following figure 3.5 and 3.6 show the numerical solution with stabilization techniques

and with mesh refinement.

Fig 3.5 Solution of Cauchy problem with Stream line artificial diffusion and with mesh refinement level 2

26

Fig 3.6 Solution Cauchy problem with Cross wind artificial diffusion and with mesh refinement level 2

Fig 3.7 Solution with stream line artificial diffusion using quadrilateral mesh with 14400 elements (left) and 60000 elements (right)

27

In figure 3.5, 3.6 and 3.7, there are no under and over shoots we can say that by using

stabilization techniques one can minimize instabilities in the numerical method, but due to the

mesh refinement the memory requirement is more when compared to the coarse mesh and

also the computational time is more.

3.2.3a Comparison with analytical solution

A major decision when one is assessing a method against other is what performance criteria to

use. There are many important criteria such as accuracy, storage and memory requirements,

flexibility etc. Some of these criteria lead to a conflict. So, we have chosen accuracy as the

main criteria in this work. So the problem has been solved in different combinations like with

different stabilization techniques and different mesh refinements and the solution is compared

with analytical solution. To compare with the analytical solution the final value of the

simulated solution has been extracted and compared with the analytical solution. The error in

the simulated solution is represented in terms of norm and for better understanding the error is

represented in the figure 3.8. The error norm is defined in our scope as

( _ _ )u sim u anaeNEQ= where u_sim = simulated solution (3.2)

u_ana = analytical solution

NEQ = No. of elements

Fig 3.8 Representation of error for refinement level 1 and without any stabilization techniques

28

Like this the error norm has been calculated for all the combinations and is tabulated and

listed in appendix 1A (Table A1 & A2). The decrease of error with different stabilization

techniques with no. of elements is represented in fig 3.9 and 3.10

Fig 3.9 Error norm for different stabilization techniques using triangular elements

Fig 3.10 Error norm for different stabilization techniques using quadrilateral elements

29

3.3a Burgers problem This problem is taken from [4]

2

2

u u uut x x

= + (3.3) Initial condition

( ) 0.1 0.5,0 ,o o oo o o

A B C

A B Ce e eu xe e e

+ += + + 0 1x (3.4) and boundary conditions

( ) 0.1 0.50, ,L L LL L L

A B C

A B Ce e eu te e e

+ += + + 0,t (3.5)

( ) 0.1 0.51, ,R R RR R R

A B C

A B Ce e eu te e e

+ += + + 0,t (3.6)

where ( )0.05 0.5 ,oA x= ( )0.25 0.5 ,oB x= ( )

0.5 0.375 ,oC x=

( )0.05 0.5 4.95 ,LA t= + ( )0.25 0.5 0.75 ,LB t= + ( )

0.5 0.375 ,LC =

( )0.05 0.5 4.95 ,RA t= + ( )0.25 0.5 0.75 ,RB t= + ( )

0.5 0.625 ,RC = The problem is a highly nonlinear so it is an adequate test for the quality of the method.

Solution approach:

1. Problem has been solved in the same way as mentioned in the section 3.2.1a

2. Initially the problem has been solved with nonlinear solver, but due to the strong

nonlinearity we faced convergence problems, to ease up these numerical difficulties

we introduced a relaxation parameter (damp) in front of the source term, then the

problem is solved with parametric solver.

3. Parametric solver solves initially with damp=0, the yielded solution is used as initial

guess for the next parametric step , this procedure goes until damp=1.

The solution to the problem has been solved with both triangular elements and with

quadrilateral elements. Fig 3.11 shows solution of the problem with both triangular and

quadrilateral elements with refinement level 1 and with stream line artificial diffusion.

30

Fig 3.11 Solution to Burgers problem with triangular (left) and quadrilateral mesh (right)

The solution begins with two wave fronts. They move from left to right and merge to form

one wavefront.

Likewise the problem has been solved with stream line diffusion and with different mesh

refinement levels and the results are listed in appendix 1A(Table A3 & A4).

3.3.1a Comparison with analytical solution

The comparison with the analytical solution has been done in the same way as the Cauchy

problem and the representation of the error norm is shown in fig 3.12. The decrease of error

norm with and with out artificial diffusion is illustrated in fig 3.13.

31

Fig 3.12 Representation of error for Burgers problem

3.13 Representation of error norm with and without artificial diffusion

32

3.4a Coupled reaction engineering problem This is a reaction-diffusion-convection system for modelling catalytic surface reaction. The

problem has the form

( )( ) 21 1 11 3 1 1 3 4 21

11u u un D u Au u ut x Pe x

= + + (3.7)

( )( ) 22 2 22 4 2 2 3 4 21

11u u un D u A u u ut x Pe x

= + + (3.8)

( ) ( ) 223 31 1 3 4 1 3 3 4 3 4 21

11 1u uAu u u D u Ru u u ut Pe x

= + (3.9)

( ) ( ) 224 42 2 3 4 2 4 3 4 3 4 21

11 1u uA u u u D u Ru u u ut Pe x

= + (3.10) Where 0 1x and 0t , with initial conditions ( )1 ,0 2u x r= , ( )2 ,0u x r= , ( ) ( )3 4,0 ,0 0u x u x= = and boundary conditions

( ) ( )1 11

1 0, 2u t r uPe x

= ( ) ( )2 111 0,u t r u

Pe x =

3 4(0, ) (0, ) 0,u ut tx x

= =

31 2 4(1, ) (1, ) (1, ) (1, ) 0uu u ut t t tt t t t

= = = =

Where 1( , )u x t and 2 ( , )u x t are nondimensionalized concentrations, 3( , )u x t and 4 ( , )u x t are

coverage of adsorbed reactants on the catalytic wall, 1Pe and 2Pe are Peclet numbers, and D1

, D2, R, A1 and A2 are Damkohler numbers. The chosen values are A1=A2=30, D1=1.5, D2=1.2,

R=1000, r=0.96, n=1 and Pe1=Pe2= 100.

The problem does not have an exact solution; we compared the solution with the one from

the article [4]

3.4.1a Solution approach

1. As we have chosen global discretization, the problem is solved in 2D that is x axis

with space and y axis with time.

2. Stationary non linear solver is chosen to solve the problem.

33

3. The problem is solved with a relative tolerance of 1E-06 and the maximum of iteration

are 40.

The concentration profiles of 4 species are shown in fig 3.14 and 3.15.

Fig 3.14 Concentration profiles for species 1(left) and 2(right)

Fig 3.15 Concentration profiles for species 3(left) and 4(right)

Interestingly the solutions obtained do not agree with the solutions given in [4].

34

So, now the problem is solved in one dimension that is discretizing in space and integrating

over time which is commonly called method of lines. FEMlab provides a Time dependant

solver which is used for solving the problem. We extracted the data files and plotted the

solution in 3D which is shown in fig 3.16.

Fig 3.16 concentration profiles for different species using time dependant solver

The above diagram is in good agreement with the solution.

Comments:

Though we got the solution but the unresolved question is whether one can use global

discretization or method of lines on finite element context. Contrary to the [1], from the test

case 3, we ca say that global discretization on finite element context yields wrong results.

35

3b TEST PROBLEMS USING FINITE DIFFERENCE METHOD

The finite difference method is a very popular engineering tool for solving differential

equations. It is derived from the idea of replacing the derivatives of the differential equation

by finite-difference approximations. As a result, the differential equation can be rewritten into

an algebraic equation, which can be easily solved to approximate the solution.

We choose the test cases that solved in chapter 3 and we introduced the modern High

Resolution Total Variation Diminishing (TVD) schemes to stabilize the convective term, that

will be discussed in the following section.

3.1b TVD Schemes It was mentioned earlier that convection term is difficult to treat and it is a potential source of

numerical troubles. Standard high order methods give rise to non physical oscillations while

the results produced by low order one are corrupted by excessive numerical diffusion.

Unfortunately, there is no way out of this dilemma as long as the discretization technique is

linear. Therefore modern high resolution schemes are typically based on a non linear

approximation of convective fluxes. Roughly speaking a high order method is employed in

regions where the solution is sufficiently smooth but in the vicinity of steep gradients it is

replaced by a non oscillatory first order scheme like upwind.

3.1.1b TVD Methodology

Consider a linear convection equation,

0,u uvt z

+ = v > 0 (3.11) In the limit of pure convection, any physically admissible solution to a scalar transport

problem proves TVD. In one dimension, TVD is defined as

( ) uTV u dxx= (3.12)

Thus, it is natural to impose the same constraint on the numerical solution, so that

( ) ( )1 ,n nTV u TV u+ where ( ) 1i ii

TV u u u+= To illustrate the derivation of classical TVD schemes, the eqn 3.11 is discretized in space by a

conservative finite difference method which yields

1/ 2 1/ 2 0i i iu f ft z

+ + = (3.13)

36

The neighbouring grid points xi and 1ix exchange the conserved quantities via numerical

fluxes 1if which are supposed to be consistent with the underlying continuous flux f vu= . Harten[5] proved that a semi discrete scheme is TVD if it can be written in the form

( ) ( )1/ 2 1 1/ 2 1i i i i i i idu c u u c u udt + += + (3.14) with non negative coefficients. So the numerical flux for a TVD method can be constructed

by blending a high order approximation 1/ 2H

if + and it low order counterpart 1/ 2L

if + as

follows

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 ,L H L

i i i i if f f f+ + + + = + 1/ 20 2i + (3.15) Here 1/ 2i is an adaptive correction factor which is referred to as flux limiter which depends on the local smoothness of the solution and on the choice of the limiter function. This

corresponds to adding a proper amount of nonlinear antidiffusion to the low order flux

approximation 1/ 2L

if so as to improve accuracy without generating spurious wiggles and

violating TVD property. If the linear flux approximations are given by

1/ 2L

i if vu = and 11/ 2 2H i i

iu uf v ++

+= then it is easy to verify that the standard upwind, central and downward discretization of the

convective term are recovered in case of 1/ 2i + =0, 1 and 2 respectively. 3.1.2b Discrete upwinding

A new approach for the design of fully multidimensional flux limiter of TVD type was

proposed by Kuzmin and Turek [6], which is based on algebraic flux correction. It is, a

centered space discreization of the convective terms is rendered local extremeum diminishing

by a conservative elimination of negative off diagonal coefficients from the discrete operator.

These algebraic manipulations can be done in the following way.

Consider equation 3.11, which is discretized by a linear high order method i.e. central

difference which yields a system of ordinary differential equations for the vector of time-

dependant nodal values and can be written as

LduM Kudt

= ( ) ,ii ij j ij i

dum k u udt

= (3.16) where ML = diag {mi} is the lumped mass matrix and K = {kij} is the discrete transport

operator. If the coefficients kij were nonnegative j i , then the semi discrete should be local

37

extremum diminishing (LED) [7]. Furthermore the higher order operator is transformed into

lower order by adding a discrete diffusion operator { }ijD d= defined by [6] max{0, , },ij ji ij jid d k k= = ii ik

k id d

=

In deed the resulting low order operator L=K+D has no negative off diagonal coefficients

Resulting lower order scheme can be written as

LduM Ludt

= where L = K+D (3.17) Moreover this modification is conservative since the diffusive terms can be represented as a

sum of skew symmetric internodal fluxes.

The Discrete upwinding technique yields the least diffusive linear LED scheme, however

linear monotonicity preserving methods are at most first order accurate. In order to prevent

excessive smearing, it is necessary to remove as much artificial diffusion as possible without

generating wiggles. To this end a limited amount of compensating anti diffusion F is added in

the next step, then the discrete transport operator can be written as K* = L+F . In practise both

diffusive and antidiffusive terms are represented as a sum of internodal fluxes which are

constructed edge by edge and inserted into the global vectors. Even though the final transport

operator K* does have negative off diagonal coefficients the flux limiter guarantees that the

discretization remains local extremum diminishing. Thus for a given solution vector u, there

should exist a matrix L* such that all off diagonal entries l*ij are non negative and L*u = K*u.

Then the so obtained nonlinear algebraic equations can be solved iteratively e.g. by a fixed

point defect correction scheme

( ) ( )( ) ( )1( 1) m m mmu u A u r+ = + , (0) nu u= (3.18) The low order operator ( )( ) ( )( )m mLA u M tL u= constitutes an excellent preconditioner which is easy to invert. The limited antidiffusive fluxes faij are evaluated edge by edge and

inserted into the global vectors ( ) ( )*1n n n nLb M u tK u u= + and ( ) ( )( ) ( )( )m m mn mr b A u u tFu = + (3.19)

The defect correction cycle may consists of following algorithmic steps

1. Compute the residual of the low order scheme r = f- Au.

38

2. Evaluate the limited antidiffusive fluxes ( )aij ji ij i jf d u u= and insert them into the global defect vector r.

3. Solve the linear sub problem Au = f and compute u = u+u

In this way this methodology is applicable to steady state problems well as to solve time

dependant PDEs reformulated as stationary space-time domain.

The above mentioned stabilization technique is used for the numerical solution of test cases

which were solved in chapter 3.

3.2b Cauchy problem A pure transport problem was considered with constant positive velocity v

0u uvt z

+ = ( ) ( )2, 0,1 ,x t = (3.20)

( )0,

, 0 1,0,

u z t= =

for 0.0 0.20.2 0.40.4 1.0

zzz

As mentioned earlier we have chosen global discretization, we choose a two dimensional

equidistant grid to discretize over space 0 1x and time 0 1t . The schematic diagram for space time grid is shown in fig 3.17. The problem is solved with the following

discretization schemes

1. Leap frog i.e. central difference in time and space

2. Implicit Euler in time and central difference in time

3. Central difference in time and upwinding in space

Fig 3.17 Schematic diagram of a space- time grid

39

Where i is the time index, j is the space index, t is the time step and x is the spatial step.

Here we introduced different finite difference stencils and they are described in the following

section.

3.2.1b Leapfrog scheme

This method makes discretization in space and time by using central difference scheme. The

global discretization of the above equation using leap frog yields 1 1

1 10.5 02 2

j j j ji i i iu u u u

t x

+ + ++ = (3.21)

it can further be written as

( ) ( )1 1 1 10.5 0j j j ji i i icourant number

tu u u ux

+ +

= 14 2 43 (3.22)

This results in a system of linear algebraic equations LFM U b= (3.23)

The nodal unknowns uij for i = 1,.1/z-1 and j = 11/t-1 approximate the solution in the

points ( ), jiz t for .iz i z= and .jt j t= . The sparse matrix for leap frog scheme with 5 point s in space and time is shown in fig 3.18

Fig 3.18 Sparse matrix with 5points in space and time for leap frog scheme

40

The matrix MLF depicts the so called discrete transport operator and the right hand side b

contains the initial conditions and spatial boundary conditions. Solving the linear system by

any linear solver like direct solvers lead to a solution exhibiting the mentioned oscillations

throughout the domain. To suppress these numerical or unphysical oscillations we

implemented the modern high resolution TVD schemes. Starting from the linear system this

method first substitutes the high order transport operator by using discrete upwinding. The

new transport matrix MDU already fulfils the TVD properties but it is also very diffusive at the

same time. Secondly, in a defect correction loop the amount of additional antidiffusive flux

for each node is detected by the help of limiter functions and then added node wise. This

solution is essentially second order accurate and still fulfilling TVD properties.

The figure 3.19 shows the 3D plot and the final value of the step function with 160 points in

space and time. In this way the problem is solved with different grid points in space and time

and the error norm is calculated and the results are listed in appendix 1B (Table B1 & B2).

The representation of error is shown in fig 3.20.

Fig 3.19 3D plot for a leap frog scheme (left) and the final value of the step (right)

41

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

Numerical Solution of stepstrdiff

coarse

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

Analytical Solution of step

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

Difference Solution

Fig 3.20 Representation of the error using Leap frog scheme

From the Appendix one can say easily that, this method needs large number of iterations in

order to converge, and also we observed that for different velocities the convergence

behaviour is different. Fig 3.21 and 3.22 shows the convergence behaviour for 0.5 and 02

velocities with different grid points.

42

Fig 3.21 Convergence behaviour for velocity = 0.5 for leap frog scheme

Fig 3.22 Convergence behaviour for velocity = 0.2 for leap frog scheme

43

From the figures 3.21 and 3.22, we observed that as the velocity is increasing the convergence

is slow. In order to secure the convergence it is worth while to perform implicit under

relaxation that divides the diagonal entries of the pre conditioner by a suitably chosen

parameter 0w1 so as to enhance the diagonal dominance. From this its possible to enhance

the convergence of the problem.

3.2.2b Discretization with CDS in time and Upwinding in space (CDS/UPW)

As mentioned earlier the discretization of convective term does not yield oscillations, the

discretization has been done with CDS in time and Upwinding in space.

The global discretization of this scheme can be written as 1 1

1 02

j j j ji i i iu u u uv

t x

+ + + = (3.24)

which can be written as

( ) ( )1 1 120.5 0j j j ji i i icourant number

tu u u ux

+ +

= 1 4 2 43 (3.25)

In this scheme the Courant number is different to Leapfrog scheme, and also one can say that

the behaviour of the discretization scheme can be depend on Courant number. The associated

sparse matrix with 5 grid points is space and time is shown in fig 3.23

3.23 Sparse matrix for CDS/PW

44

The global discretization using this scheme results in a system of algebraic equations and ca

be represented as LFM U b= (3.26)

This linear system is solved by using high resolution TVD method, as mentioned in the

previous section, in order to suppress the oscillations and numerical diffusion. Fig 3.24 shows

the 3D plot of the step using CDS/UPW

Fig 3.24 Representation of solution in 3D (left) and the final value of the step

From the fig 3.24, one can say that this scheme exhibits excessive numerical diffusion that is

due the discretization of convection term using upwind scheme.

The problem is solved on different grid points and the error is represented in terms of norm,

which is listed in Appendix 1B (Table B3).

By using CDS in time and Upwind in space the error is quiet high when compared to the

leapfrog method, its possible improve the accuracy by increasing the grid points but the

memory requirement will be more. And also in this scheme the convergence is quiet fast

when compared to the leapfrog method, which is shown in fig 3.25

45

Fig 3.25 Convergence behaviour for CDS/UPW We observed that the initial defect is low for CDS/UPW when compared to the leap frog

method thats why the convergence behaviour is fast for the CDS/UPW scheme.

3.2.3b Discretization with Implicit Euler in time and CDS in space (Impeuler/CDS)

This scheme is the discretization in time and CDS in space. The global discretization of eq

3.11 using this scheme yields

1

1 1 02

j j j ji i i iu u u uv

t x

+ ++ = (3.27)

( ) ( )1 1 1 02j j j ji i i icourant number

tu u v u ux

+

= 14 2 43 (3.28)

The resulting system of algebraic equations is solved in the same fashion as did for the leap

frog and CDS/UPW schemes. The figure 3.26 shows the 3D plot and the corresponding final

value for 160 grid points in space and time.

46

Fig3.26 3D plot of the step and the corresponding final value using imp.Euler/CDS

In the same way the problem is solved with different grid points and the results are listed in

appendix 1B (Table B4).

When compared to leapfrog method this method is also not accurate showing excessive

numerical diffusion but the convergence is fast which is shown in fig 3.27.

Fig 3.27 Convergence behaviour for Impeuler/CDS

47

To summarize, in the above three methods the leapfrog exhibits good accuracy but the

method is very slow when compared to the CDS/UPW method. In order to improve the

accuracy it is necessary to design a good preconditioner.

3.3b Burgers equation We took the same equation as did for Finite element method. The problem is as follows

2

2

u u uut x x

= + Initial condition

( ) 0.1 0.5,0 ,o o oo o o

A B C

A B Ce e eu xe e e

+ += + + 0 1x and boundary conditions

( ) 0.1 0.50, ,L L LL L L

A B C

A B Ce e eu te e e

+ += + + 0,t

( ) 0.1 0.51, ,R R RR R R

A B C

A B Ce e eu te e e

+ += + + 0,t

where ( )0.05 0.5 ,oA x= ( )0.25 0.5 ,oB x= ( )

0.5 0.375 ,oC x=

( )0.05 0.5 4.95 ,LA t= + ( )0.25 0.5 0.75 ,LB t= + ( )

0.5 0.375 ,LC =

( )0.05 0.5 4.95 ,RA t= + ( )0.25 0.5 0.75 ,RB t= + ( )

0.5 0.625 ,RC = For this problem also we implemented modern High resolution schemes based on TVD which

was explained in the section 3.2.1a.

The computational domain was chosen to be a two dimensional equidistant grid to discretize

over space 0 1x and time 0 1t . We used upwinding method to solve the problem.

48

Fig 3.28 shows the numerical solution to Burgers equation using Upwinding method for a 50

grid points in space and time.

Fig 3.28 Numerical solution to Burgers problem using FD method

49

CHAPTER 4

4. DECOMPOSITION OF N2O IN REVERSE FLOW REACTOR 4.1 Introduction

4.1.1 Reverse Flow Reactor Background

The increasingly strict governmental regulations on industrial pollution has placed increased

pressure on industry to study and evaluate the disposability of their manufacturing waste

streams, one of those pollutants is Nitrous Oxide(N2O). The interest in the control of N2O,

mainly deriving from adipic acid synthesis processes and fluidized bed coal combustion, is

related to their potentially disastrous effects on the environment such as global warming and

ozone layer depletion[15].

One of the most commonly used methods in the past, thermal oxidation served an appropriate

role in removing hazardous waste products, such as volatile organic compounds (VOCs),

from the gaseous chemical effluent stream. Commonly, thermal oxidation involves gas-phase

reactions taking place at 1000-1250 K. The need to achieve these high temperatures requires

significant energy expenses. Also, large combustion chambers designed to operate at high

temperature are costly. Therefore, thermal oxidation is considered an unattractive way to

eliminate the pollutants.

Recently, catalytic oxidation has become a popular alternative for the removal of pollutants

from industrial effluent streams. The main advantages of catalytic oxidation are significantly

lower operating temperatures (about 500 K) and smaller operating units (and therefore lower

cost). In fact, since the concept was first proposed and patented by Matros , the catalytic

reactor with flow reversal has been shown to be an effective technique for pollutant removal.

However, thermal instability of reverse flow operation is one of the major flaws that prevent

this technique from being widely used in industrial applications. According to Matros et.

al.(1994), Monsanto Enviro-Chem System Inc. is the only company located in United States

to utilize reverse flow reactors. Reverse flow reactors are designed to trap the heat of reaction

within the catalyst bed of the reactor, eliminating the need for preheating and maintaining the

entire reactor. This can be achieved by periodically reversing the flow direction. A schematic

of the reverse flow reactor is shown in Figure 4.1.

50

VALVE 1

FLOW OUT

FLOW IN

VALVE 2

4.1 Schematic Diagram of reverse flow reactor by cyclic opening and closing valve 1 and valve2

4.2 Reaction mechanism for N2O decomposition

N2O decomposition is an exothermic reaction, which can proceed either catalytically or

thermally. Much work has been done on catalytic decomposition with different catalytic

systems and they proposed different reaction mechanisms. The below reaction mechanism

was proposed by Galle [9]. Generally, the mechanism of the catalytic reaction may be represented as the adsorption of

N2O on an active site, usually consisting of a free transition metal ion on the catalyst surface,

followed by the breakdown of adsorbed molecule, resulting in the production of gaseous

nitrogen molecules and adsorbed oxygen atoms. The latter can be desorbed in the form of

molecular oxygen through recombination with one another or via the direct reaction with

further N2O molecule, the reaction mechanism is as follows

51

1. * *2 2[ ]N O M M O N+ + 2. * *2 2 2[ ]N O M O N O M+ + + 3. 22 adsO O where M* is the Active site and *M O is the occupied site. Due to the considerably higher value of activation energy, thermal N2O decomposition

requires much higher temperatures than the catalytic route. The following mechanism has

been proposed for the thermal decomposition of N2O.

4. { } { }' * '2 2N O M N O M+ + + 5. *2 2 2N O O N O+ + 6. *2 2N O O NO+

where { }M is the impingent species 4.2.1 Kinetic modelling

Classically the reaction over oxidic catalysts is described by adsorption followed by an

oxidation of active sites, and a subsequent removal of deposited oxygen. The adsorption and

desorption are generally assumed to be in under quasi equilibrium, the rate expressions for the

proposed reaction mechanism can be written as

21 1 *

. . .N O Tr k p N = *22 2. . .N O T Or k p N =

where NT is active site concentration (mol/gcat), * is the free vacant active site and *O is the active site occupied by oxygen and 1r 2r s the conversion rate of N2O (mol/s.gcat) for reaction

7 and 8.

The total conversion rate is just the combination of both rate expressions

*2 2 21 * 2. . . . . .N O N O T N O T Or k p N k p N = + (4.1)

unknown surface occupancies can be eliminated by considering steady state and sitebalnce

At steady state, r = r1 = r2

*

*

2 * 2

2 2 2 *

7.

8.

O

O

N O N

N O N O

+ +

+ + +

52

Then the expression 4.9 yields, * 1 *2

.O

kk

=

From the site balance, **1 O = + (4.2)

2*1 2

kk k

= +

Inserting *O and * in the rate expression 4.1, yields

2 2

1

1 2

2 ./ 1

TN O N O

k Nr pk k

= + (4.3)

The ratio k1/k2 equals [O*]/ [*] and so determines the state of the active sites. For k1/k2>>1

the difficult step is reaction 7 and the sites are oxidised, while for k1/k2

53

Fig 4.2 Scheme of reaction for a conventional catalyst (left) and for a structured catalyst

(right). Course of temperature T, concentration C and reaction rates (rhyb,rhom, rhet) are shown. Grey and white areas depicts active and inert section.

4.4 Reactor modelling The reactor considered contains two zones, the schematic diagram (fig 4.3) is illustrated

below, the reactor length chosen to be 1.5 m and the catalytic phase is from 45 cm to 105 cm

where both homogeneous and heterogeneous reaction takes place and the remaining sections

are inert phase where only homogenous reaction takes place.

Inert Phase

Catalytic Phase

Inert Phase

Fig 4.3 Schematic representation of the computational domain

From this geometry the behaviour of the reverse flow reactor can be used to provide the novel

technique for examining the hybrid thermal and catalytic contributions to high temperature

reactions and to yield insights into the mechanism by which they interact with one another.

54

For the initial treatment the different coupling mechanisms for the N2O was investigated and

further we tried to implement a new species and studied the interaction of that new species.

The model equations to describe the temperature and concentration distribution of N2O,

comprises two balance equations, one pseudo homogeneous energy balance and the other a

mass balance. Due to the dominating behaviour of the heat balance, the mass balance was

regarded as to be pseudo steady state. Since the gas phase is coupled to the reaction term, it

implicitly depends also on time. The resulting system can be represented as

( )( ) ( ) ( ) ( ) ( )2 21 ,ambp eff p r hybs gT T T kc c T T H r T Ct z z r = + + (4.4)

( )2 20 ,eff hybC CD v r T Cz z = (4.5)

4.5 Coupling mechanism 4.5.1 TC model

This is a pure thermal coupling which considers the two reaction path ways as being

chemically independent of one another due to the heat of reaction, thus the overall hybrid

reaction rate rhyb is simply the sum of the contributions from the two reactions paths, which

can be written as

( ) . . ( , ), .. ( , )

TC v hethet

v het

a k T Cr T C Ca k T C

= + (4.6)

hom hom.TCr k C= (4.7)

Thus the overall hybrid reaction can be written as

( ) hom. . ( , ), ( , ) .. ( , )TC v hethyb v heta k T Cr T C k T C C

a k T C

= + + (4.8)

The partial differential equations 4.4 and 4.5 are solved in FEMlab; we have given periodic

boundary conditions for the direct calculation of cyclic profiles. The temperature and

concentration profiles are plotted in the fig 4.4.

55

4.4 Concentration and Temperature profile in N2O decomposition

Fig 4.5 Reaction rate for pure heterogeneous and for hybrid reaction

56

4.5.2 TCC model

In addition to the strictly thermal coupling a chemical reaction between two reaction path

ways may also be present. For example the reactive intermediates from the homogeneous

reaction can adsorb on the catalytic surface and their withdrawal from gas phase can inhibit

the homogeneous reaction. Thus this model incorporates the reaction of oxygen radicals from

the gas phase react with a surface species.

Thus the reaction rate according to the TCC model can be reduced to that of TC model

augmented by a single correction term.

( , ) .min( , )Tcc TChyb hyb het hetr T C r r r= (4.9) The effect of reactive intermediates is shown clearly in the fig 4.6.

Fig 4.6 Comparison between TC and TCC model with a correction term gamma=0.2

4.5.3 Model with Enhancement factor

Alternatively as mentioned in the TCC model, the reactive intermediates from the

heterogeneous surface reaction can be passed into the gas phase and thus promotes the

production of free radicals which can increase the rate of homogeneous reaction. In this case

57

the enhancement was taken to into account in the simplest way, by a discrete increase of

apparent rate constant [13].

When the interactions are accounted for by a discrete increase of homogeneous reaction rate

constant after the catalytic zone of the reactor, now the rate of homogeneous reaction can be

written as

hom hom(1 ) .r k C= + (4.10)

Thus the rate of hybrid reaction can be written as

( ) hom. . ( , ), (1 ) ( , ) .. ( , )v hethyb v heta k T Cr T C k T C C

a k T C

= + + + (4.11)

The representation of hybrid reaction rate for TC model and the model with enhancement

factor is illustrated in fig 4.7

Fig 4.7 Hybrid reaction rate for TC model and with enhancement factor

4.6 Surface coverage dependant model Surface coverage model deals with the periodic changes in the state of the catalyst. Periodic

variations in the rate of certain catalytic reactions have been proposed by several authors, but

58

variations of reaction activation energy with coverage has been less well documented, but it is

reasonable to assume that as coverage increases bonding conditions should become less

favourable to the reaction [16]. For the initial treatment of the problem we implemented

surface coverage dependant parameter in order to extend the model and more over we

implemented a new species NO which is produced in homogeneous reaction.

4.6.1 Influence of activation energy on N2O decomposition

According to Valyon and Hall (1993), the oxygen binding energy on the catalyst surface is an

increasing function of the catalyst reduction degree that is the activation energy of reaction 7,

in which O2 is adsorbed on the catalytic surface, as an increasing function of the oxygen

removal from the surface. More specifically we have assumed the activation energy of

reaction 7 E1, as a linear function of the reduction degree of the catalyst, * thus the rate constant of reaction can be written as

1 1 *.

1 1 1. .oE E

o oRT RTk k e k e = = (4.12)

By inserting expression 4.12 in 4.8, its possible to extend the problem.

4.6.2 Effect of NO on N2O decomposition

In this section we extended the model by introducing NO, which is produced in a side step of

the homogeneous reaction. In order to study the effect of NO on N2O decomposition one need

to perform extensive experiments. In this work we are presenting the different ways of how to

couple the homogeneous and heterogeneous reaction, by reviewing the literature.

To study the interaction of the homogeneous and heterogeneous reaction, one need to

investigate the effect of different species on the decomposition. Recently Kaptejein[12]

studied the reaction mechanism of N2O decomposition on different types of catalyst. It was

stated that NO act as a reducing agent that is it removes the deposited oxygen from the

surface but the enhancement by NO can be expected only if they remove the oxygen

deposited by the N2O faster than occurs in the pure decomposition reaction.

59

Fig 4.8 Schematic representation of proposed interaction

In the proposed reaction mechanism NO is producing in homogeneous reaction, this produced

NO may have some effect on the catalytic decomposition, that is it removes the deposited

oxygen thus the rate of that reaction 7 increases. The proposed interaction is illustrated in fig

4.7

In fig 4.7 the upper part is homogeneous reaction and lower part is the heterogeneous reaction

and the produced NO in homogeneous reaction may have some effect on the heterogeneous

reaction. In what way it influences is of major interest. As stated earlier the produced NO acts

as a reducing agent that is it removes the deposited oxygen.

So, the proposed reaction can be written as

9. * 2 *ONO NO + + By introducing reaction 7 in our reaction mechanism, then the total conversion rate for N2O

can be written as

2

2 2

2

3 21

1 2 3 2

2 (( / ).( / )).

1 / (( / ).( / ))NO N O

N O T N ONO N O

k k p pr k N p

k k k k p p += + +

(4.13)

60

For vanishing NO concentrations the expression 4.13 reduces to expression 4.3. So the term

2/NO N Op p account for the effect of NO on N2O decomposition.

4.6.4 Production of NO

To study the effect of NO on N2O decomposition, the first step is to know how much amount

of NO is producing and if the rate of reaction 9 is higher than the rate of reaction 8, then one

can say that NO is influencing the N2O decomposition.

Fig4.9 Concentration profiles for N2O and NO

For initial treatment of the problem theoretically, we assumed the rate of reaction 9 as some

percent of rate of homogenous reaction i.e.

hom *(10 100%)NOdc r to

dt= (4.14)

Fig 4.8 shows the concentration profiles for N2O and NO for 10% of the rate of the

homogeneous reaction for one cycle.

It is theoretically much complicated to say whether reaction 8 or reaction 9 is fast, unless one

needs to conduct the experiments. But if we assume like reaction 9 is faster than reaction

reaction 8, then the produced NO removes deposited O2 faster than that occurs in pure

decomposition reaction, thus enhances the rate of reaction 8, then the rate of reaction 7

increases and also the activation energy of reaction 7 will change. In this way NO can effect

the N2O decomposition.

61

4.6.4 Kinetic oscillations in N2O decomposition

Recently many authors found isothermal kinetic oscillations on different catalysts for N2O

decomposition. The occurrence of these kinetic oscillations may lead to a new insight into the

behaviour of They reported that oscillations has a kinetic genesis but not due to mass transfer

limitations. In our system its possible to detect the kinetic oscillations. In this work we

propose the methodology of how one can predict the kinetic oscillations.

We have assumed the activation of energy of reaction 7 as a linear function of reduction of the

degree of the catalyst. We have chosen a coverage dependant law for the activation energy of the solid-solid reaction

of the mechanism. In particular the functionality assumed in eq (4.12) expects that step 8

reaction rate increases as the catalyst degree increases, i.e. increases when concentration of

one product increases. The dynamic behaviour of the catalyst is described by the following

mass action law,

Then the mass balance for , yields

* *2 2

*1 * 2 3. . . . . .N O N O NOO Ok C k C k Ct

= + + (4.15)

*

* *2 2 21 * 2 3. . . . . .O N O N O N OO Ok C k C k Ct

= (4.16) and site balance **1 O = + (4.17) From trace of the Jacobian matrix corresponding to the system of eqs. (4.15)- (4.17), that is if

the trace of the Jacobian is positive, then from bifurcation analysis of the system one can

predict kinetic oscillations in the system.

62

CHAPTER 5

5. CONCLUSION For the numerical treatment of in stationary fixed bed processes we have chosen global

discretization, the numerical solution of these problems produces steep gradients and shocks.

In order to treat these problems, initially different test problems were considered and were

solved in FEM and FD method. Three different test problems were solved on FEMlab by

introducing different types of stabilization techniques. The results for both of these methods

are listed in appendix, from the results one can say that both methods exhibit good accuracy,

but if we consider the test problem 3 that is Coupled reaction engineering problem the

global discretization method does not work well on finite element context but the method of

lines works quiet well. While FD method with modern high resolution schemes in TVD

exhibit good accuracy, but suffers from convergence problems needs a preconditioner using

implicit under relaxation method in order to enhance convergence.

For the Decomposition of N2O in reverse flow reactor, different types of coupling

mechanisms were investigated. When considering TC model, it is a pure thermal coupling and

in TCC model the interaction can be seen by assuming a value for , but the calculation of is

very difficult step and though we updated the model by using an enhancement factor but the

same problem exist, the calculation of enhancement factor . Nevertheless the interaction can

be understood by introducing a new species.

We introduce a new species NO, which is produced in homogeneous reaction and the

produced NO acts as a reducing agent, that is the produced NO effects the heterogeneous

reaction by removing the deposited oxygen. Though it is very difficult to study the interaction

theoretically, we proceeded with some assumptions. For the initial treatment, we produced

NO and the mechanism how this produced NO can effect the heterogeneous reaction is

mentioned with some assumptions.

63

References

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regenerative processes in cyclic operation, Compters.Chem.Engng., 21, 167-172

(1997)

[2]. V.K.Gupta, S.K.Gupta, Solution of cyclic profiles in catalytic reactor operation with

periodic flow reversal, Compuers. Chem.Engng., 15 (4), 229-237 (1991).

[3]. P.Deuflhard, A modified Newton method for the solution of ill conditioned systems pf

nonlinear equations with application to multiple shoo