Finite Element Simultations In Electrochemistry

Finite Element Simulations In Electrochemistry Submitted by Nicholas P. C. Stevens For the degree of PhD Of the University of Bath 1998

Copyright Attention is drawn to the fact that copyright of this thesis rests with its author. This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its author and that no quotation from the thesis and no information derived from it may be published without the prior written consent of the author.

To Elizabeth

Abstract The Finite Element Method (FEM) is formulated for electrochemical problems involving mass transport due to 1, 2 or 3 dimensional diffusion with or without convection, and kinetic complications, under steady state or transient conditions. Diffusion

The FEM is shown to be capable of accurately modelling 1 and 2

D potential step transient behaviour of planar, microband, hemicylinder, distorted hemicylinder and microdisc electrodes. Cyclic voltammetry is simulated at large planar electrodes. Diffusion-Convection FEM simulations are presented predicting combined convectiondiffusion transport in electrochemical cells. Rotating disk and channel flow cell geometries were examined using analytical approximations for solution velocities. Hydrodynamics

The FEM is applied to solve the Navier Stokes equations in 2

and 3 D. Results are presented for microwire electrodes in flow cells. Current/volume flow rate relationships are presented, obtaining an empirical equation for the current response. Novel Cell Designs

FEM codes are used to design two novel hydrodynamic cell

geometries: the Confluence Reactor and Collision Cell. Solution of the Navier Stokes equations to define the convection profiles is demonstrated, and the steady state hydrodynamic and electrochemical behaviours of the cells are investigated. Coupled Kinetics

FEM codes are described which accurately model Convection-

Diffusion-Reaction processes. The EC, ECE, DISP1 and DISP2 reactions are investigated in the channel cell, showing the FEM provides access to extremely high rate constants. The Confluence Reactor is shown to be ideal for the study of the CE reaction. Working surfaces for varying geometries and rate constants are presented. 3 D Velocity Profiles

Three dimensional simulations of velocity profiles in channel

cells are described, and a correction is proposed giving the true volume flow rate. Errors deriving from the parabolic assumption are examined. Concentration profiles and current/volume flow rate relationships are presented for full-width electrodes in channel cells. Departures from normal channel cell behaviour are discussed.

i

Table of Contents Chapter 1. Introduction ________________________________________________________ 1

The Basics of Simulation _______________________________________________________ 2 Discretisation ______________________________________________________________ 2 Approximation_____________________________________________________________ 3 Assembly _________________________________________________________________ 3 The Laws Governing Mass Transport _____________________________________________ 4 Diffusion _________________________________________________________________ 4 Convection________________________________________________________________ 6 The Navier-Stokes Equations _________________________________________________ 7 Analogue Devices in Simulation _______________________________________________ 7 The History of Digital Simulations in Electrochemistry _______________________________ 9 Finite Difference Methods ____________________________________________________ 9 Other Techniques _________________________________________________________ 16 Thesis Structure ___________________________________________________________ 17

Chapter 2. Finite Element Formulations__________________________________________ 19

The Finite Element Method in Electrochemistry ____________________________________ 19 The History of the Finite Element Technique ______________________________________ 19 Finite Element Formulations for Electrochemical Problems __________________________ 20 One Dimensional Simulations __________________________________________________ 20 Grid Generation ___________________________________________________________ 20 First Order Element ________________________________________________________ 22 Remarks on the matrices generated ____________________________________________ 30 Two Dimensional Simulations __________________________________________________ 31 Linear Triangular Elements __________________________________________________ 31 Current Calculation for a Triangle _____________________________________________ 37

ii Two Dimensional Axisymmetric Formulation ______________________________________ 39 Three Dimensional Simulations _________________________________________________ 43 Tetrahedral Elements _______________________________________________________ 44 Cubic Elements _____________________________________________________________ 44 Trilinear Cubic Element ____________________________________________________ 44 Finite Element Forms of The Navier-Stokes Equations _______________________________ 49 Two Dimensional Form of the Navier Stokes Equations ___________________________ 49 Solution Technique ________________________________________________________ 51 Three Dimensional Formulation ______________________________________________ 52

Chapter 3. Diffusional Simulations ______________________________________________ 53

One Dimensional ____________________________________________________________ 53 Potential Step Transients ____________________________________________________ 53 Cyclic Voltammetry _______________________________________________________ 60 Two Dimensional ____________________________________________________________ 64 Microband Potential Step Transients ___________________________________________ 65 The Microwire Electrode ____________________________________________________ 69 Conclusions ______________________________________________________________ 72 Two Dimensional Axisymmetric simulations _______________________________________ 73 The Cylinder Electrode _____________________________________________________ 73 The Disk Electrode ________________________________________________________ 75 Conclusion _______________________________________________________________ 78

Chapter 4. Convection-Diffusional Results with Analytical Flow Profiles _______________ 80

Steady State at a Rotating Disc Electrode _________________________________________ 80 The Channel Flow Cell and Hydrodynamic Electrodes_______________________________ 84 Steady State Measurements __________________________________________________ 86 Transients under Hydrodynamic Conditions _____________________________________ 93 The Microstrip Electrode ____________________________________________________ 97

iii Conclusion ______________________________________________________________ 102

Chapter 5. Systems Involving Non-Analytical Convection profiles ___________________ 103 Interpolation of Meshes ____________________________________________________ 104 The Wire Electrode _________________________________________________________ 105 Hydrodynamic Simulation __________________________________________________ 107 Electrochemical Simulation _________________________________________________ 108 Experimental ____________________________________________________________ 109 Hydrodynamic Results and Discussion ________________________________________ 110 Conclusions _____________________________________________________________ 121

Chapter 6. Novel Electrode Geometries _________________________________________ 122

The Confluence Reactor______________________________________________________ 122 Introduction _____________________________________________________________ 122 Theory _________________________________________________________________ 123 Hydrodynamic Simulations _________________________________________________ 124 Electrochemical Simulations ________________________________________________ 127 Electrochemical Results and Discussion _______________________________________ 128 Experimental Cell Construction _____________________________________________ 133 Conclusion ______________________________________________________________ 138 The Collision Cell __________________________________________________________ 139 Introduction _____________________________________________________________ 139 Hydrodynamic Simulations _________________________________________________ 140 Results _________________________________________________________________ 141 Electrochemical Simulations ________________________________________________ 144 Voltammetric Behaviour ___________________________________________________ 146 Conclusion ______________________________________________________________ 147

Chapter 7. Diffusion-Convection-Reaction Systems _______________________________ 149 Introduction _____________________________________________________________ 149

iv Theory ___________________________________________________________________ 150 Grid Formation __________________________________________________________ 150 Matrix Formation ________________________________________________________ 150 EC Reaction _____________________________________________________________ 151 ECE Reaction ___________________________________________________________ 153 DISP1 Reaction __________________________________________________________ 154 DISP2 Reaction __________________________________________________________ 155 Current Calculation _______________________________________________________ 155 Results and Discussion ______________________________________________________ 156 EC Reaction _____________________________________________________________ 156 ECE and DISP Reactions __________________________________________________ 157 The CE Reaction using the Confluence Reactor ___________________________________ 159 Introduction _____________________________________________________________ 159 Matrix Formation ________________________________________________________ 160 Hydrodynamic Simulation __________________________________________________ 161 CE Reaction _____________________________________________________________ 161 Results and Discussion ____________________________________________________ 162 Conclusion ________________________________________________________________ 169

Chapter 8. Three Dimensional Convection Profiles in a Channel Cell _________________ 171 Introduction _____________________________________________________________ 171 Results and Discussion for the Three Dimensional Solution of the Navier Stokes Equations _ 173 Cell Discretisation ________________________________________________________ 173 Boundary Conditions ______________________________________________________ 174 Three Dimensional Results _________________________________________________ 174 Conclusion ______________________________________________________________ 177 Two Dimensional Formulation ________________________________________________ 179 Cell Discretisation ________________________________________________________ 179 Results _________________________________________________________________ 180

v Implications for normally positioned electrodes _________________________________ 187 Conclusions _____________________________________________________________ 189

The Full-Width Electrode _____________________________________________________ 190 Introduction _____________________________________________________________ 190 Cell Discretisation ________________________________________________________ 190 Boundary Conditions ______________________________________________________ 192 Results _________________________________________________________________ 192 Conclusions _____________________________________________________________ 202

Chapter 9. Appendix _________________________________________________________ 203

One Dimensional Second Order Element Formulation ______________________________ 203 Bilinear Square Elements ____________________________________________________ 205 Grid Generation __________________________________________________________ 205 Matrix Formation ________________________________________________________ 207 Isoparabolic Rectangular Element Formulation ___________________________________ 211 Grid Generation __________________________________________________________ 211 Matrix Formation ________________________________________________________ 213 Linear Tetrahedral Element Formulation ________________________________________ 216 Linear Tetrahedron Matrix Formation _________________________________________ 218 20 Noded Hexahedral Element Formulation ______________________________________ 220 Three Dimensional Navier Stokes Formulation ____________________________________ 224 Matrix Solution Techniques ___________________________________________________ 226 The Gaussian Technique ___________________________________________________ 226 Banded Storage Methods ___________________________________________________ 228 Sparse Solution Methods ___________________________________________________ 230

1

Chapter 1. Introduction The Finite Element Method (FEM) has been widely used in the engineering community as a useful tool to aid in the solution of many types of differential equations. However, the electrochemical community has only recently begun to exploit this powerful method. This thesis aims to demonstrate how the FEM can be applied to a wide variety of electrochemical problems. The advantages of the FEM over previous simulation strategies employed are also stressed. Electrochemical measurements are often performed using simple experimental cell geometries which are not necessarily ideal for the task. In many cases this problem arises due to the complexity of solving the coupled convection-diffusion-reaction problems to gain quantitative information from experiments. In this thesis procedures for solving coupled convection-diffusion-reaction problems for arbitrary cell designs are discussed. The work described in this report has been executed with the aim of writing simulations which are not only capable of reproducing analytical results, but can also be used in situations where no analytical theory exists. In this chapter the history and development of digital simulation in electrochemistry is briefly discussed. Particular attention is given to the Finite Difference Method, a technique which has been previously used by the vast majority of workers in the electrochemical field.

Introduction

2

The Basics of Simulation Simulation is an activity which is commonly attempted where the system which is under study is sufficiently complex that no accurate analytical way of describing it's behaviour exists. In the field of electrochemistry, where one might like to investigate the diffusion of material through solutions, or the flow of such solutions, there are often highly accurate models available which can predict how such systems behave in certain geometries. However, these models tend to ignore effects such as surface roughness or changes in topography, and then one is forced to look to simulation to gain information about the behaviour of the system. The difficulty with relying on results provided by simulation is that the inaccuracy of the results can be difficult to measure. For this reason, simulations are often calibrated against known analytical models which have been proved for simple geometries. If a simulation can be shown to arrive at the formally correct answer where one exists, then a result for a similar system where no analytical model exists may be considered to be realistic. The key motivation for developing a simulation, using analogue components or numerical idealisations, is that once developed to a sufficient level of verisimilitude, the simulation can be used for experimental purposes in place of the real system, or as a means of double checking results obtained. The accuracy of the answers obtained will obviously be a function of how accurately the behaviour of the simulation follows that of the real system. The development of a simulation of a system can be described in three steps:

Discretisation Any simulation of a complex real process must begin by identifying the key parts of the real system. In simulations of structural, mechanical, or electrical systems, a mechanism might actually be composed as an assembly of discrete parts, such as a set of gears, making this process obvious. Where the quantity of interest is a bulk property, varying continuously in an area or volume, then the approach which must be adopted is to discretise the system into a set of separate subregions. This is the case for all the work presented in this thesis, and so the discussion will focus on this situation.

Introduction

3

This process of discretisation is described as the Grid Generation. The tessellation chosen must cover the whole region to be simulated, and must follow the boundaries of the region as well as possible. This is trivial in one dimension, and is not arduous for straight sided regions in two and three dimensions. However, the subdivision of the region must also take into account the likely variation of the quantity of interest. It is normal to attempt to construct a grid in which the variations of the quantity of interest across each element are roughly comparable between adjacent elements, with the grid being finest spatially where the greatest gradients in material properties are expected. Due to the finite nature of the computational resources available, the size of the grid used is always subject to limitations, and efficiency depends upon an intelligent discretisation of the region. Discussions of the different types of discretisation which can be used, and the topic of grid refinement, account for at least one quarter of the literature on the Finite Element Method, and are further discussed with respect to the current work in chapter 2.

Approximation Across each subregion or element defined, the variation of the quantity of interest must be approximated in some way. This approximation must specify at which points the value of the quantity is defined, how the value may be calculated, and how it depends on the other values in the system. This approximation must be dependant on the local geometry of the point or region considered, and allow the relationship between the local values of the quantity under study to be expressed as a function of the local geometry. The boundary conditions to the simulation must also be considered, and where these are prescribed, arrangements must be made for these to be enforced in the simulation.

Assembly The set of local approximations may be combined in a sequential fashion, iterating over each region to determine its values in a set order, or considered as a set of simultaneous equations. These may be solved as a matrix equation, to recover a set of values for the quantity

Introduction

4

under investigation. The solution of such matrix equations may be accomplished by a number of methods, and is discussed in the appendix, on page 226. There may be complications to these basic steps, where many assembly and solution steps are needed to improve initially approximate answers to a final, accurate result, but the basic steps outlined above are common to almost every simulation.

The Laws Governing Mass Transport The simulations presented in this thesis are mainly concerned with predicting the mass transport in cells. The modes of transport considered are diffusion and convection. The equations governing these effects are now discussed.

Diffusion Fick's First Law of Diffusion Fick1 showed that the diffusional flux J, which is defined as the number of moles of material diffusing through a unit area in one second is given by

J   Dx

C x

(1.1)

where Dx is the diffusion coefficient in the x direction of molecule C, given by

Dx 

kT 6a

(1.2)

where a is the radius of molecule C and  is the viscosity of the solution. Fick's Second Law of Diffusion The variation of concentration of a material in a fixed volume with time is given by

C  2C  2C  2C  Dx 2  D y 2  Dz 2 t x y z

(1.3)

where C is the concentration of the species in question, and Dx Dy and Dz are the diffusion coefficients in the x, y and z directions indicated. For all the simulations presented in this thesis, diffusion is considered to be isotropic unless otherwise noted.

Introduction

5

A one dimensional derivation from Fick's first law is as follows. Figure one shows an unbounded region of fluid through which material is diffusing in the direction shown.

Figure 1.1

Diagram illustrating Fick's second law

In a thin slab of solution of width l and area A, as shown in the diagram, let the concentration of a species at x be C at time t. The amount of material entering the slab, in the direction shown by the arrows, in time t is JA, and so the flux from the left hand side is given by

d C JA J   dt Al l

(1.4)

d C  J A  J    dt Al l

(1.5)

dC J  J  dt l

(1.6)

Similarly for the right hand side ,

giving the total flux as

From Fick's first law the terms in this equation may be found as

C x

(1.7)

C     C    Dx  C   l  x x   t  

(1.8)

J   Dx  J   Dx This gives the total flux as

dC 1 C C  2C   2C    Dx  Dx  Dx 2 l   Dx 2 dt l  x x x  x Clearly the three dimensional derivation is of exactly the same form.

(1.9)

Introduction

6

It may be worth noting that the way that Fick’s second law is derived, from setting Fick’s first law to be valid over a local region, and observing the behaviour as this region vanishes is a close analogy to the methods by which diffusive processes may be simulated.

Convection Convection is the term describing the movement of material in response to a mechanical force acting on the solution. There are two types of convection, natural convection, and forced convection. Natural convection is present in any solution, and arises from thermal gradients and other density gradients possibly created by reactions occurring. This is at best hard to predict, and at worst a chaotic effect, both in nature and magnitude, and it is usual to attempt to make it's effects as insignificant as possible. This can be achieved easily by ensuring that forced convection is large and well defined. Forced convection is deliberately introduced by an external force acting on the solution. This can include any sort of pumping, stirring or agitation of the solution, but where possible it is preferable to use methods which are well characterised, so the effects of convection may be understood, and accounted for in simulations of such systems. For this reason, such methods as gas bubbling, and mechanical shaking, which are often used to stir solutions in other types of experiment, are unsuitable for electrochemical experiments. This is due to the undefined nature of the convection they introduce. In most situations, the ideal method of introducing forced convection would be one which gives easily measurable and steady convection. Hydrodynamic electrodes meet these criteria, and are discussed in chapter 4. The governing equation for mass transport where both diffusion and convection operate, expressed in three dimensions, is

C  2C  2C  2C C C C  Dx 2  D y 2  Dz 2  Vx  Vy  Vz t x y z x y z

(1.10)

where V is the velocity in the direction noted, which can be calculated using the Navier-Stokes equations.

Introduction

7

The Navier-Stokes Equations The Navier-Stokes equations can be used to completely predict the flow of a fluid, with certain restrictions. The equations were developed in the early 19th century and, together with the equation of conservation of mass, they allow a complete characterisation of the flow of a viscous fluid, and are presented below in three dimensional form. For a complete derivation, the reader is directed to the textbook by Gerhart and Gross2.

1 p    2 u  2 u  2 u   u   u   u     u   v   w         x   x 2 y 2 z 2   x   y   z 

(1.11)

1 p    2 v  2 v  2 v   v   v   v   u   v   w         y   x 2 y 2 z 2   x   y   z 

(1.12)

1 p    2 w  2 w  2 w   w   w   w   u   v   w         z   x 2 y 2 z 2   x   y   z 

(1.13)

u v w   0 x y z

(1.14)

In these equations, u is the velocity in the x direction, v is the velocity in the y direction, w is the velocity in the z direction, p is the pressure,  is the density of the solution, and  is the viscosity. For the special case of an inviscid fluid where the viscosity, , is zero, these equations reduce to Euler's equations, by removing the right hand side term from equations 1.11 to 1.13. These equations can be solved completely for certain simple geometries, such as flow between infinite parallel plates, but must be solved numerically for most real geometries. This is discussed further in chapter 2.

Analogue Devices in Simulation The search for efficient methods of solving partial differential equations, such as those encountered in electrochemistry also led to the development of ‘analogue computers’3. Although we might not recognise these circuits as computers in the modern sense, being essentially assemblies of simple repeating units, designed so that the distribution of potential

Introduction

8

over the system will satisfy a particular set of conditions derived from a real physical system. Early electrochemical simulations were performed by such analogue electronic means4,5. A simple example is the Resistance Capacitance ladder, shown in figure 1.2

Figure 1.2

An electronic diffusion simulation by means of an RC ladder.

In this network, the movement of charge around the circuit simulates the movement of a species in solution, where diffusion is acting in one dimension only. The resistance of each resistor, R, may be considered as an analogue of the distance x between each point to be measured. The potentials at each node of the simulation may be though of as analogues of the concentrations of solute at the boundaries between the elements shown. The flow of charge from one capacitor in this network to another will be proportional to the difference in charge between them, and inversely proportional to the resistance between them, and so the movement of charge will obey Fick’s first law on a local scale. If there are many repeated units, the behaviour of the charge distribution in the overall system will begin to approach that predicted by Fick’s second law, as does the movement of solute in solution. Furthermore, the resistance of each resistor can be chosen so as to set differing distances between each point. In this way a grid could be constructed with a higher density of points in the region near to the electrode. As might be intuitively supposed, this kind of fine tuning can increase the accuracy of the simulation, as will be discussed.

Introduction

9

The main limitation of this kind of analogue computing are that the range of functions which can be exactly matched by an electronic component is small, and the versatility of the devices was compromised by the need to physically alter them to tackle different problems. With the rise of digital computation in the 1950s, the analogue computer was rapidly overtaken.

The History of Digital Simulations in Electrochemistry Finite Difference Methods The use of simulation as a technique to aid in the understanding of complex processes has a long history, which precedes the advent of the digital computer. The technique which is now known as the Finite Difference Method was first described in 19116, and was used and developed throughout the first half of this century7,8. This was despite the fact that the large amounts of repetitive arithmetic involved had to be done by hand. This was the case for the first application of this technique to an electrochemical problem in 1948, when Randles simulated a linear sweep experiment9. The Finite Difference Method has been the most commonly used numerical method in electrochemistry. Some details of this technique are therefore necessary, before any discussion of the relative merits of the Finite Difference and Finite Element Method can be attempted. In the 1960s the work of Feldberg brought digital simulation to the forefront of electrochemical research, using a basic Finite Difference algorithm to tackle problems involving diffusion, homogenous kinetics and very simple hydrodynamics10. A simple example is the formulation for the evolution of a one dimensional concentration profile with time.

C

Ci 1

xi

xi 1

Ci 1

xi 1 x Figure 1.3

Ci

x

Diagram illustrating discretisation of one dimensional function.

Introduction

10

The concentration profile is discretised in a point-wise fashion as shown in the diagram, where i is a point, with position xi and concentration Ci. The gradient of the concentration profile at point i can be estimated in three ways. These are: 

The gradient of the chord from i-1 to i. This is the backward difference method of determining the gradient, and may be expressed mathematically as

d C Ci  Ci 1  dx x 

(1.15)

The gradient of the chord from i-1 to i+1. This is the central difference method, and gives

d C C i 1  C i 1  dx 2x 

(1.16)

The gradient of the chord from i to i+1. This is the forward difference method, and gives

d C C i 1  C i  dx x

(1.17)

The second derivative of the gradient is commonly found from the difference in gradient between the forward and backward difference methods, which is the change in gradient over the interval x . This leads to the expression

d 2 C Ci 1  2Ci  Ci 1  d x2 x 2

(1.18)

To find expressions for the variation of a quantity with time, a similar method is used, giving

d C Cit t  Cit  dt t

(1.19)

Using these relationships, Fick's second law can be stated as

Cit t  Cit Cit1  2Cit  Cit1  Dx t x 2

(1.20)

which contains only one unknown which is the concentration at the new timestep. The equation can be rearranged to find this new concentration, giving

Cit t  Dx

t

x 

2

C

t i 1



 2Cit  Cit1  Cit

(1.21)

Introduction

11

Using this equation, the procedure for solving for the concentration gradient at each timestep is simply to iteratively march across the concentration profile applying the equation to each unknown point to give the values at each new time. Note that the above relationship could be expressed as a matrix equation, and the new concentrations to be determined by the solution of a matrix equation formed by combining the equations for every point.

 t Cit t   Dx 2  x 

1  2 Dx

t

x 2

Cit1  t   t  Dx C x 2  C ti   i 1 

(1.22)

This method requires that the concentrations at each extreme of the space being simulated be fixed, as they cannot be calculated using the formula, which requires a knowledge of the points on either side of the one in question. The boundary values are normally that at the electrode, where (i = 0), C is defined by an expression such as the Nernst equation, and the furthest node from the electrode is usually assumed to be at the bulk concentration. The accuracy of such a simple simulation is dependant on the size of the timestep, the diffusion coefficient and the distance between the nodes. The ratio of these, D x t x can be considered as the distance that material can travel in a timestep over the distance between nodes, and by ensuring that the ratio is minimised, accuracy can be expected. If this ratio exceeds 0.5 then oscillations appear in the solutions obtained which, by the nature of the technique, propagate through the profile with each timestep. Feldberg also showed that the extension of the finite difference technique to the second dimension follows a very similar procedure to the development for one dimension.

Ci , j 1

C

Ci , j

Ci 1, j

C i , j 1

xi 1

y xi1 j 1

xi x

Figure 1.4

Ci 1, j y j 1 yj

x

Discretisation of a two dimensional function.

y

y

Introduction

12

A similar method to that used for the one dimensional problem above may be used for the two dimensional diffusion equation,

C  2C  2C  Dx 2  D y 2 t x y

(1.23)

The resulting finite difference form, using the node naming system shown in the diagram, but assuming isotropic diffusion, D = Dx = Dy, and that x  y , is



Ci,tjt   Cit, j   Cit1, j  Cit1, j  Cit, j 1  Cit, j 1  4Cit, j



(1.24)

with   Dt x . 2

The condition for stability in two dimensions is that   1 4 , a stricter requirement than for one dimensional work. This early work was expanded upon over the next decade11-14, with the short book by Britz15 marking a point where the wider availability of computer power to scientists led to a wide increase in the practise of simulation. Britz also encouraged workers in the area to have more concern for theoretical rigour, including a chapter on ‘fudge factors’. Over the last two decades there has been a considerable expansion in the number of groups able to include simulation in their portfolio, as computer power advanced by several orders of magnitude from that with which the first pioneers had to manage. Indeed, Feldberg commented in 1990 that an IBM PS2 with a 387 co-processor represented ‘a level of computational power that is adequate for simulating many problems of interest’16. The FD technique outlined above is an explicit technique, which is to say that the solution to the equation is arrived at in one step, with the known values giving the unknown, going straight from one value of time to the next. Another possible method is to shift the time at which the differential term is evaluated, usually half way between time steps. This is more properly called the Crank-Nicholson routine17,18, and was used by Randles9 in the first ‘paper and pen’ electrochemical simulations (so named because all the arithmetic had to be done manually). The change of time frame for the equation requires knowledge of the concentration at a point at the intermediate time, and a linear interpolation is assumed.

Introduction

C

1 t  t 2 i

C 

t i

 Cit t 2

13



(1.25)

This allows the finite difference expression to be stated as

 2 C t t 2  1 Cit1  2Cit  Cit1  Cit1t   2Cit t   Cit 1t    2 2 x x 

(1.26)

This leads to the full equation for Fick's second law at time t  t as

Cit t   Cit Dx Cit1  2Cit  Cit1  Cit1t   2Cit t   Cit 1t    2 t 2x 

(1.27)

This can be rearranged to give the value of interest.

Cit 1t  

2   1Cit t   Cit 1t   Cit1  2  1 Cit  Cit1  

(1.28)

where   Dt x This expression can be solved simultaneously for all the nodes in a problem by usual methods, and is stable for all values of  . Expressed as a matrix equation we have

1 1  2  1   C 0t t    C 0t  2  1  C1t  C 2t       1  2  1  1  C1t t    C1t  2  1  C 2t  C3t        C 2t t        t t         C3  

(1.29)

in which all the concentrations at the new time step are found from those at the old, just as for the explicit scheme. Note that at the electrode, C0, and at the other extreme of the region, Cn, a boundary condition must be applied. The difference is that for this implicit scheme, the matrix equation is the other way round, with the matrix multiplied by the new concentrations giving a vector containing functions of the old. We have imposed a set of constraints on the values the new concentrations can take, relating them to each other, as well as to the concentrations at the old time step. This adds greatly to the stability of the scheme. Attempts to improve the finite difference technique included the ‘hopscotch’ algorithm19, introduced to electrochemical work by Shoup and Szabo20- 22, and the Alternating Direction Implicit method, first suggested by Peaceman and Rachford23 and introduced into

Introduction

14

electrochemical work by Heinze24,,27. The rationale behind this development was that it would obviously be desirable to apply the implicit technique to two dimensional simulations. This unfortunately leads to a matrix equation with five unknowns, and requires far more computer time to solve, slowing the simulation greatly. The alternating directions method, is a conceptually simple way to gain the benefits of extra stability offered by implicit methods, by using an explicit method for one direction and an implicit method for the other. This leads to matrix equations with three unknowns which can be solved as for a one dimensional implicit problem. In the next timestep the implicit method is used in the other direction, and so on. This gives more stability to a simulation, without an undue overhead in computer time. For the case where convection operates only in the x direction, but diffusion in all three dimensions is important, the governing equation may be stated as

C  2C  2C  2C C  Dx 2  D y 2  Dz 2  Vx t x x y z

(1.30)

The ADI form of equation 1.30 may be formed in three dimensions as follows. If the three coordinates of any point in a regular cubic mesh are given by x  ix , y  jy ,

z  kz , where i, j and k are the indices of the points, which form a series from 0 to the size of the grid in each direction, ni, nj, nk. The grid spacings are found as x  x r ni , y  y r N j ,

z  z r nk , where xr, yr and zr are the sizes of the grid in each direction. The governing equation may be stated as 1 t

C

t  t i , j ,k

    D y C  D z C  u 2x C

 C it, j ,k  D x 2 C it1,tj ,k  2C it,j,kt  C it1, tj ,k

  

2

t  t i , j 1, k

 2C it,j,kt  C it,jt1,k

2

t  t i , j , k 1

 2C it,j,kt  C it,j,kt 1

t  t i 1, j , k

 C it1, tj ,k



(1.31)

Introduction

15









Collecting the constant terms we can state that  x  Dt x 2 ,  y  Dt y 2 ,





 z  Dt z 2 , and  c  ut 2x  . This allows the rearrangement of equation 1.31 to yield at time t  t 3







C it, j ,k   z C it, j ,k 1  2C it, j ,k  C it, j ,k 1   y C it, j 1,k  2C it, j ,k  C it, j 1,k



  x   c  C

t  t 3 i 1, j , k

  2

x



 1 C

t  t 3 i , j ,k

  

x



 c  C

t  t 3 i 1, j , k

 (1.32)



This has three unknowns after the equals sign, with only known factors before, and so may be formulated into a tridiagonal matrix and solved using the Thomas algorithm28. It may be seen that this equation is implicit in the x direction, and explicit in the other directions. At the next step, to t  2t 3 an equation which is implicit in the y direction but explicit in the x and z direction, may be formed, being given by







Cit,j,kt 3   z Cit,j,kt 31  2Cit,j,kt 3  Cit,j,kt 31   c Cit1,tj ,3k  Cit1, tj ,3k



  x Cit1,tj ,3k  2Cit,j,kt 3  Cit1, tj ,3k











(1.33)





  y Cit,j21t,k3  2 y  1 Cit,j 2,kt 3   y Cit,j 21,tk3



Finally, for the z direction the equation shown below may be formed, implicit in z and explicit in x and y.







Cit,j2,kt 3   y Cit,j21t,k3  2Cit,j2,kt 3  Cit,j21,tk3   c Cit12, j ,tk3  Cit12, j ,tk3



  x Cit12, j t,k3  2Cit,j2,kt 3  Cit12, j ,tk3













(1.34)



  z Cit,j,kt 1  2 z  1 Cit,j,kt   z Cit,j,kt 1



The ADI method proceeds by iteration over time, and may be used to find steady state behaviour only by iterating until no further change is observed. The advantages of this method are the ability to incorporate an implicit component into the solution mechanism, and the facility with which the tridiagonal matrices formed may be solved. The complexity of programming this method is exacerbated by the need to use three different types of timestep. Other finite difference formulations exist which allow the steady state to be found directly, including the Backwards Implicit (BI), which relies on marching across the solution region from a completely known boundary, and the Strongly Implicit Procedure, SIP,29,30 which

Introduction

16

abandons the tridiagonal matrix structure for a pentadiagonal form. These increasingly complex FD schemes are not much simpler to execute than the FE methods which are described in the next chapter. As computer power has advanced, more complex implicit Finite Difference schemes, such as the ‘Fast Implicit Finite Difference’ method developed by Rudolph3132, with marginally higher computational requirements, but which have higher potential accuracy when tackling demanding problems, have gained favour. The study of diffuse double layers and electrical migration has also received recent attention from and Rudolph33, using newer modifications of the FD technique, and has been incorporated into a treatment of the Rotating Disk Electrode by Feldberg and Rudolph34, authors of the Digisim program. The use of the ADI technique to model the response of a Scanning Electrochemical Microscope35 by Bard and Unwin36 has opened a new field of application for simulation techniques. More recently the Compton group has contributed greatly to the application of the Finite Difference techniques mentioned to problems involving mass transport3748-51 and coupled kinetics52-60 and to the subsequent development of new variations on previous Finite Difference techniques in the context of electrochemical work, including the Strongly Implicit Proceedure 29 and Multigrid methods61,62. Gavaghan63-65 has recently given a thorough examination of optimum mesh generation and errors present in the FD analysis of the disk electrode using the ADI method.

Other Techniques The application of the other numerical techniques which have gained acceptance in other fields to electrochemical problems has been somewhat fragmented, possibly inhibited by repeatedly successful extensions to the Finite Difference Method, which has not been upstaged by any of the newer techniques presented so far. The main alternatives to the Finite Difference technique in engineering practise have been the Finite Element Method and (particularly in fluid engineering), the Finite Volume66 method, which has received extremely limited attention in the context of electrochemical simulations. The Boundary Element Method67,68 has also received

Introduction

17

recent attention69-74. It is surprising however that there is not more diversity in the methods used in electrochemistry, as the Finite Difference method is still hugely dominant in the current literature. Other techniques advanced have included the Finite Analytical technique, applied to inlaid disk and band electrodes by Jin75 et. al. and the Multidimensional Upwinding Method, applied by Van Den Bossche76,77 et. al. to multi-ion systems at the rotating disk electrode. Included in this category must also be the many analytical approaches used, which attempt to solve the governing equations by formal mathematical methods. Much of the work carried out by digital simulation has sought to verify the predictions made using analytical techniques. Examples include the equivalency of the band and hemicylinder microelectrodes, first proposed by Amatore78, the work on disk electrodes as detectors in chromatography of Anderson79, Fleischmann and Pons’ investigation of microdisk and microring electrodes80 and the theoretical work of Aoki81. The contribution made by Oldham to the theoretical understanding of microelectrodes82-87, electrodes of unusual shape88-93, electrochemistry without supporting electrolyte94,95, with migration effects96, and to the general practice of electrochemistry97-100, have also been very influential. The recent review by Anderson, Coury and Leddy101 gives a comprehensive view of current research across all aspects of dynamic electrochemistry.

Thesis Structure This report is composed of eight chapters and an appendix. The second chapter details the background and formulation of the FEM to solve the governing equations applicable to a variety of electrochemical problems. The third chapter is concerned with the application of the FEM to problems where mass transport is entirely due to diffusion, and chapter four introduces the use of convection where analytical convection profiles may be employed. In chapter five the solution of the Navier Stokes equations for non-analytical flow profiles is discussed, and the resulting velocity profiles for the flow over a cylinder electrode are used to model the steady state hydrodynamic behaviour of the device.

Introduction

18

Chapter six discusses the use of the FEM as a design tool to create two new cell geometries, the Confluence Reactor and Collision Cell. The current/volume flow rate behaviour of each cell has been characterised, and the results for the confluence Reactor are shown to agree with independent experimental observations. Chapter seven shows how homogenous kinetic complications may be incorporated into the model, affording new data for the common reaction schemes in the channel flow cell at extreme rate constants. The Confluence Reactor is shown to be ideal for the investigation of the CE reaction, and working surfaces are presented. Chapter eight reveals the results of three dimensional investigations of the channel flow cell, leading to a corrected volume flow rate equation. The effects of non-uniform convection on electrodes spanning the width of the cell are also discussed. Finally there are appendices containing supplementary information, and examples of the Fortran programs used in this work.

Chapter 2. Finite Element Formulations The Finite Element Method in Electrochemistry The Finite Element method was first used in electrochemical simulations by Penczek et. al.102-104, who demonstrated its accuracy in one dimensional simulations. The one dimensional form was also used by Puy and Galceran105 to model polarographic processes. Galceran106 et. al. went on to demonstrate two dimensional axisymmetric simulations of disc electrodes, using a correction for the singularity encountered at the electrode edge. More recently, Ferrigno107 used a commercial Finite Element package to investigate recessed and protruding microband electrodes, under hydrodynamic conditions, and Bartlett examined the response at an inlaid microdisc108. Speiser109-113 has produced possibly the greatest number of publications on the Finite Element method to date, using the orthogonal collocation formulation, detailed in a recent review article114.

The History of the Finite Element Technique The technique now known as the Finite Element Method is a descendant of the methods of stress analysis used by structural engineers in the 1940s, in which solid structures were treated as assemblies of rods, and planes with simple idealised behaviour115,116. The energy stored in these lattices or frameworks could then be related to the stresses and strains across each individual element and the loads imposed, and the minimisation of this energy then predicted the displacements observed for each part. The Finite Element Method can therefore trace lineage back to the classical Variational methods of analysis which it has now been proved to closely resemble in operation117. The classical Rayleigh-Ritz method of solving partialdifferential equations may be considered to be the immediate parent technique. In this method the linear combination of possibly complex trial functions which satisfies an equation over an entire region is sought. Applying the finite element method, one seeks a set of simpler trial functions, each only defined over discrete regions.

Finite Element Formulations

20

The technique as it is known today was described first by Turner, Clough, Martin and Topp118. Subsequent development in the engineering community showed it to be a special case of the method of weighted residuals119,120, and as a general method for the solution of partial differential equations. As such it has been applied successfully to problems arising in fields as diverse as soil mechanics121, aerodynamics122, heat transfer, and electric circuit analysis123, and is the subject of a vast number of publications. One of the most respected authorities in the field is Zienkiewicz124, whose very comprehensive book, first published in 1967, reached its 4th edition in 1991. Other books which may be recommended include those by Rao125, Dhatt and Touzot126, and Cuvelier, Segal and Steenhoven127.

Finite Element Formulations for Electrochemical Problems The Finite Element Method is quite simple in its basic approach to the solution of a partial differential problem, following the three stages of Discretisation, Approximation and Assembly outlined in Chapter 1. In this chapter, the specific formulations used for the work presented in this thesis will be described. In this section, the concentration is assumed to be the quantity of interest, and the two terms may be used interchangeably. All the methodology presented is applicable to any quantity governed by Poisson’s equation however, heat flow being the obvious analogy125.

One Dimensional Simulations Grid Generation The grids used in the Finite Difference method described in Chapter 1 discretised space into a grid of points, with all the ‘mass’ of the system assumed to be present at these points. In the Finite Element method, the system must be composed of elements which are continuous functions covering the whole space considered. A one dimensional space must therefore be discretised into an assembly of rod-like elements, as shown in figure 2.1.


Figure 2.1

21

Possible element types for a one dimensional simulation.

For each element, the variation of the quantity of interest across the element must be completely defined by the values of the quantity at the nodes. In theory the variation could be assumed to follow any function which can be uniquely defined by a set of points, but the discussion will be restricted to polynomial functions, which are by far the most commonly used in practise. The derivations for the first order element will be given in this chapter, with the second order formulation discussed in the Appendix on page 203. The actual positions and sizes of the elements across the region to be examined may be set arbitrarily, provided that the whole region is covered. To increase the accuracy of the simulations performed, an expanding grid is used. The form of the grid is a geometric expansion with element sizes x, ax, a 2 x, a 3 x  a n 1 x where n is the number of points in a series.

Figure 2.2

Node and element numbering in one dimension.

The algorithm for setting the sizes of each element takes the size of the first element as its input the size of the first element, x, the number of nodes in the series, n, and the total size of


22

the region, s. The size of each element is determined as a i 1 x , where i is the index of the point in the series 1 to n, with the position of each node being given by





x a i  1 a  1

(2.1)

The value of a is determined by iteration. Taking an initial value greater than 1 for a0 1

 a  1s  n a1   0  1 x  

(2.2)

is convergent to the correct solution provided that the solution has a > 1, or if this criterion is not met a converges to 1. Note that if a is given a value too close to 1, a divide by zero error may occur in a program implementing equation 2.1, and the positions of the nodes can be given by s i  1 n .

First Order Element Matrix Formation For an element with only two nodes, the only possible choice of polynomial function which can be used for the variation of the quantity of interest across the element is a straight line. The concentration within the element must therefore vary as shown.

Cj

C x 

C

Ci Nj

Ni xi Figure 2.3

l

xj

A linear element for a one dimensional simulation.

The function describing the variation of the concentration with x, C(x), may be thought of as being an interpolation between the value of C at node i, Ci and the value of C at node j, Cj. This can be formally expressed as

C x   N i x Ci  N j x C j

(2.3)


23

where N i  x  is an interpolation function, which expresses the dependence of the concentration at a point in the same element as node i on the value of the concentration at point i. By

 

examining the diagram above, it can be deduced that N i  xi   1 , N i x j  0 , and

 

correspondingly N j xi   0 , N j x j  1 . From these observations it can be shown that the interpolation functions for a linear element in one dimension are defined as

N i x  

1 x j  x l

(2.4)

N j x  

1  x  xi  l

(2.5)

This now gives a way of relating the concentrations at each node to the local geometry in a systematic way. We can express these relationships in a matrix equation for one element as

 C  NC 

(2.6)

  Ci  N  N i x  N j x  , with the arrow embellishment denoting a vector. and  C j 



where C  



If we examine a series of linked elements, we see a situation like that shown in figure 2.4.

C

C

1 2

1 3

N3 C3

4

N 5 6

0 7

Figure 2.4

1

2

3

4

5

A series of linear elements for a one dimensional simulation.

The interpolation functions for the even and odd nodes are highlighted on separate axes. The interpolation function for a node is non zero only over elements which actually contain a particular node. Note also that the interpolation function for a particular node within


24

an element is independent of the interpolation function for the same node within a different element. The variation of the interpolation function within each element is always between 0 and 1 for these linear elements. It is only one each interpolation function is multiplied by the nodal concentrations that a piecewise approximation to the whole distribution is recovered. This summing process may be expressed as before as ne   C   N eC e

(2.7)

e 1

The governing equations of any partial differential equation involved in the various problems of interest to us may now be expressed as functions of the positions of the grid of nodes created during the discretisation process by substituting in the above summation in place of the concentration. Ignoring the triviality of solving a simple steady-state diffusional problem in one dimension, we have as a governing equation

Dx

 2C 0 x 2

(2.8)

where D is the diffusion coefficient and Cx is the concentration. The first step is to express this in the form of a weighted residual. This is equivalent to finding the sum of all the errors when we substitute our set of approximations into the governing equation we are interested in. These errors are weighted, or multiplied by some other function. Different forms of the Finite Element are derived depending on the choice, but one of the most popular is the Galerkin method, in which the function chosen is the interpolation function. The function generated by this error summing process is set to zero, and solved. The weighted residual form of equation 2.8 for one element is xj   2C D x  N 2 dx  0 x xi

This is next integrated by parts, and the minus signs cancelled, to give

(2.9)


25

 j N C   C  Dx  dx  Dx  N  x x  x  i xi xj

(2.10)

 C N  Noting equation 2.6 it can be seen that  C and so equation 2.10 can be rewritten as x x    N N C  Dx  dx C  D x N x x x xi

 

xj

j i

(2.11)

We must recognise that the vector of nodal concentrations is independent of x, and can be removed from the integral on the left hand side. Furthermore, the gradient term on the right side is a boundary condition which cancels out on all inter-element boundaries (as node j of one element is node i of the next). At the edges of the grid defined, this becomes a flux boundary condition, and may be set to 0, if no flux is set in the statement of the problem. Referring back to equations 2.4 and 2.5, and remembering that to integrate a constant between xi and xj is to multiply by l,

 N  1 1   x  l l 

Dx l

 1  1   1 1 C  0  

(2.12)

(2.13)

The matrix created is known as the characteristic matrix K  for the element. The flux boundary condition is discussed below. As equation 2.13 is valid for one element, the problem can be solved for summing the left hand matrices and right hand vectors for all elements, to create a global matrix equation, which can be solved for all nodal concentrations. The contributions to the right hand side form



the load vector P , giving the global equation





K C  P

(2.14)

Unsteady Problems To simulate many processes of interest the evolution of a transient concentration profile with time is required. This is achieved by the derivation of another element matrix: the


26

element capacitance matrix. This originally arose from the consideration of the dynamics of motion of structures128, and the Lagrange equations129 for the kinetic and potential energy stored in them. Where time dependant problems are encountered, the governing equation for a one dimensional problem may be expressed as

C  2C  Dx 2 t x

(2.15)

  2 C  C Dx  N 2  N dx  0 t x xi

(2.16)

The weighted residual form is as before xj

The solution proceeds as before, accompanied by the substitution

C     NC where t

   C i t  C . C j t    N N     C   Dx  C  NN Cdx  D x N x x x xi

 

xj

j

(2.17)

i

The characteristic matrix for diffusion is as given in equation 2.13 , with the evaluation of the capacitance matrix giving xj   1  x l 2 KT    xi 1  x l  x l 

1  x l x l  dx  l 2  6 1 1  x l 2 

1 2

(2.18)

By using an implicit finite difference scheme for temporal discretisation, we can rearrange the basic equation

     K T C  KC  P

(2.19)

in a manner akin to the Crank-Nicholson technique. This is achieved by approximating

      C  C1  C0 t where C1 is evaluated at time t  t 2 , and C0 is evaluated at time





t  t 2 . As the time derivative is being calculated at the centre of a timestep, so must the other




















27



quantities, giving C  C1  C0 2 and P  P1  P0 2 . Substituting these approximations into equation 2.19 gives

  2     2      K  K T C1    K  K T C0  P1  P0 t t    

(2.20)

This equation contains only one unknown, the concentrations at the new time, and can be solved as before. It may be noted that although the right hand side of this equation contains the product of a matrix and the vector of old concentrations, the matrices themselves do not change necessarily with time, for constant time steps. Variable Time Stepping In order to most efficiently use the computational facilities available, it is desirable to discretise the time dimension in a manner akin to the physical space being simulated, in which the finest divisions occur at the points where the rate of change is highest. For this reason it is desirable to use very small time steps at the start of a simulation and advance in progressively larger increments as the solution reaches a steady state. The same algorithm can be used for the time stepping as is described on page 20 for the geometric advancement, with the size of the first timestep tf, the number of timesteps nt and the total time tt being supplied as input. Examining equation 2.20, it can be seen that there are two options. The first choice is

  2    2   K T  and   K  K T  to be formed into two matrices, which can easily be t t    

for  K 

done if t is to stay constant throughout the simulation. The other option is to generate the





global matrices for K and K T separately, and then recreate the terms needed at each timestep. Although there is an immediate cost per timestep of this approach, it is soon repaid by the ability to use fewer, larger timesteps at the end of a transient. A typical transient problem over 1s might be attempted with 250 timesteps. Using a geometrically expanding function for t, the initial timestep could be set to 5  10-6s, with the size of each timestep being 3.62 % larger than the previous one, to have the last timestep equal to 0.035 s. Comparing this to the alternative of each timestep being fixed at 0.004 s, the


28

accuracy gained by the timesteps being two or three orders of magnitude smaller at the very start of the experiment greatly outweighs any loss of accuracy at the end of the run when they become up to one order of magnitude larger. Convective Transport Although there are not many systems of interest in which a one dimensional approximation may be valid, the Rotating Disk electrode156 is amenable to a one dimensional simulation, as used in the popular commercial package Digisim161. The governing equations become

Dx

 2C C u 0 2 x x

(2.21)

where u is the velocity away from the electrode, and the Galerkin form is therefore

     N N N C  Dx   uN dx C  D x N x x x x xi

 

xj

j i

(2.22)

The extra term reflecting the addition of convection may be solved as

 u  1  1 KC   2  1 1 

(2.23)

This matrix is treated exactly as is the matrix for diffusion, for steady state and transient problems. Sources and Sinks In many systems, the species under study may be subject to process which cause it to decay, or it may be formed from other species. These processes may occur homogeneously in solution, or heterogeneously at interfaces. For the case of a flux at an interface, the gradient boundary term arising from the right hand side of 2.10 becomes non-zero. The flux boundary condition simply evaluates to

ki

C  D x  C  0  for a flux at node i, or k j for a flux at node j, where the k term is the   x  D x  x  0 

strength of the flux. Referring to equation 1.1 if the unit of C is Molcm-3, and the unit of length


29

is cm, and that of time s, the flux has units Molcm-2s-1. Note that the fluxes defined by gradient boundary conditions at the boundary may be positive or negative. Fluxes over the whole of the element may be of two types. The flux may be a function of the local concentration of the species, or not, and may be positive or negative. In the work presented in this thesis, no examples of positive fluxes dependent on the local concentration of the same species have been simulated, although any self-catalytic reaction would require such a term. It seems physically unreasonable that any negative flux can be truly independent of the local concentration of a species however, or else negative concentrations may result. In the presence of a positive flux of magnitude ks, independent of the local concentration of the species in question, the governing equation becomes

Dx

 2C  ks  0 x 2

(2.24)

and the Galerkin form is therefore

   xj  N N C  Dx  dx C   k s N dx  D x N x x x xi xi

 

xj

j i

(2.25)

The element load vector formed may be added to the global load vector on the right hand side of the equation and has the value

k s l 1 2 1

(2.26)

Where a species decays or is formed with a rate which is a function of the local concentration, the governing equation becomes.

Dx

 2C  kd C  0 x 2

(2.27)

where kd is given the sign appropriate for a species decaying. The Galerkin form is therefore

    N N C  Dx   k d NN dx C  D x N x x x xi xj

 

j i

(2.28)

The matrix formed is identical to the capacitance matrix given in equation 2.18, multiplied by the rate of decay kd.


30

Remarks on the matrices generated In the generation of these matrices, a number of practical points may be noted. Firstly, the matrix for diffusion and the capacitance matrix are symmetrical, but that for convection is not. Therefore if a simulation does not incorporate convective transport, efficiencies in the storage of the matrix are possible, and a range of extra techniques may be used for the solution of the matrix. Secondly, in checking programs for errors, the fact that in a diffusive matrix every row or column sums to zero, for a convection matrix every row sums to zero, and for a capacitance matrix, the sum of all the coefficients in the whole matrix is one. An element load vector also always sums to one. These characteristics form a very helpful check when larger matrices are derived.


31

Two Dimensional Simulations In two dimensions, the choice of elements admissible to discretise a region is very large. As in one dimension, the choice of element dictates the polynomial order of the function that will be used to approximate the concentration inside the element. The three elements that have been investigated and used in the course of these studies are the linear triangular element, the bilinear rectangular, and isoparabolic rectangular elements. The matrices and details of these elements are presented. For the sake of keeping the two dimensional simulations presented consistent with the three dimensional equivalents, the two dimensional derivations choose z to be the vertical direction.

Linear Triangular Elements Grid Generation A simple example of a two dimensional triangular grid is presented, for a simple flat electrode geometry.

Figure 2.5

A triangular mesh for a two dimensional simulation.

The elements are assembled together to form a set covering a rectangular region over and either side of the electrode. The grids expand away from the electrode edges, and the same algorithm is used to generate the expanding grids as in the one dimensional simulations. The first element sizes over each region, the number of element over them and the sizes of each region are supplied as input to the programs used. The nodes may be numbered in any scheme, but computational efficiencies result from a choice of numbering which minimises the


32

difference between the largest and smallest numbers within an element. A normal scheme of numbering each row or column sequentially is satisfactory. The nodes are also named within each element in an anticlockwise direction. Note that each node which does not lie in a boundary region is now shared between six elements and that the relationship between element numbering and the node labelling is not as simple as in one dimension. Figure 2.6 shows one corner of mesh with nine elements in the first row, with nodes and elements numbered from left to right in each row, starting from the bottom row.

Figure 2.6

Triangular mesh numbering.

The numbers of the elements are shown in italic, along with the global node numbers and local labels of the nodes within each element. In the process of grid generation for a system like this, an array is normally created at the start of the program to relate the local and global node labels. In the programs in this thesis, this array is simply called NODE. For the grid shown in figure 2.6 the elements shown would be included as shown below, with the global node number corresponding to a local node label given for each element. Element

i

j

k

1

1

2

11

2

11

2

12

3

2

3

12

19

11

12

21


33

Matrix Formation The interpolation functions for a triangular element may be defined in terms of local coordinates just as for the one dimensional elements described previously.

Figure 2.7

Triangular natural coordinates.

Figure 2.7 shows that the local coordinates of a point p in a triangular system, with respect to one of the vertices of the triangle, may be defined as the area of the triangle formed by that point and the other two vertices of the triangle, divided by the area of the whole triangle. If the point p is moved away from a vertice, the triangular subsection formed diminishes, until it reaches zero at the opposite side of the triangle. It can also be seen that the maximum value for the local coordinate is 1 if p is at a node of the triangle. Looking again at these characteristics it can be seen that this local coordinate and the interpolation function for a triangular element are one and the same function. If A is the area of the whole triangle,

Ni 

Ai A

Nj 

Aj A

Nk 

Ak A

(2.29)

where A is the area of the whole triangle. It is obvious from figure 2.7 that these functions sum to 1 at any point, and must be equal to 1 at the associated node and zero at the other two nodes, and are specified over the element by the values at these nodes. Local coordinates with the property that all coordinates only vary from zero to one are known as natural coordinates. It is also clear that there are only two degrees of freedom, and so a point may be specified uniquely by any two of these natural coordinates.


34

An elegant derivation of these interpolation functions in real space is to state that for any point p the x and y coordinates may be found from the nodal coordinates as

1   1  x   x    i  z   z i

1 xj zj

1  Ni   x k   N j  z k   N k 

(2.30)

This is then inverted to obtain

 Ni   ai  N   1 a  j  2A  j  N k  a k

bi bj bk

ci  1  c j   x  c k   z 

(2.31)

where ai  x j z k  xk z j , a j  x k z i  xi z k , a k  xi z j  x j z i , bi  z j  z k , b j  z k  z i ,

1 xi 1 bk  z i  z j , ci  x k  x j , c j  xi  x k and c k  x j  xi , and A  1 x j 2 1 xk

zi zj zk

These interpolation functions may be visualised as shown in figure 2.8. C x

Cj

C(x,z)

Cj Nj

z Ci

Nk Ck

Ck xj, zj

xk, zk

xk, zk xi, zi

Figure 2.8

xi, zi

Triangular interpolation functions.

The general form of the equation governing the flux of any species through the two dimensional region shown, where diffusive and convective transport operate, and there are sources and sinks where the species is generated or decays is given by


35

C  2C  2C C C  Dx 2  Dz 2  u w  kd C  ks t x z x z

(2.32)

where C is the concentration of the species, u and w are the velocities in the x and z directions and ks and kd are the rates of generation and decay of the species at the point in question. The Galerkin form of this equation is

        N  N  N N N N D  D  u N  w N  k N N d x d z C  d A x x x z z z x z   C  C k N d x d z  D N cos  d s  D N s x z A S x S z sin  ds

(2.33)

where the angle  is the angle between the x axis and the normal to the surface at a point where a gradient boundary condition is applied. For each triangular element the matrices and the load vector may be evaluated, assuming diffusional isotropy, where D = Dz = Dz as

 bi2  ci2 K   D bi b j  ci c j 4A bi bk  ci c k 

bi b j  ci c j b 2j  c 2j b j bk  c j c k

bi bk  ci c k   b j bk  c j c k  bk2  c k2 

(2.34)

bi u K C   bi 6 bi 

c i bk   w bk   ci 6 c i bk  

cj cj cj

ck   ck  c k 

(2.35)

bj bj bj

2 1 1  kd A  K T   1 2 1   12 1 1 2 1  Dk si li   Dk sj l j P 1  2   2 0

(2.36)

0  1 1 1  Dk sk l k 0  k s A 1   2   3  1 1 1

(2.37)

where kd is the size of the rate of decay, ksi, ksj and ksk are gradient boundary conditions for sources which operate on only the edges of the element (on the sides indicated), and ks accounts for sources over a whole element. A is the element area as defined as for equation 2.31, and l is the length of the side indicated. The complete equation is given by





K   K C   KT C  P

(2.38)


36

  N N In deriving the matrices above, the values of and are obvious from their x z



definition in equation 2.31, but the integration of N over the element may be greatly simplified by considering a special property of natural coordinates. The one, two and three dimensional cases are presented: xj

L

 i

Lj dx 

xi

LLL

dA 

LLLL

dV 

A

V

 i

 i

 j

 j

 k

 k

 l

!! l     1!

(2.39)

2!! ! A       2!

(2.40)

6!! !! V         3!

(2.41)

where Li is the natural coordinate which is 1 at node i, l is the length of a one dimensional region, A is the area in two dimensions, and V is a three dimensional volume. The factorial terms may be evaluated easily and their reciprocals are given below.

    1!

      2!

        3!

!!

2!! !

6!! !!

-

2

3

4

-

-

3

6

10

-

-

-

4

10

20

1

1

-

-

6

12

20

2

1

-

-

12

30

60

3

1

-

-

20

60

140

1

1

1

-

-

60

120

2

1

1

-

-

180

420

3

1

1

-

-

420

1120

2

2

-

-

30

90

210

2

2

1

-

-

630

1680

1

1

1

1

-

-

840

2

1

1

1

-

-

3360









1

-

-

2

-

3

Returning to the example above, the integral term for the capacitance matrix is


 N i2  A  N i N j Ni N k 

Ni N j N 2j N j Nk

37

Ni Nk   N j N k  dA N k2 

(2.42)

As the interpolation functions are natural coordinates of the elements, the matrix can be found as shown in equation 2.36. Note that an edge integral for the side of a two dimensional region described in natural coordinates may be given by equation 2.39, and a surface integral for a three dimensional region may be given by equation 2.40.

Current Calculation for a Triangle The gradient in the direction normal to an element surface may be derived by considering the direction of the normal to be of the form n  cos  x  sin  z giving

C C C  cos   sin  n x z

(2.43)

where  is the angle measured anticlockwise between the x axis and the outward normal to the l

surface. The total flux can be derived as

C

 n ds where l is the length of the side. For simple 0

elements where the gradient is invariant over a side this gives just l

C . n

Referring back to the definition given for a triangular interpolation function in equation 2.31, the concentration gradient normal to the element surface may be found as



C 1 cos  bi  n 2 A

bj





bk  sin  ci

cj

 C1  c k C 2  C 3 



(2.44)

The complete expression for the current as a sum over each element from 1 to ne, where ne is the number of elements over the electrode, may therefore be given by ne

I   nFDwC Bulk  l e e 1

C e n

(2.45)


38

Where the electrode is uneven and the grid used is not perfectly regular this scheme is the only one which can really be relied upon, where the interpolation function used to derive the current is exactly the same as the one used to generate the concentration profile. For situations where a much more regular grid is used, with a flat electrode, using a square interpolation function has been found to show a small improvement. In this case the current is given by a sum evaluated over each square pair of elements immediately over the electrode.

Figure 2.9

Square pair of elements used for current calculation.

Using the triangular flux calculation scheme, the flux across the bottom side of element 1 in

 C1  l  l 0 l  C 2  for a perfect right angled triangle, which is figure 2.9 would be given by 2A C11  equal to

l C11  C1  , using the global numbers for the nodal concentrations, which is the bj

obvious expression for the gradient along the left side of element 1. If the assumption of a square interpolation function akin to that described on page 205 is made then the flux may be approximated as

l C11  C12  C1  C 2  2b j This expression for the flux may be substituted into equation 2.45.

(2.46)


39

Two Dimensional Axisymmetric Formulation The use of axisymmetric coordinates allows seemingly three dimensional forms such as the disk, hemispherical or cylindrical electrode to be modelled without resorting a full three dimensional representation. The coordinate axes used in an axisymmetric system are shown in figure 2.10, with the cross section of the region to be simulated shown.

Figure 2.10

A cubic discretisation of a volume surrounding an electrode.

The discretisation of an axisymmetric region into linear triangular elements may be accomplished using a numbering scheme exactly as shown in figure 2.6 for two Cartesian dimensions. The interpolation function in the radial direction r and vertical direction z may be found as

 Ni   ai  N   1 a  j  2A  j  N k  a k

bi bj bk

c i  1  c j   r  c k   z 

(2.47)

where ai  r j z k  rk z j , a j  rk z i  ri z k , a k  ri z j  r j z i , bi  z j  z k , bi  z k  z i ,

1 ri 1 bk  z i  z j , ci  rk  r j , c j  ri  rk and ck  r j  ri , and A  1 r j 2 1 rk

zi zj zk

The equation governing the diffusive flux of the species of interest through the two dimensional region shown is given by

C  2 C D C  2C  Dr 2  r  Dz 2  k d C  k s t r r r z The first order term may be combined with the second derivative in r, as

(2.48)


40

1   C    2C  Dr  r    Dz 2  0 r  r  r   z

(2.49)

Using integration by parts, the factors of r cancel, and the volume integral can be expressed as

dV  2r dr dz , where r is the radial distance to the centroid of the element, found by expressing r in terms of the natural coordinates r  ri N i  r j N j  rk N k . For the purposes of evaluating the characteristic matrix for diffusion, as no other functions of the natural coordinates appear, it is possible to find the centroid of the triangle as the integral

r

2

dA . This evaluates as

r

 r 1 6

i

2

 r j2  rk2  ri r j  ri rk  r j rk



1 2

(2.50)

The Galerkin form of this equation is then given as

      N N N N 2r  Dr  Dz  k d NN drdz C  A r r z z  C  C Dr  N cos  ds  Dz  N sin  ds S S r z

(2.51)

Note the surface integrals are given as integrals over dS, where the surface S may be found as

S  2r ds , where s is the two dimensional surface. Assuming diffusional isotropy as before, the characteristic matrix may therefore be given as

 bi2  ci2 K   r D bi b j  ci c j 2A bi bk  ci c k 

bi b j  ci c j b 2j  c 2j b j bk  c j c k

bi bk  ci c k   b j bk  c j c k  bk2  c k2 

(2.52)

The capacitance matrix is given in terms of the Natural coordinates, but as these are also employed to determine r , they appear in terms up to the third order, rather than just the second order, as for the Cartesian triangle. The matrix is therefore given by

6ri  2r j  2rk k d A  K T   2ri  2r j  rk 30   2ri  r j  2rk  where kd is the rate of decay.

2ri  2r j  rk 2ri  6r j  2rk ri  2r j  2rk

2ri  r j  2rk   ri  2r j  2rk  2ri  2r j  6rk 

(2.53)


41

The load vector also differs from that for the Cartesian triangular element, as indicated above

 2r  r j   Dk si li  i   Dk sj l j P r  2 r i j   3 3  0 

 0    Dk sk l k 2r j  rk   3 r j  2rk   

2ri  rk   0   k s A   6 ri  2rk 

2ri  r j  rk    ri  2r j  rk  ri  r j  2rk    (2.54)

where, ksi, ksj and ksk are gradient boundary conditions for sources which operate on only the boundaries of element on the sides indicated, and li, lj and lk are the lengths of the sides. Current Calculation The calculation of the flux at the electrode requires the evaluation of the expression

C

 n dS

where S is the surface of the annular element. For a linear triangular element this is

given by 2r l

C , where l is the length of the side in the r direction, and the radial distance to n

the centroid r is as given in equation 2.50. The gradient may be calculated as in equation 2.44. To use an interpolation function akin to that given in equation 2.46 for the current calculation is desirable though, where the gradient is assumed to vary linearly across the element edge. Using an example numbering system as shown in figure 2.11, the flux may be more accurately derived.

Figure 2.11

Flux calculation for a linear triangle in axisymmetric coordinates.

At distances far from the axis of symmetry, the gradient may be approximated as being equal to the mean of the gradients at each edge. This ignores any weighting due to the radial coordinate though. Assuming a linear variation of the gradient along an element edge the gradient may be expressed using the interpolation functions as

C j Ci C ,  Ni  Nj n n n


where

C j n

42

is the gradient at node j. Recalling that the surface in axisymmetric coordinates

may be given as S  2r ds , where ds is the edge of the element in two dimensions and that the r coordinate may be expressed in terms of the natural coordinates as

r  ri N i  r j N j  rk N k . This gives for the gradient



2 N i2 ri  N i N j r j  N i N k rk

 Cn  N N r  N i

i

j i

r  N i N k rk

2 j j

 Cn

j

ds

Referring to equation 2.40 to evaluate the terms in natural coordinates, this gives

C j  C 2l  13 ri  16 r j  16 rk  i  16 ri  13 r j  16 rk  n n 

  

(2.55)

where l is the length of the element side in the r direction, and the gradients at each node may be evaluated using equation 2.44. The current may then be evaluated using

I   nFDC Bulk J where J is the flux found in equation 2.55.

(2.56)


43

Three Dimensional Simulations The development of three dimensional formulations for the equations governing mass transport follows an identical scheme to the steps taken in two dimensions, with the governing equation being

C  2C  2C  2C C C C  Dx 2  D y 2  Dz 2  u v w  kd C  ks t x y z x y z

(2.57)

The Galerkin form of the equation is

           N N N N N N N N N V Dx x x  D y y y  Dz z z  u x  v y  w z  k d NN dxdydz C   C  C  C   k s N dxdydz  D x  N lds  D y  N mds  D z  N nds V S S S x y z (2.58) where l  n.i , m  n.j , n  n.k , where i , j , and k are the unit vectors in the directions of the x, y, and z axis, and n is the outward surface normal. Grid Generation The most obvious unit for discretising three dimensional regions is the cube. A schematic of the sort of grids that may be used in the region around an electrode is shown below. The same expanding series of elements may be used as for the one and two dimensional elements (page 20).

Region over Element

Figure 2.12

A cubic discretisation of a volume surrounding an electrode.


44

Tetrahedral Elements The particular strength of a tetrahedral mesh is that terahedrons are suited to tessellating regions of complex geometry. A tetrahedron may also be subdivided by the addition of an internal point into four smaller elements, without changing the faces of the parent shape, allowing considerable scope for automatic mesh refinement. Where the geometric requirements allow the use of rectangular elements however, they have been found to be marginally more efficient than tetrahedrons, and therefore although much early work in this project employed tetrahedral element meshes, the final results presented in this thesis have not used them. The FE formulation of tetrahedral meshes is therefore given in the appendix, on page 216.

Cubic Elements Trilinear Cubic Element A trilinear cubic element with nodes at each corner may be formed as shown in figure 2.13, where a, b and c are the dimensions of the cube in the x, y and z directions.

Figure 2.13

Node numbering for a trilinear cube.

The trilinear cubic element may be defined in terms of a simple local coordinate system, exactly as for the rectangular two dimensional elements with the addition of a third term derived from the z coordinate.




2 x  xm  a



2  y  ym  b

45



2 z  z m  c

(2.59)

The basis function which can be defined using eight nodes in three dimensions is

 B  1 x

y

z

xy xz

yz

xyz . This can be seen to be a symmetrical choice from the

pyramid of three dimensional terms in figure 2.14. 1

Constant

y x

z

Linear

yz

Quadratic

y2 xy x2

z2

xz y3 zy2

xy2 x2y

yz2

xyz

x3

x2z

xz2

Cubic z3

y4 xy3 x2y2 x 3y x4 Figure 2.14

y3z xy2z

x2yz x3z

y2z2 xyz2

x2z2

Quartic yz3

xz3

z4

Pascal’s pyramid of three dimensional terms.

Using equation 9.15 the interpolation functions can be shown to be

Nn 

1 8

1   n 1   n 1   n  

n  1 8

(2.60)

where  n is the value of the coordinate of node n. It is obviously more difficult to visualise these three dimensional interpolation functions. It may be useful to consider that these three dimensional interpolation functions give functions identical to the two dimensional ones defined previously, when evaluated on each face of the cubic element. Figure 2.15 shows a cubic interpolation function, evaluated on the three faces of a cube for which it is non-zero.


Figure 2.15

46

A cubic interpolation function, evaluated across the surfaces of a cube.

The matrices for this element are

4 - 4  - 2  Dbc 2 K   36a 2  - 2 - 1  1

-4 4 2 -2 -2 2 1 -1

-2 2 4 -4 -1 1 2 -2

2 -2 -4 4 1 -1 -2 2

2 -2 -1 1 4 -4 -2 2

-2 2 1 -1 -4 4 2 -2

-1 1 2 -2 -2 2 4 -4

1  -1 -2  2 2  -2 -4  4 

4 2  - 2  Dac - 4  36b 2  1 - 1  - 2

2 4 -4 -2 1 2 -2 -1

-2 -4 4 2 -1 -2 2 1

-4 -2 2 4 -2 -1 1 2

2 1 -1 -2 4 2 -2 -4

1 2 -2 -1 2 4 -4 -2

-1 -2 2 1 -2 -4 4 2

-2 -1 1   2 -4  -2 2  4 

4 2  1  Dab 2  36c - 4  - 2 - 1  - 2

2 4 2 1 -2 -4 -2 -1

1 2 4 2 -1 -2 -4 -2

2 1 2 4 -2 -1 -2 -4

-4 -2 -1 -2 4 2 1 2

-2 -4 -2 -1 2 4 2 1

-1 -2 -4 -2 1 2 4 2

-2 -1 -2  -4 2  1  2  4 

(2.61)


- 4  - 4  - 2  - 2 K C   ubc  72 - 2  - 2  - 1   - 1

47

4

2

-2

-2

2

1

4

2

-2

-2

2

1

2

4

-4

-1

1

2

2

4

-4

-1

1

2

2

1

-1

-4

4

2

2

1

-1

-4

4

2

1

2

-2

-2

2

4

1

2

-2

-2

2

4

- 4  - 2  - 2   yac - 4  72 - 2  - 1  - 1   - 2

-2

2

4

-2

-1

1

-4

4

2

-1

-2

2

-4

4

2

-1

-2

2

-2

2

4

-2

-1

1

-1

1

2

-4

-2

2

-2

2

1

-2

-4

4

-2

2

1

-2

-4

4

-1

1

2

-4

-2

2

- 4  - 2  - 1   wab - 2  72 - 4  - 2  - 1   - 2

-2

-1

-2

4

2

1

-4

-2

-1

2

4

2

-2

-4

-2

1

2

4

-1

-2

-4

2

1

2

-2

-1

-2

4

2

1

-4

-2

-1

2

4

2

-2

-4

-2

1

2

4

-1

-2

-4

2

1

2

-1  -1  -2  -2  -2  -2  -4   -4 2  1  1  2  4  2  2   4 2  1  2  4  2  1  2   4

(2.62)


8 4  2  V 4 K T    162 4  2 1  2

48

4

2

4

4

2

1

8

4

2

2

4

2

4 2

8 4

4 8

1 2

2 1

4 2

2

1

2

8

4

2

4

2

1

4

8

4

2

4

2

2

4

8

1

2

4

4

2

4

1 1 1 1 1 1 0  1        1 0  0  1        Dk s1 ab 1 Dk s 2 ac 0 Dk s 3bc 1 k sV 1 P    4 0  4 1 4 1 8 1        0  1 0  1 0  0  0  1        0 0 1 1

2 1  2  4 4  2 4  8 

(2.63)

(2.64)

where V the volume of the element. For the element load vector, only three sides are shown for the gradient boundary conditions. Current Calculation for a Trilinear Cube As in two dimensions flux is given by the integral of the concentration gradient in the normal direction, here given by

     C N N N  C l d S  D l d S  D m d S  D n d S x y z S n    S x  S y S z  

(2.65)

where l  n. i , m  n.j , n  n.k , where i , j , and k are the unit vectors in the directions of the x, y, and z axis, and n is the outward surface normal. Assuming a regular rectangle aligned with the coordinate axes the flux for one element can be evaluated for the trilinear element as

Je 

ab C5  C6  C7  C8  C1  C2  C3  C4  4c

(2.66)

The total flux is found as a sum over all the elements over the electrode surface, and the current can be calculated from I   nFDC Bulk J .


49

Finite Element Forms of The Navier-Stokes Equations The Navier Stokes Equations and continuity equation for a fluid are given in equations 1.11 to 1.14. There exist in the literature a number of methods of formation of these equations127, some of which attempt to eliminate the pressure from the equations, and thereby decrease the time required for the computations. These methods are not available for three dimensional calculations, and present extra difficulties in programming when compared to the basic Finite Element form presented.

Two Dimensional Form of the Navier Stokes Equations Continuing to define the z coordinate as the vertical direction for consistency between two and three dimensional work, the two dimensional Navier Stokes equations and continuity equation may be given as:

   2 u  2 u  1 p  u   u   2  2    u   w   0   x z   x  x   z 

(2.67)

   2 w  2 w  1 p  w   w   2  2    u   w   0   x z   z  x   z 

(2.68)

u w  0 x z

(2.69)

where  is the viscosity and  is the density of the fluid.



The Galerkin forms of these equations are found using N as the interpolation









function for u and w, and M for the pressure p. U , W and P are the vectors of u, w and p values at the nodes of the elements. The gradient boundary terms and element load vector are not given, as they may be assumed to be zero for all the circumstances discussed.

        N  N    N N  N N 1  M A  x x   z z  uN x  wN z dxdz U  A  N x dxdz P  0 (2.70)         N  N    N N  N N 1  M A  x x   z z  uN x  wN z dxdz W  A  N z dxdz P  0 (2.71)


50

   N   N  M d x d z U  M d x d z W 0 A x A z

(2.72)

These equations must be solved simultaneously, and so may be assembled as shown

 

K a   K g  0   K e  



K b  U    K c   K h  K d  W   0  0

K 

(2.73)

0   P 

f

where , the kinematic viscosity, is defined as     . The zero in the pivotal position of the third row of the above equation imposes certain conditions of the range of interpolation functions which can be used for the pressure and the x and z velocities. In practice only the use of elements with more degrees of freedom for the velocities than the pressures can ensure non-singular matrices. The interpolation functions chosen are therefore an eight noded rectangle for the velocities, and a linear four noded rectangle for the pressure. The matrices K a  and K c  can be seen to be both equal to the characteristic matrix (multiplied by  rather than D) for the isoparabolic element given in equation 9.22, and the

 

matrices K g and K h  can each be seen to be equal to the matrices for convection in the x and z direction respectively for the element, given in equation 9.23. The other matrices are given by

 1   M   dxdz b    N  A  x  1 - 8  1  b - 6 b 36 2  - 4 2  - 6

-1

-2

8

4

-1 6

-2 6

-2

-1

4

8

-2

-1

6

6

(2.74)

2 -4 2  -6 1   -8 1   -6


51

 1   M   dxdz d    N  A  z  1 - 6  2  a - 4 d 36 2  - 6 1  - 8

2

-2

-6

6

1 -8

-1 8

1

-1

-6

6

2

-2

-4

4

(2.75)

-1 6  -2  4 -2  6 -1  8 

   N   dxdz e    M  A  x  

- 7  b - 2 e 36 2  1

-6 -6 -6 -6

-2 -7 1 2

4 8 -8 -4

-2 -1 7 2

(2.76)

6 6 6 6

-1 -2 2 7

8 4  -4  -8

   N   dxdz f    M  A  y  

- 7  a 1 `f  36 2  - 2

8 -8 -4 4

-1 7 2 -2

6 6 6 6

-2 2 7 -1

(2.77)

4 -4 -8 8

-2 2 1 -7

-6 -6 -6  -6

Solution Technique The procedure to solve the above matrix equations is necessarily iterative, due to the

   N  N presence of the terms containing a quantity multiplied by its derivative, e.g. uN ,  wN x z in the governing equations. It is therefore necessary to use some sort of iterative method, the simplest being the method whereby a solution is found for some initial assumed value of u, v, and w and the solution then proceeds using the results of the previous iteration. Suitable initial guesses may be zero, or the result of a previous simulation for a similar set of parameters. This is known as the Picard iteration technique127. This is not the most efficient method available in


52

computational terms, requiring the global matrix to be reassembled at least in part on every iteration, but has been found to be adequate.

Three Dimensional Formulation The three dimensional formulation of the Navier-Stokes equations is conceptually identical to that of the two dimensional form shown above. The full derivation and matrices are therefore given in the appendix, on page 224.

Chapter 3. Diffusional Simulations In this chapter, simulations involving diffusional transport only are considered. The first section details the results of simulations of transient experiments for geometries which may be modelled in one or two Cartesian dimensions. These are then followed by simulations of transients at axisymmetric electrodes, including transients tending to the steady state, which can be achieved at microdisc electrodes.

One Dimensional Potential Step Transients Consider the one electron transfer reaction

A  e  B

(3.1)

If migration is neglected then the mass transport of species A to the electrode will be governed only by convective and diffusive transport. For a large planar rectangular electrode it has been established that the mass transport in directions other than the direction normal to the surface is insignificant, and the equation governing the mass transport is simply

C C 2 D 2 t z

(3.2)

where z is the distance from the electrode into the solution, and C is the concentration of the species in question. Simulation of diffusional transport under electrochemical perturbation using the finite element method was performed by dividing the region under investigation into a series of elements. A schematic of a one dimensional grid as used in the simulations is shown in figure 3.1.

Reaction Systems

Figure 3.1

54

Discretisation of a one dimensional region.

Node number one is assumed to lie on the surface of the electrode, and a one dimensional grid extending in a direction orthogonal to the electrode surface to z = zr (where zr is the height of the simulated region above the electrode surface) is generated. The number of nodes in the z direction is given by nz, with the size of the elements being given by the expanding algorithm described in Chapter 2, on page 20, with the size of the first element adjacent to the electrode given by the parameter zf. One Dimensional Boundary Conditions On the surface of the electrode, it is assumed that complete conversion of species A to species B occurs. The concentration at the node at the top of the region simulated is assumed to be equivalent to the bulk value, CBulk. The initial condition of C = CBulk over the whole region was also set. The boundary conditions are: z=0

C1 = 0

(3.3)

z = zr

Cnz = Cbulk

(3.4)

It must be noted that for experiments with no convective transport introducing material through the top edge of the simulation, the specification of the fixed boundary condition is optional. By virtue of the surface integral term introduced into the Galerkin form of the governing equations, if no other boundary condition is set, an edge of a simulated region will be governed by a no flux boundary condition

C  0. z

Reaction Systems

55

One Dimensional Current Calculation The program calculates the concentrations of the species under study at each node point. These are used to derive the current by calculating the flux to the electrode during each time step by evaluating the concentration gradient at the electrode surface. The formal

 N 1  C1 , expression used to derive the gradient of the concentration profile at the electrode is z with the 1 subscript indicating that this is evaluated for element 1 only. For a linear element this gives  1l

1 l



C1  C  C1 , or in the obvious form 2 , where l is the element length and C1 and  l C 2 

C2 are the nodal concentrations at the electrode and at the first node above. As the length of the first element is defined as zf this gives for the current at the electrode

I  nFDAC Bulk

C 2  C1 zf

(3.5)

where A is the electrode area, CBulk is the bulk concentration of the A species, F is Faraday’s constant, D is the diffusion coefficient, and n is the number of electrons transferred by each molecule of species A which is reduced. One Dimensional Potential Step Results The Cottrell equation130,1 predicts for a reversible one electrode reduction,

I 

nFAD 1 2 C Bulk 1 2 t 1 2

(3.6)

Finite element simulations were performed and the results compared with those predicted by the Cottrell equation. Typical results are shown in figure 3.2 for simulations with nz = 12, zr = 0.02 -0.06 cm, nt = 200, tt = 1 s, and zf = 0.000 4-0.001 cm, for an electrode area of 0.4 cm2 and CBulk of 1 mmol The diffusion coefficients are shown in cm2s-1.

Reaction Systems

56

1.4

D / cm2s -1

1.2

1 x 10-5 2 x 10-5 4 x 10-5 8 x 10-5

1.0

-I /mA

0.8 0.6 0.4 0.2 0.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Time /s Figure 3.2

Simulated current transients and Cottrell response for varying diffusion coefficients.

Excellent agreement is observed between analytical theory and the finite element simulations. In all cases the currents differed from those predicted by the Cottrell equation by no more than 0.5% after the first 0.1s had elapsed. The error during the first few timesteps was lower than 4% in all cases. The lesser accuracy during the first few timesteps is due to the very abrupt change at the first timestep, where the initial condition of bulk concentration at all nodes is modified by the sudden imposition of the boundary condition at the electrode. The simulation therefore requires a short period of about 20 timesteps to attain full accuracy. This is a trivial requirement as, for the parameters chosen, the whole 200 timesteps take 0.3 s. If 2000 timesteps are used, the simulation reaches accuracy of better than 0.5% at 5  10-3 s of simulated time or 10 timesteps, with the whole simulation taking 0.7s. In practice for transient problems, full accuracy should not be expected during the first 10 – 20 timesteps, and the investigator must allow for this in the parameters chosen as input to the programs. The use of the variable timestepping scheme described on page 20, allows the region over which the simulation settles to full accuracy to be compressed in time to very small fractions of the total time over which a simulation is run.

Reaction Systems

57

The expanding grid algorithm, used in all the work done in this thesis, has been described on page 20. The one dimensional simulations may be used as an example to illustrate the necessity of such a routine. Figure 3.3 shows the transient given above for D = 1  10-5 cm2s1

compared to the results of the same simulation done using a regular, rather than expanding

grid. 0.70 0.60

-I /mA

0.50 0.40

Cottrell Expanding Grid Regular Grid

0.30 0.20 0.10 0.00

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Time /s Figure 3.3

Current transients using expanding and regular grids.

It may be noted that while the simulation using regular grid converges to almost the correct current at long times, it underestimates the current severely at short times. Figure 3.4 shows the concentration profile at two times during the simulation.

Reaction Systems

58

1.0

0.1s 0.8

1s

C/CBulk

0.6

0.4

Regular Grid Expanding Grid

0.2

0.0 0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

z coordinate /cm Figure 3.4

Concentration profiles using regular and expanding grids.

The inaccuracy in the simulation using the regular grid is due to the inability of the regular mesh to accurately reflect the concentration profile when the whole of the diffusion layer in concentrated into a very small region. Figure 3.4 illustrates that at 0.1 s, the regular grid has only one element wholly within the diffusion layer, and the flux to the electrode is seriously underestimated. At 1 s, five elements lie within the diffusion layer, allowing better approximation of the true concentration profile. A regular grid with 50 elements could be used to discretise the region, but this would be a far less elegant solution than the expanding grid. Second Order Elements Repeating the above sets of experiments using second order elements produced no appreciable differences in the concentration profiles generated for the expanding grids detailed, but the use of these elements is complicated by considerations relating to the current calculation. Deriving an expression for the gradient at the electrode from the interpolation function gives

 C1  1  1  2  4 1  2C 2  2l C 3 

(3.7)

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This is a function of the local coordinate, as the gradient must vary linearly over the element, if the concentration varies as a second order function. Evaluated at =-1, the local coordinate of the first node, and recognising that for this case C1 is always zero, this gives

C 1 4  1 2  . This is dependant on the concentration at the third node, as well as at the 2l C 3  second node. In effect this expression fits a parabolic curve for the concentration profile through the three points of the element to evaluate the derivative. It is obvious that this may overestimate the current, as shown in figure 3.5. 1.2 1.0

C/CBulk

0.8 0.6 0.4 0.2

Parabolic Linear

0.0 0.0

0.2

0.4

0.6

0.8

1.0

z coordinate Figure 3.5

Parabolic element current calculation.

Figure 3.5 shows a possible set of nodal concentrations over a second order element. 2 Using equation3.7 is the equivalent to assuming a concentration distribution of C  3z  2 z ,

the equation of the parabola shown and a corresponding gradient of 3 at the electrode surface is found. Assuming a linear variation between the first two nodes and ignoring the third would be equivalent to an assumption of C  2 z , giving a gradient of 2. This may seem a worst case scenario for the possible errors arising from using a second order method for current calculation, but it actually represents the situation found at short times for every transient simulation of this type, no matter what parameters are chosen. It has been found that overestimation of the current at short times can approach close to 50% if such a flux calculation method is used.

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The linear method of current calculation may be applied to simulations using a parabolic interpolation function to generate the concentration distribution, with results which are every bit as accurate as those from a properly converged simulation using linear elements. More importantly, current overestimation is eliminated by the choice of a linear method of flux calculation. This is useful, as simulations almost always converge to the true current from below, allowing a proper assessment of the extent to which a simulation has converged, by whether increasing the grid refinement increases the current. Any factor which may make it harder to ensure that a simulation is properly converged must be eschewed, and so higher order approximations to the flux have not been widely employed, and absolutely not for simulations of transient processes.

Cyclic Voltammetry The technique of cyclic voltammetry is one of the most well explored in the field, and has been dealt with by many methods of simulation. A brief illustration of how it may be performed and simulated in one dimension is given. Cyclic voltammetry entails performing repeated linear sweeps back to back, cycling the potential at the electrode repeatedly between the two limits EMin and EMax. Figure 3.6 shows the variation of the electrode potential, E, measured against the equilibrium potential for the redox couple concerned E0, with time for four different scan rates, shown in mV/s.

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Scan rate /mVs-1 -400

400 200 100 50

E -E0/mV

-200

0

200

400 0

4

8

12

16

20

24

28

32

Time /s

Figure 3.6

Potential variation with time for cyclic voltammetry.

For the simple reaction A  e   B , assuming identical diffusion coefficients for the oxidised and reduced forms gives B  1  A at the electrode, and all points in solution. The Nernst equation is

E  E0 

RT A ln nF B

(3.8)

This may be used to derive the correct boundary condition at node 1, which lies on the electrode surface at z = 0.

 nF    C1  1  exp E 0  E  RT    





1

(3.9)

This boundary condition must be recalculated at every timestep, reflecting the continuous sweeping of the potential. The boundary condition at the top of the simulated region is best left not explicitly set, as for a sufficiently small region of simulation, and length of scan, it is possible to completely convert between the two forms of the redox couple under consideration. In this case, the implicit no flux boundary condition is the most appropriate. The methods of current calculation are identical to those for a potential step experiment described in equation 3.5.

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Figure 3.7 shows the current response for the same set of scan rates as shown in figure 3.6 for an electrode of area 1.0 cm2. The grid parameters used were nz = 20, zr = 0.08 cm, the number of timesteps over each sweep was nt = 2000, zf = 10-4 cm, D = 10-5 cm2s-1, CBulk = 1 mmol, and the electrode area was 1 cm2. Each run took 0.9s on a Pentium 100 MHz PC.

0.5 0.4 0.3

-I /mA

0.2 0.1 0.0

Scan Rate /mVs -1

-0.1

400 200 100 50

-0.2 -0.3 -0.4 400

300

200

100

0

-100

-200

-300

-400

E - E0 /mV Figure 3.7

Current against potential for cyclic voltammetry at varying scan rate.

These results clearly fit the normal behaviour expected for a reversible system.

Figure 3.8

The features of a cyclic voltammogram.

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63

The anodic and cathodic peaks, E AP and ECP are separated by 59.13/n mV at 298K, where n is the number of electrons transferred.



The size of the anodic and cathodic peaks, I AP and I CP , measured as shown in the previous diagram, are equal.



The peak currents, I AP and I CP , are proportional to the square root of the scan rate.



The potential at which the forward and reverse peaks, E AP and ECP , occur is independent of the scan rate. Thin Layer Cyclic Voltammetry The voltammograms presented above are typical of the behaviour of electroactive

species dissolved in solution, where the diffusion layer at the electrode is permitted to grow without limit during the timescale of the experiment. Different behaviour is observed for species which are confined in a region very close to an electrode. This may be found for species dissolved in a solute in a particularly thin cell, but it is also possible to observe similar responses for species forming an adsorbed layer on an electrode. 0.4

zr /cm 0.3

0.01 0.006 0.004 0.003 0.002 0.001 0.005

0.2

-I /A

0.1 0.0 -0.1 -0.2 -0.3 -0.4

400

300

200

100

0

-100

-200

-300

0

E - E /mV Figure 3.9

Cyclic voltammograms for a shrinking thin layer cell.

-400

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64

The simulation parameters used were as for figure 3.7, with a scan rate of 200 mV/s and zr varying as shown in the legend, in cm. The imposition of a limit on the growth of the diffusion layer has two effects. Firstly it decreases the current response on the forward scan, from the point that the diffusion layer reaches the limit of the cell. This diminishing effect on the current is progressive, increasing in magnitude as the concentration at the top of the region falls from the bulk value. Therefore if the diffusion layer reaches the limiting distance before the normal potential at which the peak current is reached, the position of the peak current observed is shifted back to the equilibrium potential. This may be rationalised by examining the change in the surface concentration with potential, from the relationship given in equation 3.9. It can be observed that the rate of change of surface concentration with potential is greatest at the equilibrium potential. Therefore, as the concentration throughout the region containing the electroactive material follows the concentration at the electrode surface more closely the more the diffusion layer is limited, the potential at which the maximum current occurs will be shifted towards the equilibrium potential. It may appear that the size of the reverse peak is increased, but if measured as shown in figure 3.8, it can be seen that this is not the case, and the reverse peak can also be seen to move back to the equilibrium potential. Finally as complete conversion is achieved on each sweep between potentials, the two halves of the cyclic voltammogram become completely symmetrical.

Two Dimensional Two dimensional simulations form the vast majority of the literature with regard to electrochemistry. The vastly greater range of situations which may be efficiently approximated by two dimensional approaches – both analytical and numerical – may explain why there are no popular electrode designs that are not amenable to two dimensional simulation, either by using axisymmetric coordinates or by considering one direction as infinite.

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The simulations described in the first section of this chapter have so far been concerned with electrodes large enough that only diffusion in the z direction normal to the electrode need be considered. In the remainder of the chapter, the influence of diffusion in the x direction will be shown, for planar electrodes of small dimension, and where the electrode is distorted up and curved towards a cylinder electrode.

Microband Potential Step Transients The microband electrode is the first microelectrode that will be discussed. A schematic of a microband is shown in figure 3.10.

Figure 3.10

A microband electrode.

The key characteristic of a microelectrode is that it has a dimension which is of the order of 50 m or less, and the term ‘ultramicroelectrode’ may also be used for electrodes in which there is a dimension in which the electrode measures less than 20 m. For the microband electrode, the length of the electrode, xe is the micro-dimension, with the width of the electrode, w, normally a few mm or more. The advantages of microelectrodes over larger electrodes have been well explored131-133, and led to their use in all aspects of mechanistic electrochemistry. The very small surface area of these electrodes results in the currents involved in experimentation often being of the order of nanoamps , allowing the IR drop in solution to be minimised. The IR drop is the disparity between the potential difference between the potential drop measured between the working and counter electrodes, and the potential drop which actually exists across the electrical double layer at the working electrode. This arises because of the resistance to the current flowing in solution, which is proportional to IR.

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The minimisation of the IR drop also facilitates experimentation in electrolytes with extremely small dielectric constants134,135, expanding the range of systems open to the experimentalist. In voltammetric experiments, the extremely small IR drop, and small double layer capacity allow very fast scan rates, up to 104 V/s to be employed. These ultrafast scan rates allow events on the nanosecond timescale to be investigated using techniques such as cyclic voltammetry136. The characteristic which most differentiates microelectrode geometries from equivalent electrodes of normal dimension (i.e. having no dimension in the sub-50 m range) is the influence of diffusion in directions other than the axial direction, the ‘edge effect’. This effect arises in situations where the electrode size is comparable to the diffusion layer thickness, where the effect of the diffusional flux to the edges of the electrode in the lateral directions parallel to the electrode surface augments the diffusion in the axial direction to a measurable degree. This effect has also received much attention, and both numerical25,137,138 and analytical139,140 approaches to the characterisation of this effect have been advanced. The transient current response to a potential step has been characterised by these investigators as being equal to the Cottrell current plus a correction at short times, but at long times tending to the inverse logarithm of the time. Aoki published both a pair of rigorously derived expressions for the current response. The first, giving the behaviour at the limit of the shortest times 141, is given as



I  nFDC Bulk w  

 12

1



(3.10)

The second predicts the response at long times142, taking the form of large integral series requiring numerical evaluation, given as

 1  0.577 1.312 I  nFDC Bulk w      2 3 ln   3  ln   3 ln   3 

(3.11)

Aoki also published a very compact approximate empirical expression139, given as

   9.90 1 I  nFDC Bulk w   2  0.97  1.10 exp   ln12.37 

   

(3.12)

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where   Dt xe2 . This is quoted by Aoki as being accurate to within 0.8% for  < 108, and is therefore sufficiently accurate to serve as a good check on any numerical procedure. Finite Element Discretisation of a Microband. The grid used to discretise a cell containing a microband electrode is shown in figure 3.11.

Figure 3.11

Discretisation of a cell containing a microband electrode.

The dimensions of the grid are denoted by the variables in x and z, with the number of elements set across a region given by n, using subscripts indicating particular regions. An expanding grid is used, with the size of the smallest element across a region, and the total number of elements being set as shown, and a geometric series previously detailed on page 20used to define the subsequent element sizes. Boundary Conditions The initial condition at t = 0 for the transients is, as before, that of C = CBulk at all points. The boundary conditions at t > 0 appropriate to the problems of interest are:

z=0

xr < x < xr + xe

C=0

z=0

x < xr

dC/dz = 0

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z=0

x > xr + xe

dC/dz = 0

z = zr

all x

dC/dz = 0

all z

x=0

dC/dx = 0

all z

x = 2xr + xe

dC/dx = 0

68

Current Calculation The current may be found using equation 2.45, incorporating the square interpolation function given in equation 2.46. Results The results of a set of simulations are compared to the Aoki equation139 and the Cottrell equation in figure 3.12. 16

Cottrell Simulation Aoki

14

-I /nA

12 10

0.04 cm 8 6

0.02 cm

4

0.004 cm 2 0

0.0

0.2

0.4

0.6

0.8

1.0

Time /s Figure 3.12

Potential step transients at a microband electrode.

The parameters used were nz = 25, nr = 12, ne = 12, all first node sizes 1  10-5 cm, zr = 0.4 cm, xe varies as indicated, xr = 0.2 cm, w = 0.6 cm, D = 1  10-5 cm2s-1, nt = 250, tt = 1 s, tf = 5  10-6 s. The simulations match Aoki’s empirical theory to within 1% at all points after the first ten timesteps. The current enhancement due to the axial diffusion to the electrode is clear, with the Cottrell equation being lower due to the extra lateral diffusion to the microband. This ‘edge

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69

effect’ is initially slight, being a function of the relative size of the diffusion layer and the electrode length across the short dimension. At very short times, the current follows exactly the Cottrell equation, but diverges above it as the diffusion layer grows outward from the electrode surface.

The Microwire Electrode Another microelectrode of wide use in the experimental community is the microwire electrode. As one of the simplest possible electrode geometries, the cylinder electrode has attracted wide attention from investigators. As with some of the other classic problems of diffusion, the first applicable work derived from the consideration of heat conduction in solids143. The concentration contours around the electrode are completely axisymmetric, and so the radial dimension can be considered as uniform, as well as the long dimension, greatly aiding in both analytical144 and numerical145 approaches. While comparisons between the microband and microwire electrode have been made145, due to the very similar transient response, the intermediate geometries represent an unexplored region. This may derive from the difficulty of mapping finite difference grids onto unusual geometries, and the level of complexity this introduces146,147. As a demonstration of the ability of the finite element method to probe geometries where accurate analytical predictions are difficult, the grid shown in figure 3.13 may be constructed.

Figure 3.13

Discretisation of a cell containing a microband electrode.

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The bottom edge of the central section may be scaled by setting the value of z e x e to vary between unity for a perfectly hemicylindrical electrode, and zero for a completely flat microband, with the surface of the electrode having coordinates given by the equations

  i 1    z e z  sin   n  1 c   

  i 1    xe x  cos   n  1 c   

(3.13)

where i is the number of the point on the electrode surface, numbered from i to nc. The angle  was set to 1 radian, 63.7 for all the simulations performed, although varying this angle over the range 0.5-1.25 had a negligible effect on the results of the simulation. The normal expanding grid algorithm described on page 20 was used, fixing the first node size xf, shown above. The first node size xc is found by assuming an even grid over the top of the central section of the mesh. This value is then used to determine the expanding grid across the top of the outer sections. The reason for using a grid of the form shown is to enable the modelling of cylindrical electrodes in flow cells, where the inclusion of the cell walls is desirable. This development is detailed in chapter six. The results of a set of potential step transients for values of z e x e varying between 0 and 1.0 are shown in figure 3.14. The parameters used were nr = 20, nc = 20 nz = 20, xr = 0.04 cm, xe = 20 m, zr = 0.04 cm, xf = 1  10-5 cm, w = 0.5 cm, CBulk = 1 mmol, nt = 300, tt = 1.0 s and tf = 1  10-5 s.

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5.0

ze/xe

4.5

1.00 0.95 0.90 0.85 0.80 0.75 0.70

-I /A

4.0

0.65 0.60 0.55 0.50 0.45 0.40 0.35

0.30 0.25 0.20 0.15 0.10 0.05 0

3.5 3.0 2.5 2.0 1.5 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Time /s

Figure 3.14

Potential step transients for a microcylinder to microband transition for varying values of z e x e .

The legend shows the values of z e x e for each transient. It may be observed that the transition appears very linear, and comparisons between all the geometries at different times, shown in figure 3.15, confirm that this is essentially the case. The transient shown for the microband geometry agrees to within 0.033% with the Aoki approximation given in equation 3.12. The result for the microcylinder may also be compared to another empirical curve found by Aoki, given by



I  nFD2wC Bulk   2  0.422 0.0675 log   0.0058log   1.47  1

2

 3.14

   for log   1.47,    for log   1.47 where   Dt xe2 . Note that for the band xe was the entire width of the electrode, but for the cylinder xe is defined as the radius, not the diameter. The transient shown for a hemicylindrical electrode falls within 1.4 % of the Aoki equation for the hemicylinder.

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t /s 0.01 0.02 0.05 0.1

12

-I /A

10

0.2 0.5 1.0

8

6

4

2 0.0

0.2

0.4

0.6

0.8

1.0

ze/xe Figure 3.15

Potential step transient currents at times indicated in s.

Conclusions The conclusion of the work on two dimensional diffusion in normal coordinate systems is that the finite element method offers an accurate and flexible methodology with which the transient behaviour of electrodes may be found. The method is very flexible with regard to electrode geometry, and affords the ability to probe extremely short timescales efficiently using expanding discretisation schemes in both space and time. The ability to reproduce analytical behaviour is demonstrated, along with the facility to explore previously unfamiliar geometries.

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Two Dimensional Axisymmetric simulations The property of axial symmetry is exploited in simulation whenever possible. The transformation of a three dimensional problem in Cartesian coordinates into a two dimensional problem in polar coordinates reduces the computational load by a large factor. Electrodes possessing axial symmetry have also proved amenable to formal mathematical description, leading the existance of accurate analytical equations to predict their behaviour. The matrices and formulation of axisymmetric diffusional problems are given in chapter two on page 39. The expanding timestepping described on page 20 was also employed in these simulations. Two different applications of this formulation are discussed.

The Cylinder Electrode Another electrode geometry which allows an axisymmetric description is the cylinder electrode. Optionally, the cylinder electrode may be modelled as infinite in the axial direction, which would allow a one dimensional formation to be used. Alternatively, the axial direction may be preserved, admitting variations in the radial geometry of the cylinder or cell, and it is this formulation which is presented here. Aoki144 has shown that the response of a cylinder electrode to a potential step may be empirically predicted by equation 3.14. Cell Discretisation The cylinder cell may be discretised as shown in figure 3.16.

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Figure 3.16

74

Axisymmetric discretisation of a cylinder.

An expanding grid is used in the r direction, but not in the z direction. As before, no variation in the annular direction can be accommodated. Boundary Conditions The boundary conditions are

z=0

all r

dC/dz = 0

z = zr

all r

dC/dz = 0

all z

r=0

C=0

all z

r = rr

dC/dr = 0

Note that the grid begins at a distance re from the axis, and so the explicit boundary at the left side of the grid in this case does not give rise to a singularity. Results and Discussion The results of potential step transients for four electrode sizes are shown in figure 3.17. The four electrode radii are given in the legend, with the other parameters used being,

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nr = 20, nz = 5, re varies as shown in the legend, rr = 0.06 cm, zr = 0.2 cm, rfr = 5  10-5 cm, D = 1  10-5 cm2s-1, CBulk = 1 mmol, nt = 1000, tt = 10 s, tf = 1  10-4 s. 10

0.1 cmSimulation 0.04 cmSimulation 0.01 cmSimulation 0.001 cmSimulation 0.1 cmAoki 0.04 cmAoki 0.01 cmAoki 0.001 cmAoki

-I /mA

1

0.1

0.01

0.001

0

1

2

3

4

5

6

7

8

9

10

Time /s

Figure 3.17

Potential step transients for cylinder electrodes.

The transients all agree with the empirical theory of Aoki to within 1.3%. The runs took 57 s for the 1000 timesteps used on a Pentium 75 PC with 80 Mb of memory, with the global matrix occupying less than 32 kb of memory, due to the low number of elements used in the z direction.

The Disk Electrode Introduction The microdisk electrode is one of the most common types of axisymmetric electrode, and due to the ease with which such electrodes may be fabricated, much experimental work has been done using them. The microdisk electrode also has the property that a steady state is reached under diffusion only conditions. The current flowing under such conditions has been evaluated analytically148 by using Bessel functions149, for a disk of radius re as

I s  4nFC Bulk D re

3.15

The transient response has also been investigated widely, and was the target of much early work in the field of digital simulation14,150-152. These investigations have also extended beyond potential step transients to cover most other types of voltammetry153-155. The most

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commonly used equation at present is the analytical equation derived by Shoup and Szabo20, which is valid over all times and electrode radii, and more importantly does not take the form of large series expansion, as do those proposed by Aoki150,151. Shoup’s function is given by equation 3.16, where re is the electrode radius, and t is the time in s.



I S  2C Bulk FDre re Dt 

 12

 1  4   1 e 0.391re

2

Dt





3.16

Cell Discretisation The cell may be discretised into triangular elements as shown in figure 3.18. The grid used expands away from the disc edge in the radial direction r, and away from the disc in the vertical direction, z. The assumption of axial symmetry dictates that the disc must be totally regular in the annular direction, .

Figure 3.18

Grid discretisation and schematic of a disk electrode.

Boundary Conditions Using the standard initial condition of bulk concentration over the whole region simulated, the appropriate boundary conditions are

z=0

re < r

C=0

z=0

r > re

dC/dz = 0

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z = zr

all r

dC/dz = 0

all z

r=0

dC/dr = 0

all z

r = 2rr + re

dC/dr = 0

77

Note that the no flux boundary condition at the axis of symmetry is essential, to avoid a singularity in the concentration profile. Results A set of potential step transients are shown in figure 3.19. The results for three different electrode sizes can clearly be seen to agree well with the Shoup equation, 3.16, over this timescale. The parameters used were ne = 15, nr = 15, nz = 30, re varies as shown in the legend, rr = 0.08 cm, zr = 0.08 cm, zfe = rfe = rfr = 1  10-5 cm, D = 1  10-5 cm2s-1, CBulk = 1 mmol, nt = 2000, tt = 10 s, tf = 1  10-4 s. 2.0 1.8

0.04 cm Simulation 0.01 cm Simulation 0.02 cm Simulation 0.04 cm Aoki 0.02 cm Aoki 0.01 cm Aoki

1.6 1.4

I/mA

1.2 1.0 0.8 0.6 0.4 0.2 0.0

0

1

2

3

4

5

6

7

8

9

10

Time /s

Figure 3.19

Potential Step transients for disk electrodes.

The steady state at the achieved at the disk under diffusion only conditions is not achieved at times on the order of those in figure 3.19. The transient for a 0.01 cm radius disk electrode was simulated over a timescale three orders of magnitude longer, using parameters ne = 15, nr = 15, nz = 30, re = 0.01 cm, rr = 0.5 cm, zr = 0.5 cm, zfe = rfe = rfr = 1  10-5 cm, D =

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1  10-5 cm2s-1, CBulk = 1 mmol, nt = 104, tt = 104 s, tf = 1  10-4 s. The results are shown in figure 3.20. 52 48 44

-I / A

40 36

Simulation Aoki Cottrell

1 0.8 0.6 0.4 0.2 0.1 0.08

0

2000

4000

6000

8000

10000

Time /s Figure 3.20

Potential Step allowed to decay to a steady state at a disk electrode of radius 0.01 cm.

The simulation can be seen to offer results which dip marginally below the Shoup equation between 100 and 2000 s, after which the two traces converge. The final current found at 104 s was 38.906 A, close to the value of 38.594 A predicted by equation 3.15 for the steady state, indicating that in excess of 104 s is required for a disk of this size to reach the steady state. The bottom part of figure 3.20 shows the Cottrell response showing that the where diffusion is only one dimensional, no steady state is found. The simulations took approximately 7 minutes for each run of 1000 timesteps on a Pentium 75 PC with 80Mb of memory, although the matrices used occupied less than 0.5 Mb of storage.

Conclusion The conclusion of this work is that the axisymmetric formulation offers an efficient method of accessing a range of interesting geometries, which are widely used in experimental practise. The finite element formulation is able to exploit the axisymmetric model, and give accurate results for an exceptionally modest computational cost.

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Chapter 4. Convection-Diffusional Results with Analytical Flow Profiles In this chapter, the use of analytically derived flow profiles in FEM simulations will be explored. The range of geometries for which good approximations exist is limited, and in this chapter two cell types are considered: the rotating disk electrode, and the channel cell. In the channel cell, two electrode types, the microband and microstrip are considered, along with intermediate rectangular microelectrodes. For each geometry the simulations of the current responses at varying flow rates are compared with analytical theory. For the microband where relatively accurate theoretical predictions exist, these are confirmed, and extended to smaller sized electrodes than previously published. For the microstrip, where previous descriptions have been purely exploratory, a set of empirical relations are proposed to allow experimentally useful predictions.

Steady State at a Rotating Disc Electrode The only common example of a system where convective transport and diffusive transport may both be considered to only operate in the direction axial to the electrode is that of the rotating disk electrode156. A schematic of the velocity profile is shown in figure 4.1.

Figure 4.1

Schematic of velocity profiles at the RDE.

The RDE is formed from a cylindrical electrode within a solid insulating casing. The electrode and surround may be rotated, causing the solution to be dragged up to the electrode surface, and expelled out in the lateral direction.

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This electrode geometry has received a lot of attention from experimentalists, and is possibly the most popular hydrodynamic electrode extant157. It has also been popular with practitioners of simulation, and was the focus of much attention during the early development of Finite Difference models10,158 due the simplicity with which one dimensional processes may be modelled, and the modest computational resources demanded. The more recent modelling work has included many much fuller treatments of the geometry, using axisymmetric two dimensional models77,159. The governing equation for such an electrode is

D

 2C C w 0 2 z z

(4.1)

with identical boundary conditions and current calculation methods to those described for a potential step experiment in equations 3.3 to 3.4. Note that due to the influence of convection, the specification of a fixed boundary condition at the top node of the simulated region is mandatory. The velocity profile above the RDE has been studied in depth, and the z component is often 3

approximated as w  0.510 2 

 12

, first suggested by Levich160. This actually represents a

slight overestimation of the true velocity profile, and results (analytical or simulated) generated using this approximation are less accurate at low Schmidt numbers, Sc   D , when the diffusion layer is very thin, and lies entirely within the region where this velocity approximation is most accurate. More accurate treatments of the velocity profile have been offered, with one of the most accurate being that used by Feldberg161 in the commercial Digisim 2.0TM program. The calculation of this convection profile may be simplified by the assumption of a dimensionless z 1

1

coordinate,   2  2 z . The velocity profile is then given by

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82

1 1  0.88447S  w  2  2    0.88447  S 

S  0.51023 2  0.32381 3  0.34474 4  0.25972 5  0.16214 6

(4.2)

 0.07143 7  0.019911 8  0.0030405 9  0.00019181 10 The concentration and velocity profiles for a simulation using the above approximation are shown in figure 4.2. Also shown is the velocity profile calculated using the Levich approximation given above, which can be seen to overestimate the velocity further from the electrode.

1.0

0.005

C/CBulk

0.004

0.6

0.003

0.4

0.002

0.2

Velocity /cms -1

C/CBulk

0.8

0.001 -1

Velocity /cms Simple Velocity

0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.000 0.35

z coordinate /cm

Figure 4.2

Concentration and velocity profiles for the RDE.

The concentration profile can be seen to be almost linear inside the diffusion layer, with convective transport limiting the size of the diffusion layer and maintaining bulk concentration at regions beyond. This almost linear diffusion profile within the diffusion layer results in little benefit accruing from the use of grids with very small first element sizes. Referring to figure 3.1 for the discretisation of a one dimensional region, the parameters used to generate the profile shown above were nz = 20, zr = 0.3 cm, zf = 4  10-3 cm, D = 1  10-5 cm2s-1  = 0.1 rad s-1,  = 0.1 cm2s-1, re = 0.5 cm, and CBulk = 1 mmol, where re is the electrode radius,  is the disc rotation rate, and  is the kinematic viscosity,     , where  is the real

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viscosity, given in g s-1cm-1, and  is the density, given in gcm-3. The parameters nz, zr, and zf, are the number of elements over the whole grid, the total size of the grid, and the size of the first element respectively. The Schmidt number for the above simulation is 104 yet the diffusion layer can still be seen to reach out far enough, 0.12 cm, for the difference between the simple and accurate approximations to the velocity profile to be clearly observable. The Levich equation for the RDE160, which derives from the approximate treatment of the velocity is 2

1

1

I L  0.62048nFC Bulk r 2 D 3  6  2

(4.3)

Finite Element simulations performed using the same approximation agree extremely well with the Levich equation given above, with errors less than 0.01%, using the same parameters as detailed below. The availability of superior approximations to the true velocity profile at the RDE has led investigators to suggest corrections to the Levich equation for the current. The Newman formula162 has been accepted as being accurate to with 0.1% of the correct value for Schmidt numbers of greater than 100. This correction has the form



I N  I L 1  0.2980Sc

 13

 0.14514Sc

 23



(4.4)

Simulations performed using and Feldberg’s approximation for the velocity profile and ne = 30, varying the size of the first element and total grid to accommodate different values of  agreed with the Newman equation given above to within 0.1%. Figure 4.3 shows a comparison of the simulated current responses at the steady state with both the Levich and Newmann current predictions, using the velocity profile given in equation 4.2. The coincidence of the simulated and Newmann current response is clear, with the Levich velocity approximation slightly overestimating in comparison. The parameters used for the simulation were nz = 20, zf = 5  10-3 to 2  10-6 cm, zr = 2.0 to 0.01 cm D = 5  10-3 cm2s-1,  = 0.1 cm2s-1, re = 0.5 cm, CBulk = 1 mmol.

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32 28

-I /A

24 20 16 12

Simulation Levich Newmann

8 4 0 0

1

2

3

4

5 1/2

6

7

8

9

10



Figure 4.3

Steady state current response for the RDE.

The Channel Flow Cell and Hydrodynamic Electrodes The channel flow cell is hydrodynamic cell in which a variety of electrodes may be incorporated. By far the most common is the rectangular family of electrodes, illustrated in figure 4.4.

Figure 4.4

Schematic of a channel cell with a rectangular large electrode.

Included in this family are the microband, where xe is of the microelectrode dimension, below 50 m, and the microstrip, where w is the micro-sized dimension. Note that this distinction is only meaningful in the context of hydrodynamic experiments. In the microband, the flow direction is across the shortest dimension of the electrode, but for a microstrip the flow direction is aligned along the long axis of the electrode. If neither dimension

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of the electrode is very small, then the electrode can be treated as a planar electrode, with none of the edge effects that characterise microelectrodes. The governing equation for the transport of material through the channel cell, assuming that diffusion is isotropic, is

D

 2C  2C  2C C C C  D  D u v w 0 2 2 2 x y z x y z

(4.5)

where u, v, and w are the velocities in the x, y and z directions. This may be reduced in dimension by assuming that the cell is effectively infinite in the y direction, in which case the concentration profile in that direction may be considered to be uniform, and need not be explicitly simulated. This reduces the governing equation to

D

 2C  2C C C  D u w 0 2 2 x z x z

(4.6)

The effect of the hydrodynamic transport on the simulations which may be performed are twofold. Firstly, the transient response decays to a steady state (as seen at the Rotating Disk Electrode, page 80) due to the growth of the diffusion layer being limited by convective transport maintaining the bulk concentration upstream. This steady state is far easier to experiment upon and model, and hydrodynamic electrodes have been found to be a powerful tool for kinetic analysis, which is discussed in chapter seven, page 149. The second effect of the convective transport, is that as it operates in the x direction, parallel to the electrode surface, rather than in the z direction, as in the case of the Rotating Disk Electrode, the diffusion layer across the electrode will vary in the x direction, and any simulation must be at least two dimensional, in the xz plane, assuming the concentration profile to be invariant in the y direction. If this last condition is not valid, then a full three dimensional model must be employed. This case is explored further in chapter eight. For the case of microelectrode geometries, any direction in which an edge effect operates must be explicitly considered, as the diffusion layer will be hemispherical or ovoid rather than planar. In the case of the microband, the x direction is already required in the model, due to convective effects. The microstrip, which has its short dimension aligned in the y direction, will therefore experience strong diffusional

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effects in y as well as z, and convection in x, and will therefore require a full three dimensional model.

Steady State Measurements The steady state at a large rectangular electrode was also considered by Levich160, pioneer of many hydrodynamic techniques, who assumed that diffusion in directions other than the z direction was negligible, and that the diffusion layer would be sufficiently small for the variation of the velocity within it to be effectively linear. This approximation derives from consideration of heat flow in pipes, and is the Lévêque approximation163, in which the true



velocity profile, u x  u0 1  z r 2  z 

2

z r 22 , where u is the velocity in the x direction

and u0 is the velocity at the centre of the cell at z = zr/2, is approximated near the electrode as

u  2u 0 z h . The Levich equation derived using this approximation160 is 1

I Levich

 Vf 3  0.925nFwDC Bulk xe D  2  h d  2 3

2 3

(4.7)

Discretisation and Boundary Conditions With the inclusion of convective effects, a two dimensional discretisation as shown in figure 3.11 may be employed, with boundary conditions

z=0

xr < x < xr + xe

C=0

z=0

x < xr

dC/dz = 0

z=0

x > xr + xe

dC/dz = 0

z = 2h

all x

dC/dz = 0

all z

x=0

C = CBulk

all z

x = 2xr + xe

dC/dx = 0

The concentration at any edge where convection operates to inject material must be explicitly defined, rather than allowing the no flux condition as for the transients previously described.

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Results A series of simulations were performed, over a range of electrode sizes between 0.4 cm and 0.1 m, and over volume flow rates between 1.6  10-5 and 1.6  103 cm3s-1. The upper limit of flow rate therefore represents a maximum centre velocity of 105 cms-1, or 1 kms-1 ,or Mach 3. While this may seem an extraordinary speed, Compton has reported a high speed channel electrode capable of flowing solution over an electrode at up to 150 mph164. As in any branch of simulation, finding the limits of the performance of a program is of interest when assessing the range over which accuracy may be expected. Finally, this velocity is the answer to the question ‘What flow rate will eliminate the edge effect for a 0.1 m wide microband?’. To illustrate the behaviour of a large planar electrode, the response of the 0.2 cm wide electrode is shown in figure 4.5. 0.6

0.5

-I /A

0.4

0.3

0.2

Simulation Levich

0.1

0.0 0

2

4

6

8

10

12

Vf(1/3) Figure 4.5

Current response for a large planar electrode in a channel flow cell.

The simulated current can be seen to be a uniformly linear function of Vf1/3. It must be noted that while the simulation may behave according to the theory up to the extremely high volume flow rates simulated, the current response for a real experimental cell would certainly not. The massive current density predicted at the fastest flow rates, over 10 A/cm2, could not be sustained in a real electrolyte.

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Figure 4.6 shows the normalised currents found.

32 16 8 4

I/ILev

2 1 0.5 0.1 0.001 0.01

1 0.1

u0 /cms-1

Figure 4.6

10 1

10

100 1000 10000 100000

100

xe /m

1000

Normalised current surface for a channel flow cell.

The parameters used were nr = 40, ne = 40, nz = 40, xr = 0.1 cm, z = 0.04 cm, D = 1  10-5 cm2s-1, d = 0.6 cm, w = 0.2 cm, zf = 1.5  10-7 cm, xfe = 1.5  10-7 cm, xfr = 1.5  10-7 cm. For centre velocities up to 100 cms-1 the whole cell was considered, above this rate only the bottom 0.002 cm were simulated. Finally, for the 0.1 m electrode, using only 20 elements over each half of the electrode was found to be satisfactory, rather than 40 regular elements of 1.25  10-7 cm. All the simulations were run on a Pentium 100 PC, using less than 48Mb of memory, with execution times just under a minute for each run. The surface in figure 4.6 shows three distinct regions, with a flat, triangular central section extending from the highest flow rates to the lowest flow rates only for the 0.02 and 0.04 cm electrodes, within which the normalised current is very close to unity, indicating the Levich equation is obeyed. There is a small section at low flow rates and large electrode sizes where the normalised current falls substantially below one, in which not only does the Lévêque approximation fail, but the thin layer condition is encountered. The remainder of the surface

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forms a steep slope at the smallest electrode sizes and flow rates, with the normalised current rising to above 35, indicating a massive edge effect operating. Examining the results in more detail, figure 4.7 shows the results for the six largest electrode geometries. 1.4

xe /cm 0.01 0.02 0.04

1.2

0.1 0.2 0.4

I/ILev

1.0

0.8 1.01 1.00

0.6

0.99 0.4

0.001

0.01

0.1

1

10

100

1000

10000

100000

10 -1

100

1000

10000

100000

u /cms

Figure 4.7

The beginning of the edge effect for a channel flow cell.

The Lévêque approximation163 is only valid if the time taken for material to diffuse vertically across the cell, which may be expressed as 4h 2 D is far greater than the time taken for material to convect across the electrode in the x direction, xe u 0 . Rearranging, it can be stated that for the Lévêque approximation to be valid, 4h 2 u 0 D xe must be far in excess of unity. This may be rationalised by considering that at the highest flow rates the diffusion layer will extend only short distances from the electrode in the z direction normal to the surface, and will therefore experience only the most linear portion of the convection profile. This is the situation for which the Lévêque approximation is derived, and therefore where it might be expected to be most accurate. Referring to the inset in figure 4.7, the 0.4cm electrode only comes within 1% of the Levich current for 4h 2 u 0 D xe = 64000, and the 0.2 cm electrode at a value of 32000, and

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25600 for the 0.1 cm example. The 0.04 cm electrode however shows a very unusual response, showing a minimum of 0.95 in the response at u0 = 10 cms-1. It appears that for electrodes of this size and below, the extra transport made possible by lateral diffusion, the ‘edge effect’ becomes important at a flow rate sufficiently high as to mask the effects of the Lévêque approximation breaking down. A 0.04 cm wide electrode is not normally considered a microelectrode, being almost an order of magnitude outside the normal range of 50 m or below, but at slow enough flow rates it appears it is able to adopt characteristics of these smaller sizes. It is also notable that even for very fast flow rates, the simulations converge to a value 0.3% below the Levich current. This is also true of the smaller geometries explored, at flow rates sufficiently fast so as to eliminate edge effects. Figure 4.8 shows the development of the edge effect with decreasing electrode size. It can be observed that for a sufficient level of miniaturisation of an electrode, the edge effect is almost impossible to outrun by increasing the rate of convection.

32

xe /cm 0.00001 0.00002 0.00004 0.0001 0.0002 0.0004 0.001 0.002 0.004

16

I/ILev

8

4

2

1 0.001

0.01

0.1

1

10

u0 /cms Figure 4.8

100

1000

10000

100000

-1

The development of the edge effect for a microband in a channel flow cell.

The edge effect has been studied by various groups, and analytical expressions for the magnitude of the effect have been advanced. These have included the suggestion by Aoki 165 that

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91

3

I I Levich  1  0.5644 p 4  0.2457 p



where p  Dh 2u 0 xe2

(4.8)

. 2 3

Newman offers a different value for the first coefficient, 3

I I Levich  1  0.874 p 4  0.2457 p

(4.9)

while Compton and Alden45 propose a pair of values, 0.7737 for 0 < p < 5.5 and 0.837 for 0 < p < 0.2. 3

By plotting I I Levich against p 4 , the value of this coefficient may be determined by linear regression, as shown in figure 4.9.

I/ILevich - 1 + 0.2457p

8

6

4

2

Simulation Linear Regression 0 0

2

4

6

8

10

(3/4)

p

3

Figure 4.9

Determination of the coefficient of p 4 for a 0.001 cm microband

It may be observed that the gradient appears higher between 0 and 0.5, and this may explain Compton’s adoption of a two part approximation. Repeating the analysis for each electrode size where an edge effect may be detected, over the ranges adopted by Compton, values are found as shown in the table below for the 3

expression I I Levich  1  0.2457 p  a p 4  c . The values of r and  shown refer to the linear regression fitting.

Reaction Systems Electrode Size

a

c

Maximum p 4

r

40 m

0.78609

0.00094

1.5

0.99981

2.5

0.99982

5

0.99996

0.5

0.99988

4

0.99995

0.4

0.99999

5

0.99997

0.5

0.99999

5

0.99998

0.5

0.99999

4

0.99998

0.4

0.99998

5

0.99999

0.5

0.99999

5

0.99997

0.5

0.99996

 0.00586

20 m

0.77588  0.00391

10 m

0.77143  0.00224

0.82298  0.0092

4 m

0.76586  0.00284

0.82123  0.00223

2 m

0.76405  0.0019

0.8073  0.00196

1 m

0.75797  0.00187

0.7964  0.00152

0.4 m

0.74586  0.00185

0.77342  0.00315

0.2 m

0.74008  0.00158

0.74991  0.00153

0.1 m

0.72754  0.0027

0.72356  0.00348

3

92

 0.00305

0.00101  000305.

0.00493  0.00404

-0.00609  0.00029

0.0033  0.00442

-0.00895  0.00045

-0.00358  0.00338

-0.01239  0.00044

-0.00944  0.00368

-0.01859  0.00039

-0.02333  0.00303

-0.03105  0.00069

-0.04467  0.0033

-0.04635  0.00067

-0.0749  0.00637

-0.06957  0.00123

These data sets agree very well with Compton’s approximations for the larger electrode sizes. Compton appears to have plotted all the data for the three electrode sizes studied, (5, 1 and 0.2 m) on the same graph before performing the linear regression. This would lead to the result being dominated by the highest values, which would explain why the

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93

figure quoted fits almost exactly with the results presented here for the 4-10 m range. The data presented here show that a more detailed analysis of the data for electrodes of sizes below 5 m 3

is possible, and that the coefficient of p 4 is weakly dependant on the electrode size xe.

Transients under Hydrodynamic Conditions For a macroelectrode Aoki166 has shown that all plots of I/ILim , where I is the transient current at time t and ILim is the limiting current at the steady state, against normalised time t', where

 4 Du 02 t'  t  h2x2 e 

   

13

(4.10)

should be identical. A series of potential steps, normalised against the results of the corresponding steady state simulations, give the results shown in figure 4.10. Also shown are the analytical values derived by Aoki at five points. The parameters used were nz = 20, nr = 18, and ne = 20, h = 0.02 cm, xr = 0.8 cm, xe = 0.4 cm, xfe = 0.000 5 cm, xfr = 0.000 5 cm, zf = 0.000 05 cm and D = 10-5 cm2s-1. The transients were all calculated for 1 000 uniform time steps over tt = 0.5 s and with central velocities, u0 as shown, in cms-1.

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2.6

u0 /cms-1

2.4

10.0 5.0 2.0 Analytical

2.2 2.0

I/ILim

1.8 1.6 1.4 1.2 1.0 0.8 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Normalised Time t' Figure 4.10

Normalised potential step transients for a macroelectrode in a channel flow cell.

The normalised transients are extremely close to being collinear, and are in excellent agreement with the analytical theory. The time dependant form of the program may be also used to simulate potential step transients for microelectrodes, as for the macroelectrode transients shown above. This complex problem has been approached previously via an ADI Finite Difference approach with reasonable success37. Figure 4.11 shows potential step transients normalised to the limiting current, for a range of flow rates and two electrode sizes, with a comparison to results from the ADI finite difference method 37.

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95

4.0

u0 /cms-1

3.5

FE 0.1 FE 0.8 FE 8.0 FD 0.1 FD 0.8 FD 8.0

I/ILim

3.0

2.5

2.0

1.5

1.0 0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

Time /s Figure 4.11

Graph showing normalised potential step transients against time for a microelectrode of size 0.0031 cm

The parameters used were nz = 20, nr = 12, and ne = 12, zr = 0.04 cm, xr = 0.4 cm and D = 10-5 cm2s-1, xfe = 1.510-6 cm, xfr = 1.510-6 cm, zf = 1.510-6 cm, tf = 10-7 s, tt = 0.14 s, 0.1 s and 0.02 s and nt = 2000. The results are also shown without normalisation in figure 4.12.

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96

10

u0 /cms-1 FE 0.1 FE 0.8 FE 8.0 FD 0.1 FD 0.8 FD 8.0 Aoki Cottrell

9 8

-I /A

7 6 5 4 3 2

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Time /s

Figure 4.12

Graph showing potential step transients against time for a microelectrode of size 0.0031 cm

The form of the transients can be clearly observed, and is determined by three effects. Initially the current decays in a Cottrellian manner, as the diffusion layer must be planar at very short times. The ‘edge effect’ then begins to operate, as diffusion in directions parallel to the electrode surface brings in extra material to the electrode. This causes the current to rise above that predicted for the planar electrode, and at this stage, the current agrees with Aoki’s equation for the transient at a microband, equation 3.12. Finally, as the diffusion layer grows out from the edge of the cell further, the influence of convection acts to maintain the concentration, and the diffusion layer stops growing, with the system reaching a steady state. It can be seen in figure 4.12 that if the flow rate is rapid, the steady state may be reached in fractions of a second at a microelectrode. Good agreement between the data from the FE and FD method is noted, with the FE method predicting a slightly higher current response.

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The Microstrip Electrode The microstrip electrode is akin to the large planar electrode shown in figure 4.4, except that the width in the y direction, w, is of microelectrode dimensions. As discussed in the introduction to this chapter, the microstrip electrode requires the use of a full three dimensional model to fully reflect the mass transport involved, as in equation 4.5. This has been achieved using eight noded cubic elements as described in the appendix, page 211, for the steady state response of a microstrip in a channel flow cell. The behaviour of this electrode type is similar to that of a microband, with an edge effect developing at low flow rates, as the diffusion layer grows out in laterally in the y direction normal to the electrode edges as well as up and downstream in the x direction as is the case for a microband. Bearing in mind that an edge effect operates to increase diffusion in the direction normal to an electrode edge, it can be seen that for a microband, any extra diffusional transport in the axial, x direction operates along the electrode edge extending across the cell in the y direction. As the concentration profile is assumed to be uniform in the y direction, therefore the magnitude of any edge effect for a microband is independent of the size of the electrode in the y direction. Similarly, the extra lateral diffusion in the y direction to a microstrip electrode would occur along any electrode edge extending downstream in the x direction. As the concentration distribution over any electrode in a channel cell is non-uniform in the x direction, the magnitude of the edge effect at a microstrip electrode must depend not just on the small, microscopic dimension in y, but also on the length in x. At a microstrip electrode the edges along the x direction allow higher mass transport to the electrode along its whole length, allowing the current density to remain higher than for a large electrode moving downstream. This results in the current enhancement by lateral diffusion being larger for longer electrodes. This current enhancement may be expressed by normalising the steady state currents found for the microstrip

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98

against the Levich equation160, given as equation 4.7, or an accurate value for a microband of similar dimension if the length of the electrode is small. The simulations performed were checked against the results of previous work using a finite difference approach., and then extended to the systematic exploration of a range of microstrip geometries. Cell Discretisation Figure 4.13 shows the outline of the cubic grid used, with the darker shaded section at the centre representing the electrode region.

Figure 4.13

Discretisation of microstrip flow cell using cubic elements

Boundary Conditions The boundary conditions are

xr < x xr + xbe + xe

dC/dz = 0

z=h

x < xr

dC/dz = 0

z = 2h

All x

dC/dz = 0

All z

x=0

C = CBulk

All z

x = xr + xbe + xe + xae

dC/dx = 0

Electrochemical Results and Discussion Electrochemical Results The Confluence Reactor offers the possibility of mixing two streams of material, under very well defined conditions. Perhaps the simplest example of this is the case when one

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129

inlet contains reactant, in solution with a suitable background electrolyte, and the other contains only the background electrolyte. At very low flow rates the concentration profile of the species in solution becomes uniform in z before the electrode is reached. Under these conditions the response seen at the electrode is the same whether the reactant is introduced through the upper or lower inlet, and may be approximately predicted using the Levich equation, equation 4.7. Preliminary simulations were performed for various geometries and flow rates. At very low flow rates the Confluence Reactor behaves as an analogue of a channel cell of height 2h. This situation changes as the flow rate is increased. A point is reached where the concentration profile does not become uniform in z before the electrode is reached. If the reactant is introduced through the lower inlet the predicted current at the electrode tends to the Levich response for a channel of height h. If the reactant is introduced through the upper inlet the current tends to zero as the flow rate rises, as in the case for the opposite sided double electrode cell, as the electroactive species is carried past the electrode before diffusion can carry it across the cell to the electrode. Figure 6.5 shows two simulated steady state current/voltage plots for the Confluence Reactor, showing the divergence in the response between two species introduced through the top and bottom channels as the flow rate is increased, where I Lim . The value of E0 for the species introduced through the bottom inlet is set to zero, and that for the species flowing in through the top inlet is set to +400mV. The parameters used were as given in the description of figure 6.13.

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130

1.0

0.8

I/ILim

0.6

0.4

Vf = 0.0032 cm3s -1

0.2

Vf = 0.0064 cm3s -1 0.0 -200

0

200

400

600

800

E /mV Figure 6.5

Current Voltage plot at two flow rates for the Confluence Reactor.

Sample concentration profiles may also be examined to illustrate the effect of increased convection. Figure 6.6 shows a typical concentration profile, at a flow rate of 4.86  10-5 cm3s-1, under conditions where no electrode reaction occurs. 0.08

z /cm CBulk 0

0.04 0 0

0.1

0.3

0.2 x /cm

0.4

Reaction Systems 0.4 0.08

0.04

z /cm

131

0

0.3 0.2 x /cm 0.1

Figure 6.6

Concentration profile in the Confluence Reactor at a flow rate of 4.86  10-5 cm3s-1.

z /cm 0.08

CBulk 0

0.04 0 0

0.1

0.2 0.4 0.08

x /cm

0.3 0.04

0.4 z /cm

0

0.3 0.2 x /cm 0.1

Figure 6.7


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132

z /cm 0.08

CBulk 0

0.04 0 0

0.1

0.2

0.4 0.08

0.4

0.3

x /cm

0.04

z /cm

0

0.3 0.2 x /cm 0.1

Figure 6.8


The effect of the increased convection in lengthening the distance necessary for the concentration profile to approach uniformity in the z direction is obvious. This new geometry is also found to be sensitive to the diffusion coefficient of the species. Figure 6.9 shows the effect of varying the diffusion coefficient of the species introduced through the top channel, using the same parameters as for the results in figure 6.5, but with the diffusion coefficients shown, and Vf = 0.0032 cm3s-1 in all three cases.

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Current /A

40 35

3.2x10-5 cm2s -1

30

2.4x10-5 cm2s -1 1.6x10-5 cm2s -1

25 20 15 10 5 0 -200

0

200

400

600

800

E /mV Figure 6.9

Current Voltage plots showing the effect of varying diffusion coefficients.

From the simulation data available, for every combination of geometry and diffusion coefficient, the range of flow rates over which the response at the electrode will vary most strongly is found, and from these possible sets of parameters, those which are most suited to experimental application were chosen. Initial simulations showed that a gap between the junction of the two inlets and the electrode needed to be greater than 1cm for an appreciable current to be measured at the electrode when the electroactive species was introduced through the upper channel, at flow rates sufficiently high as to be experimentally practical.

Experimental Cell Construction The experimental work on the Confluence Reactor, described in the following section, was carried out by Qiu Fulian*.

*

Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford University,

Oxford, OX1 3QZ.

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A schematic of the confluence reactor is shown in figure 6.10 and can be seen to be constructed from two rectangular ducts separated in the non mixing region with aluminium foil. The two ducts were fabricated from synthetic fused silica (Heraeus silica and Metals Byfleet, Surrey) and cemented together using a low melting point wax185. Platinum electrodes of typical dimension 4mm  2mm (Goodfellows Metals Ltd) were employed for electrochemical detection. The electrode was cemented onto one face of the cell and polished to 0.25 µm smoothness using progressively decreasing sizes of diamond lapping compound. Electrical contact was made to the electrode via a small hole in the cell which was sealed using epoxy resin. A fully constructed cell had typical internal dimensions of height 0.08 cm, width 0.6 cm and length 2.5 cm.

Outlet Inlet Feeds from capillaries

Detector Electrode Aluminium Plate

Upper Duct Figure 6.10

Lower Duct

The Confluence Reactor

The confluence reactor was held in a gravity fed flow system, the essential details of which have been noted previously185. Two separate reservoirs were employed to hold the appropriate electrolyte solutions. The inlet solution flow rates were controlled by the use of two capillaries placed upstream of the reactor which enabled solution flow rates in the range 10-3 to

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10-1 cm3s-1 to be accessed. A silver pseudo-reference wire was also placed upstream of the cell. A Platinum gauze counter electrode 80 mm x 30 mm, (52 mesh) was located downstream of the reactor to ensure counter products did not enter the reaction vessel. The apparatus was prepared for the experiment by flushing degassed Argon solvent through the system which acted to remove any oxygen. Electrolyte solutions were prepared using acetonitrile (Fisons) dried over alumina and distilled. Tetrabutyl ammonium perchlorate, TBAP, (Fluka, purum) was employed as background electrolyte (typically 0.1 mol dm-3 concentration). Tris-bromophenyl amine (TBPA)(Aldrich) and ferrocene (Aldrich) where used as received. Experimental Results and Discussion To evaluate the new arrangement experiments were performed using various different operating modes. In the first set of measurements (mode 1) a single reactant tris(p-bromophenyl)amine186 (TBPA) of the same concentration is placed in both reservoirs. Typical conditions employed were 1.1 mmol TBPA and 0.1M tetrabutylammonium perchlorate (TBAP) acetonitrile solutions with cell parameters of, d = 0.6cm, h = 0.041 cm, xbe = 1.22 cm, xe = 0.419 cm and xae = 0.2cm. The mass transport limited current from the oxidation of TBPA to the corresponding cation were then recorded as a function of the volume flow rate. For all measurements undertaken both outlets were fixed at to the same volume flow rate. Using mode 1 the results were in excellent agreement with the predicted values using the Levich equation for a channel cell160,187 with a bulk solution concentration of 1.1 mmol. This can be rationalised since under the experimental conditions of operation the length of cell required after the inlet region to re-establish fully parabolic flow is of the order of 1 mm. The supply of the same material from each of the outlets simply results in an experiment essentially identical to those reported previously for the channel flow cell177. Next experiments were performed with only one of the outlets containing the electroactive material, 2mM Ferrocene188, which shall be referred to as mode 2. Figure 6.11 shows typical experimental results along with values predicted using the numerical calculations

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for the case where material is introduced from the outlet on the far side of the cell in relative to the electrode. All parameters were as for Figure 6.13 except d = 0.42cm. 20

15

10

I /A

Experimental Simulation

5

0

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Vf

Figure 6.11

Results and simulation for mode 2 operation.

As can be seen the current drops as a function of increasing volume flow rate. This can be explained since the transit time of material in the cell must be comparable to the time taken for material to diffuse across the cell width and be detected. As the flow rate is increased the current therefore gradually drops reflecting the fact that less material is able to diffuse over from the far side of the cell. The opposite is true when material is allowed to enter from the bottom inlet only. The upper line in figure 6.13 shows this case for both experimental and theory values of a 1.0 mmol TBPA solution. Now as the flow rate increases the detected current also rises. Of course this is a result of the material now staying closer to the wall in which the detector is mounted. At sufficiently high flow rates the current response becomes identical to that of the a cell operated in mode 1, since the concentration near the surface is effectively the same as that leaving the bottom inlet. The final mode (3) of operation examined in this chapter is the case where different reactants are introduced from the two inlets. In mode 3 two electrochemically well defined materials, TBPA as above and ferrocene were examined. Acetonitrile solutions of TBPA and

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2.0 mmol ferrocene, both containing 0.1 M TBAP were degassed in separate reservoirs. The experiments were performed with both outlets fixed at the same flow rate and a steady state voltammogram was then recorded. Figure 6.12 shows a typical voltammogram recorded using at . 0.43 mmol TBPA and 11 mmol ferrocene 25

20

I /A

15

10

5

0 0

200

400

600

800

1000

1200

1400

E /mV

Figure 6.12

Experimental current/voltage plot for mode 3 operation.

The two materials can be seen clearly as separate waves on the voltammogram. This procedure was repeated for a range of volume flow rates. A typical set of experimental results obtained for the variation of the mass transport limited current as a function of the volume flow rate are noted in figure 6.13 for 1.0 mmol TBPA in the adjacent channel and 2.0 mmol ferrocene in the opposite channel. Again numerical results are also presented and it is apparent that excellent agreement is noted between the experimental and computational predictions. The parameters used were xr = 0.4cm, xbe = 1.22cm, xe = 0.419cm, ye = 0.355cm, d = 0.6cm, xae = 0.2cm, za = 0.04cm, zb = 0.04cm, nxr = 10, nxbe = 10, nxe = 10, nxae = 10, nya = 10, nyb = 10, nyc = 10, nyd = 12.

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40 35 30

I /A

25

Experimental Opposite Side Experimental Same Side Simulated Opposite Side Simulated Same Side

20 15 10 5 0 0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

Vf Figure 6.13

Results and simulation for mode 3 operation.

Conclusion The Confluence Reactor has been shown to be of great interest as a novel cell geometry allowing rapid mixing of materials under precisely defined hydrodynamic conditions. The simulation of the behaviour of the device has allowed both rapid development of this system, and verification of the experimental results obtained. Future applications of this geometry are many and further work may easily include applications in kinetic resolution and photoelectrochemistry.

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The Collision Cell Introduction In the investigations into the Confluence Reactor described in this chapter, it is apparent that if the investigation of homogenous kinetic processes is attempted, as described in chapter seven, there is likely to be a limit to the rate of reaction which may be measured using this geometry, due to the fact that as the flow rate is increased, the product of any reaction cannot be detected until further downstream from the union of the two inlets. This occurs because the direction in which the solution flows is parallel to the electrode, and therefore the mass transport from where any intermediate might be formed, at the union of the two inlets, to the electrode is not aided by the direction of the convection employed. While the abilities of the Confluence Reactor may prove sufficient to probe a wide range of rate constants, it would be desirable to create a new cell type where the convection necessary to mix the two streams of reactant also transported the product of the reaction to the electrode. This requirement is addressed by the collision cell, shown in figure 6.14.

Figure 6.14

Schematic of the Collision Cell.

This cell is a logical development of the collision cell, in which solution is introduced in through two inlets at the top of the cell, flows round the dividing plates through a central region where solution mixing and reaction can occur, and any product species exit through the channels at the bottom part of the cell. Any homogenous kinetic processes will occur in close proximity to the electrode, and so studies of transient intermediate species might be anticipated.

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Hydrodynamic Simulations The mass transport in the collision cell is dependant on the convective transport of solution around the cell. As for the microwire and confluence reactor, this convection profile is cannot be specified by analytical methods, and must be found by simulation. The assumption that the convection profile is uniform in the y direction is made, allowing a two dimensional approach to be employed. Cell Discretisation The cell may be discretised for hydrodynamic simulations as shown in figure 6.15.

Figure 6.15

Grid used for hydrodynamic simulations of the Collision Cell.

Note that the central region xc corresponds to 2xbe + xe, the gap between the plates dividing the cell. Boundary Conditions The boundary Conditions appropriate to the problem are

x

z

u

w

p

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141

x=0

z < zb

du/dx = 0

dw/dx = 0

dp/dx = 0

x = xr

z < zb

du/dx = 0

dw/dx = 0

dp/dx = 0

x=0

z > zb

u = u1(z)

w=0

dp/dx = 0

x = xr

z > zb

u = u2(z)

w=0

dp/dx = 0

all x

z=0

u=0

w=0

dp/dz = 0

x < xr

z = zb

u=0

w=0

dp/dz = 0

x > xr + xc

z = zb

u=0

w=0

dp/dz = 0

all x

z = za + zb

u=0

w=0

dp/dz = 0

Where u1(z) and u2(z) are parabolic profiles applied at the two inputs to the system, as for the confluence reactor The point at which the pressure is specified was at the centre of the lower channel in the z direction, two elements before the boundary on the right side.

Results Hydrodynamic Behaviour The hydrodynamic behaviour of the collision cell is more complex than that of the confluence reactor, with the size of the central gap being of critical importance in determining the convective profile. To visualise these flows, a regular mesh of rectangles over the domain of interest may be allowed to deform in the direction of flow of the solution. This is identical to the flowlines shown in figure 6.3, but with the addition of lines pointing roughly in the x direction. This is necessary as over parts of the cell, the flow in the collision is oriented mainly in the y direction. Figures 6.16 to 6.19 show the flowlines for a cell with geometry za = zb = 0.04 cm and xr = 0.1 cm, at varying values of xc between 0.04 and 0.12 cm, using a centre velocity of 5  10-2 cms-1 in each of the inlet channels, giving a total volume flow rate of 1.6 10-3 cm3s-1, for a cell with d = 0.6 cm. The other parameters used were nxr = 4, nxc = 8, nza = 6, nzb = 8, nzc = 8, nzd = 6, zfa = 0.005 cm, zfb = 0.001 cm, zfc = 0.001 cm, zfd = 0.005 cm, xfr = 0.004 cm, xfc = 0.001 cm.

Reaction Systems

0.08 0.07

z coordinate /cm

0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00

0.04

0.08

0.12

0.16

0.20

0.24

x coordinate /cm Figure 6.16

Flowlines through the collision cell, xc = 0.04 cm

0.08 0.07

z coordinate /cm

0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00

0.04

0.08

0.12

0.16

0.20



0.24

0.28

142

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143

0.08 0.07

z coordinate /cm

0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28



0.08

z coordinate /cm

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28

0.32



As the gap size xc is increased, the flow profiles can be seen to evolve from a simple curve with the maximum flow at the centre of the gap to a double humped curve, with the maximum solution velocities observed at around 0.02 cm from the end of the dividing plates. Using cell heights za = zb = 0.04 cm at wider gaps than 0.12 cm a central, semi-stagnant region forms where the solution velocities become very low, which would make for less efficient mixing between the two inlet streams. This ability to tune the velocity profile impinging on the electrode implies a degree of versatility that may aid experimentation. The diffusion layer near the electrode is compressed and shaped both by the velocity profile and electrode position. This

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144

gives rise to the possibility that the distribution of current density over the electrode may be tuned by the judicious choice of cell geometry. The effects of increasing the flow rate through the collision cell are shown in figure 6.20, which uses the same geometry as figure 6.18, with parameters were nxr = 9, nxc = 7, nza = 5, nzb = 8, nzc = 8, nzd = 5, zfa = 0.008 cm, zfb = 0.001 cm, zfc = 0.001 cm, zfd = 0.008 cm, xfr = 0.001 cm, xfc = 0.001 cm, and a centre velocity of 10 cms-1. This is equivalent to a total volume flow rate of 0.32 cm3s-1.

0.08 0.07

z coordinate /cm

0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28


Flowlines through the collision cell, xb = 0.08 cm

The flowlines just under the dividing plates at the centre of the cell show that recirculation is induced at these higher flow rates, with eddies forming just as for the wire electrode at the same flow rates.

Electrochemical Simulations Cell Discretisation The collision cell may be discretised for the purposes of diffusional simulations using linear triangular elements as shown in figure 6.15, and the concentration profile found using the formulation and matrices given in chapter two for the triangular element. Note the explicit inclusion of the electrode placement, which is not necessary for the hydrodynamic simulations, as the electrode placement has no effect on the convection profile, only on the concentration profile.

Reaction Systems

Figure 6.21

145

Schematic of the Collision Cell Discretisation for diffusional problems.

Concentration Profiles Simulated concentration profiles, shown in figure for the collision cell demonstrate the behaviour of the cell when the same reactant is introduced in the cell through both inlets. The parameters used were nxr = 10, nxbe = 4, nxe = 8, nza = nzb = nzc = 8, nzd = 10, xr = 0.1 cm, xbe = 0.02, xe = 0.04 cm, ha = hb = 0.04 cm, zfa = 0.002 cm, zfb = zfc = 5  10-4 cm, zfd = 5  10-4 cm, xfr = xfbe = 0.001 cm, xfe = 1  10-4 cm, D = 1 10-5 cm2s-1. z /cm 0.08 0.04 0

CBulk

0

Figure 6.22

0.05

0.1

x /cm

0.15

Concentration profile in the collision cell.

0.2

0.25

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146

0.25 CBulk

0.2 0.15 0.1

x /cm

0.05 0 0.08

0.06 z /cm

Figure 6.23

0.04

0.02

0

Concentration profile in the collision cell.

Voltammetric Behaviour The voltammetric response of the collision cell can be studied, under conditions as above, where an electroactive species is introduced into the cell through both inlets. Figure 6.24 shows the response for two electrode sizes, 0.04 and 0.02 cm. The cell parameters used were nxr = 10, nxbe = 6, nxe = 10, nza = 8, nzb = nzc = 10, nzd = 12, xr = 0.1 cm, xbe = 0.02, xe = 0.04 cm, ha = hb = 0.04 cm, zfa = 0.002 cm, zfb = zfc = 5  10-4 cm, zfd = 2  10-5 cm, xfr = xfbe = 0.001 cm, xfe = 1  10-4 cm, D = 1 10-5 cm2s-1, w = 0.2 cm, d = 0.6 cm. The hydrodynamic data came from a file created using the same parameters as given for figure 6.18. The centre velocities set at the boundaries vary so as to give the volume flow rates shown.

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147

0.9

0.04 cm Levich 0.04 cm Simulation 0.02 cm Levich 0.02 cm Simulation

0.8 0.7

-I /mA

0.6 0.5 0.4 0.3 0.2 0.030

0.035

0.040

0.045

0.050 0.055 3 -1

0.060

0.065

0.070

Vf /cm s

Figure 6.24

Schematic of the Collision Cell Discretisation for diffusional problems.

The simulated responses for the collision cell were a factor of 0.8 to 0.85 away from the Levich behaviour calculated for an electrode situated in a channel cell of the same height as the inlet channel, with a flow rate equivalent to that in each inlet channel

Conclusion The conclusions to be drawn from these preliminary investigations are that the collision cell promises to offer similar benefits to the confluence reactor, with well defined hydrodynamics and versatility, but with a much shorter path between the mixing region and the electrode, and with the added benefit that the transport from this mixing region to the electrode is aided by convection. This should make it suitable for the investigation of extremely rapid kinetic processes. The behaviour may be expected to vary greatly with the relative gap sizes and electrode sizes employed, necessitating the use of simulation as an interpretative tool. The generation of a set of working surfaces for a range of geometries is expected to be straightforward however, which should enable wider use of this cell type.

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A further development of the collision cell envisaged is the addition of another electrode opposite the first across the centre of the cell. This would then allow double electrode experiments, with the transport from generator to the detector either aided or impeded by the solution flow. With the computational techniques demonstrated, the extension of the current work into these new areas is expected to be rapid.

Chapter 7. Diffusion-Convection-Reaction Systems Introduction The study of chemical kinetics using hydrodynamic electrodes, including the channel electrode, shown in figure 4.4, is well established41,60,189-190, with the inference of kinetic parameters from simple voltammetric experiments being both straightforward and highly accurate. The facility to achieve a steady state through the use of convection simplifies both the experimentation and the modelling of such systems, and a variety of computational techniques, chief among them the finite difference method, have been applied. The finite element method has been shown to be suitable for simulating the behaviour of a range of systems where convective and diffusive transport are critical183, and in this chapter, the theory and application of the technique to homogenous kinetic processes is shown. The EC, ECE and DISP processes have been well characterised previously41, and the FE method is shown to be capable of reproducing the analytically established behaviour, using only modest computational resources. The particular suitability of the FE technique to simulations where the rate constants are extremely large is demonstrated. The ECE and DISP reactions are characterised by the apparent transfer of more than one electron to each reactant molecule, resulting from the further reaction of the initial product A to form a species B which then accepts another electron. This second electron may come from the electrode in the ECE mechanism, shown in step 3 below, or from the DISP (disproportionation) reaction, shown in step 4 below. 1.

A + e-  B

2.

BC

3.

C + e- D

4.

B+CA+D

Rate k2

Rate k4

Compton’s notation41 of kn and k-n for the forward and backward rate constants of reaction n will be adopted. In the DISP mechanism the rate determining step may be either

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150

reaction 2 (DISP1) or 4 (DISP2). In both cases the reactions 2 to 4 may be approximated as 2B  A + D, but with a first order rate constant of k2 for the DISP1 mechanism, or a second order rate constant of k 2 k 4 k 2 for the DISP2 case. The EC mechanism is described by steps 1 and 2 above, where step 2 is irreversible. The number of electrons transferred at the limit is therefore always one, but the depletion of B at the electrode caused by subsequent reaction shifts the potential required for the reduction of A to B towards more positive potentials. This is commonly described by measuring the half wave potential; the potential at which the current is equal to half of the transport limited current. This characterisation is appropriate, as the shape of the overall wave does not vary with the rate of the reaction.

Theory Grid Formation The channel electrode may be described as in figure 4.4, with the parabolic convection profile shown. The concentration profile over the electrode is therefore also two dimensional provided that the width of the electrode, w, is substantially greater than the length xe. The region of interest may therefore be discretised into triangular elements as shown in figures 3.11 and 6.4. The grids chosen are highly non uniform, with the elements near the electrode being typically over four orders of magnitude smaller than those furthest away, forming a geometric series as previously described on page 20. The grid parameters of the simulations are shown on the figures, with those of the type x or z denoting the size of the grid over the regions shown, and the size of the smallest elements in each direction denoted by the same variables with an extra f subscript. The number of elements over each region is given by those parameters of the form nsub where the subscript indicates the region.

Matrix Formation The general form of the equation governing the flux of any species through the two dimensional region shown, where diffusive and convective transport operate, and there are

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sources and sinks where the species is generated or decays is given by equation 2.32, which is restated below.

C  2C  2C C  D 2  D 2 u  kd C  ks t x x z where C is the concentration of the species, u is the velocity in the x and z directions and ks and kd are the rates of generation and decay of the species at the point in question. The full derivation of the finite element matrices for the triangular grid used is given in equation 2.34 to 2.37. For the channel electrode, the fully developed flow in the central region of the channel follows the analytical relation



u  u0 1  h  y  h 2 2



(7.1)

where u is the velocity in the x direction, h is the half height of the cell and u0 is the centre velocity. It should be noted that the sources for each species are dependant on the local concentrations only of other species, but are independent of the local concentration of the species in question, but that the rates of decay of the species concerned are always functions of their local concentration. If the rate of decay of a species is a higher order function of the local concentration, then an iterative method must be employed, in which the first order rate of x reaction is approximated as k eff  k d C x where C is the local concentration at the previous

iteration, raised to the necessary power. The iterative routine then repeats until a sufficiently small change in the concentration profile is found.

EC Reaction For the EC Reaction the exact equations governing the system are

 2C A  2C A C D D u A  0 2 2 x x z D

 2CB  2CB C  D  u B  k 2CB  0 2 2 x x z

(7.2)

(7.3)

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D

152

 2 CC  2 CC CC  D  u  k 2CB x x 2 z 2

(7.4)

with the boundary conditions z=0

x < xr

dCA/dz = 0

dCB/dz = 0

dCC/dz = 0

z=0

xr < x < xr + xe

CA = exp CB

dCB/dz = -dCA/dz

dCC/dz = 0

z=0

xr + xe < x

dCA/dz = 0

dCB/dz = 0

dCC/dz = 0

z = 2h

all x

dCA/dz = 0

dCB/dz = 0

dCC/dz = 0

all z

x=0

CA = CA Bulk

CB = 0

CC = 0

all z

x = 2xr + xe

dCA/dx = 0

dCB/dx = 0

dCC/dx = 0

 F  E  E0  , and CX is the concentration of species X. The rate constant k2  RT 

where   

appears as a decay term of the form kd in equation (7.3), but a source term of the form ks in equation (7.4). The gradient boundary condition for species B at the electrode is expressed in the form of a ks1 contribution to the element load vector. The interdependence of the boundary conditions at the electrode surface requires an iterative scheme to be employed to solve the equations. The initial values chosen for the concentrations at the electrode for species A and B



were found as C A  1  k 2  1e  



1



and C B  1  e   k 2



1

, proceeding from the

assumptions that CA + CB + CC = 1, CB/CC = k2, and CA/CB = e-. This estimate was found to give a good initial approximation to the final answer. An iterative procedure was then followed to relax these initial estimates to the true solution. This entailed calculating the distribution of A, using the values of the concentration of B from the previous iteration, and calculating a new distribution of B using those values for A. The values of B used in the next iteration are then set by the formula

C B  1  re C B 0  reC B1

(7.5)

where re is a relaxation constant normally varying between 0.4 and 0.02, CB0 is the value carried forward from the last iteration, and CB1 is the value just calculated. This is repeated until the

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153

largest change at any point, measured as ch  C B 0 C B1  1 fell below a set tolerance, normally 1  10-3 to 1  10-5, depending on the level of complexity of the simulation. Only points at which CB1 was at least 5  10-4 CBulk were considered in this check, below which level numerical noise becomes evident. The value of re was also progressively decreased as the value of ch fell, normally halving for every decade past 0.01, which was found to bring the simulation to convergence smoothly. To further minimise computational requirements in the iterative schemes employed, where a potential or rate constant was sequentially increased, the converged concentration distribution found for each run was used as the initial distribution for the next. To predict the current response for the EC reaction, it is only necessary to consider species A and B. As species C is not electroactive, it does not affect the observed steady state response in any way, and it may be calculated after converged values of A and B are found.

ECE Reaction The ECE reaction is similar to the EC reaction defined above for species A and B, with the further contribution that species C is itself electroactive. The fact that neff varies with flow rate has led to this reaction normally being studied at potentials sufficient to cause complete conversion of A to B at the electrode. If the reduction potential for C is sufficiently negative of the reduction potential of A then C may also be considered to be completely converted to D. The governing equations are therefore

D

 2C A  2C A C  D u A  0 2 2 x x z

(7.6)

 2CB  2CB C D D  u B  k 2CB  0 2 2 x x z

(7.7)

 2 CC  2 CC C  D  u C  k 2CB  0 2 2 x x z

(7.8)

 2CD  2CD C  D u D  0 2 2 x x z

(7.9)

D

D

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with the boundary conditions z=0

x < xr

dCA/dz = 0

dCB/dz = 0

dCC/dz = 0

dCD/dz = 0

z=0

xr < x < xr + xe

CA = 0

dCB/dz = -dCA/dz

CC = 0

dCD/dz = -dCC/dz

z=0

xr + xe < x

dCA/dz = 0

dCB/dz = 0

dCC/dz = 0

dCD/dz = 0

z = 2h

all x

dCA/dz = 0

dCB/dz = 0

dCC/dz = 0

dCD/dz = 0

all z

x=0

CA = CA Bulk

CB = 0

CC = 0

CD = 0

all z

x = 2xr + xe

dCA/dx = 0

dCB/dx = 0

dCC/dx = 0

dCD/dx = 0

Obviously, as C contributes to the overall current in the ECE mechanism, it must be explicitly simulated. As the concentration of A at the electrode surface may be taken as 0 at sufficiently high potentials, and is not a function of the concentration of B, it is not necessary to use an iterative method to reach a solution. The solution procedure is therefore to sequentially find A, then B, then C. D does not need to be simulated to find the current response, but if required, may be found as given.

DISP1 Reaction In the DISP1 reaction, the formation of A from B necessitates the use of an iterative procedure, and the assumption that the species C is a short-lived intermediate with no appreciable concentration at any point further modifies the above equations, to become

 2C A  2C A C D D  u A  k 2CB  0 2 2 x x z

(7.10)

D

 2CB  2CB C  D  u B  2k 2 C B  0 2 2 x x z

(7.11)

D

 2CD  2CD C  D  u D  k 2CB  0 2 2 x x z

(7.12)

with the governing equation for species D as in equation 11, and with the boundary conditions: z=0

x < xr

dCA/dz = 0

dCB/dz = 0

dCD/dz = 0

Reaction Systems z=0

xr < x < xr + xe

CA = 0

dCB/dz = -dCA/dz

dCD/dz = 0

z=0

xr + xe < x

dCA/dz = 0

dCB/dz = 0

dCD/dz = 0

z = 2h

all x

dCA/dz = 0

dCB/dz = 0

dCD/dz = 0

all z

x=0

CA = CA Bulk

CB = 0

CD = 0

all z

x = 2xr + xe

dCA/dx = 0

dCB/dx = 0

dCD/dx = 0

155

The initial conditions chosen are that CB = 0 at all points, and the concentration of A is determined. The solution procedure is to then find B using the initial distribution of A. These initial values found are then refined by iteration, where new values for A and B are calculated, using the values of A and B from the previous iteration in both cases. The values used for the next iteration are then calculated as before by relaxing the values previously used.

DISP2 Reaction The DISP2 mechanism is formulated similarly to the DISP1 mechanism, except that the rate of the overall reaction is a second order power of the local concentration of B. As the boundary conditions at the electrode for either DISP process require iteration, the solution procedure is unaffected. The governing equations are

 2C A  2C A C k k D D  u A  2 4 C B2  0 2 2 x k 2 x z

(7.13)

 2CB  2CB C k k D D  u B  2 2 4 C B2  0 2 2 x k 2 x z

(7.14)

 2CD  2CD C k k  D  u D  2 4 C B2  0 2 2 x k 2 x z

(7.15)

D

The boundary conditions are as for the DISP1 process, as is the transport equation for D.

Current Calculation Using a square interpolation function as discussed on page 38, for the grid used the current is given by a sum evaluated over the elements immediately over the electrode

Reaction Systems

l C n,k  Cn1,k  C n,i  C n, j  n 1 2b j

156

ne

I  nFDwC Bulk 

(7.16)

where Cn ,k is the concentration at node k of element n. Where more than one electroactive species is considered, the current is evaluated separately for each species and summed.

Results and Discussion EC Reaction Compton and Coles191 showed that the shift of the current-potential curve for an EC reaction is a function of the flow rate and the normalised rate constant



K 1  k 2 h 2 x e2 4u 02 D



1 3

(7.17)

The effect of the decay of the B species is to reduce the equilibrium concentrations of both A and B at the electrode, as the potential applied fixes the ratio of the A and B concentrations. This reduction of the concentration of the A species at the electrode surface leads to an increase in the current response, compared to that observed where B does not decay. The effect of increasing the rate of decay of B is therefore to shift the potential required to achieve a given current response in the negative direction. The effect of increasing the volume flow rate, for a constant rate of decay, is to increase both the production of species B, and the speed at which B is transported away from the electrode, resulting in the concentration of B at the electrode remaining closer to that observed in the absence of any decay. As the shape of the currentpotential curve remains constant with the variation of K1, the position of the curve may be characterised by the half wave potential, at which the current observed is half that found as the potential tends to infinity.

Reaction Systems

157

2.4

FE Simulation FD Simulation

FD E1/2 /RT

2.0

1.6

1.2

0.8

0.4

0.0 -1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

log K1 Figure 7.1.

The shift in half-wave potential for an EC reaction.

Figure 7.1 shows the calculated variation of the shift of the half wave potential, plotted against the log of the normalised rate constant, compared to the values given by Compton et. al. using the BI finite difference method41, and excellent agreement is noted. The grid parameters used were nz = 40, nxr = 30, nxe = 30, zf = 4  10-7 cm, xfr = 4  10-6 cm, xfe = 4  10-7 cm, h = 0.04 cm, xr = 0.2 cm, xe 0.4 cm, D = 1 10-5 cm2s-1, w = 0.4 cm, d = 0.6 cm, Vf = 0.01 cm3s-1. The iterative process used re values starting at 0.8 and descending to around 0.2, as the solution converged. The convergence tolerance was set to 1  10-3 for all runs, and between 5 and 35 iterations were required for each run to converge, taking 50s per iteration, on a 100MHz Pentium, using less than 32Mb of memory.

ECE and DISP Reactions For the ECE and DISP reactions the shift of the current-potential wave for the reduction of A is obscured by the reduction of the C species, and the standard characterisation of the process is therefore to examine the variation of the current response at the limiting potential where complete reduction occurs.

Reaction Systems

158

As the rate of decay of B increases, the effective number of electrons transferred at the electrode for every molecule of A which is reduced, neff, varies from 1, for k2 = 0, to 2 as k2  , in which case the C species is formed within a sufficiently small distance of the electrode to be completely reduced. Compton et. al. revealed the ability of the channel electrode geometry to distinguish between the ECE and DISP reactions by examining the flow rate dependence of the transport limited current, using neff as the diagnostic factor. Previous investigations have shown how working curves can be constructed for the factor neff for channel cell geometries, and these are reproduced using the finite element method. The parameter of neff has been shown to a unique function of K1 for the ECE or DISP1 mechanisms, or K2, given below, for the DISP2 mechanism.



K 2  2k 2 k 4 k  2 C Bulk h 2 xe2 4u 0 D



1 3

(18)

2.0

1.8

I /ILim

1.6

FE ECE FD ECE FE DISP1 FE DISP1 FE DISP2 FD DISP2

1.4

1.2

1.0 -2

-1

0

1

2

3

4

5

6

log K Figure 7.2.

The variation of neff with the normalised rate constant for ECE, DISP1 and DISP2 reactions.

Figure 7.2 shows the calculated values of neff for the different types of reaction discussed, plotted against the appropriate normalised rate constant. The grid parameters used for the ECE

Reaction Systems

159

process were nz = 40, nxr = 30, nxe = 30, zf = 5  10-7 to 2  10-8 cm, xfr = 2  10-6 cm, xfe = 2  10-6 cm, h = 0.04 cm, xr = 0.2 cm, xe 0.4 cm, D = 1 10-5 cm2s-1, w = 0.4 cm, d = 0.6 cm, Vf = 0.01 cm3s-1. For the DISP1 and DISP2 reactions, nz = 30, nxr = 20, nxe = 30, for DISP1 zf = 5  10-8 cm and for DISP2 zf = 2  10-7 cm in all cases, and other parameters were as for the ECE reaction. The re values used ranged from 0.8 to 0.08, and around 15-50 iterations were required for almost the whole range of rate constants, rising up to around 100 for the highest rates, each taking around 30s per iteration on a 100 MHz Pentium with 96Mb RAM. Convergence tolerances were set between 1  10-4 and 1  10-5. Good agreement between the FE results presented, and previous FD work is noted. The FE method is also shown to be capable of handling extremely high rate constants, over two orders of magnitude above those previously demonstrated, at a very modest increment in computational time.

The CE Reaction using the Confluence Reactor Introduction The Confluence Reactor, shown in figure 6.1, has recently been developed and characterised within our group192 to further enable the investigation of kinetic processes, and is discussed in Chapter 5, on page 122. The ability of the Confluence Reactor to mix separate streams of solution under precisely defined, stable and reproducible hydrodynamic conditions makes it an ideal tool to investigate any reaction scheme where solution mixing forms the initial step. The ability to tailor the position of the detector electrode and the flow rate employed add to the versatility of the reactor in tackling a range of reactions. The modelling of homogenous kinetic processes using this novel geometry and the application to the CE reaction is shown. To model the CE reaction two different species are introduced into the confluence reactor through the two inlets. These mix and react at some point after the union of the inlets,

Reaction Systems

160

producing an electroactive species C, which is reduced at an electrode placed downstream of the union of the inlets. The reaction scheme may be stated as shown. A+BC

1.

Rate kCE

C + e-  D The Confluence Reactor geometry was investigated to determine the range of rate constants of a CE reaction which could be distinguished over a practical range of volume flow rates and cell geometries. In this type of experiment, the A and B species are introduced into separate inlets of the reactor, and react on mixing to form an electroactive species C, which is reduced at an electrode set into the wall of the cell at a distance downstream. As the generation of the electroactive species occurs in the centre of the cell, away from the side of the cell where the electrode is mounted, the confluence reactor behaves similarly to the opposite sided double electrode as the volume flow rate is raised. At low flow rates and high rates of reaction the C species is generated well before the electrode in the x direction, and is able to diffuse across to the cell. If complete conversion of A to B occurs the current response is identical to that for a channel electrode of the same dimensions. As the volume flow rate is raised, the production of the electroactive C species increases, the region over which C is generated becomes larger in the x direction, and the distance downstream before appreciable quantities of C may be detected at the upper and lower walls of the cell increases. This leads to a current response which initially rises, in line with the predicted behaviour for a channel cell, and then falls off back to zero, at flow rates high enough to prevent C diffusing across to the electrode. This is analogous to the situation previously demonstrated, where an electroactive species is introduced into the upper inlet of a confluence reactor192.

Matrix Formation The general governing equation for a species in the Confluence Reactor is

C  2C  2C C C  D 2  D 2 u w  kd C  ks t x z x z

(7.19)

Reaction Systems

161

where C is the concentration of the species, u and w are the velocities in the x and z directions and ks and kd are the rates of generation and decay of the species at the point in question. The full derivation of the finite element matrices for the triangular grid used is given in equation 2.34 to 2.37. It may be noted that w is non zero only in the central regions of the Confluence Reactor, around the transition between the two inlet parameters and the single channel.

Hydrodynamic Simulation The convective profiles for the Confluence Reactor are more complex, and are found by the solution of the Navier-Stokes equations, also using a finite element method. The Navier Stokes equations are given in equations 2.67 to 2.69, and are solved over similar two dimensional grid to that used for the diffusional simulations, with eight noded rectangular elements being used, with the values of u and w being evaluated over all eight nodes, and the pressure being evaluated only at corner nodes. The derivation of the finite element form of these equations is given in equation 2.73 and the section following.

CE Reaction The CE reaction is formulated in a similar manner to the other reactions studied, with the addition of convection in the z direction, active over the central region of the cell.

 2C A  2C A C C  D  u A  w A  k CE C A C B  0 2 2 x z x z

(7.20)

 2CB  2CB C C D D  u B  w B  k CE C A C B  0 2 2 x z x z

(7.21)

D

D

 2 CC  2 CC C C  D  u C  w C  2k CE C A C B  0 2 2 x z x z

(7.22)

The boundary conditions are z=0

x < xr + xbe

dCA/dz = 0

dCB/dz = 0

dCC/dz = 0

z=0

xr + xbe < x < xr + xbe + xe

dCA/dz = 0

dCB/dz = 0

CC = 0

z=0

xr + xbe + xe < x

dCA/dz = 0

dCB/dz = 0

dCC/dz = 0

z=h

x < xr

dCA/dz = 0

dCB/dz = 0

dCC/dz = 0

Reaction Systems z = 2h

all x

dCA/dz = 0

dCB/dz = 0

dCC/dz = 0

0 xr

all y

z=0

C=0

x < xr

all y

z=0

dC/dz = 0

all x

all y

z = 2h

dC/dz = 0

x=0

all y

all z

C = CBulk

x = xr + xe

all y

all z

dC/dx = 0

all x

y=0

all z

dC/dy = 0

all x

y = ye

all z

dC/dy = 0

Results Using the eight noded elements, a series of simulations were run over centre velocities 6.25  10-3 to 62.5 cms-1, in a cell of width 0.6 cm, with cell heights 0.04, 0.06, 0.08 and 0.1 cm, and for a 0.04 cm high cell, this range of velocities gives volume flow rates of 1  10-4 to 1 cm3s-1. Simulation parameters used were ny = 12, nxr = 12, nxe = 15, nza = 14, nzb = 8, yf = 0.001 cm, xfr = 1  10-5 cm, xfe = 1  10-5 cm, zfa = 1  10-5 cm, zfb = 1  10-5 cm, yr = 0.3, xr = 0.05 cm, xe = 0.2 cm, D = 1  10-5 cm2s-1. For each set of parameters, two simulations were run, identical in every respect, except that one was performed using the a velocity profile generated by the simulations discussed in the previous section, and the other was run using a parabolic velocity profile defined by equation 8.1. Concentration Profiles Owing to the three dimensional nature of the problem, it is problematic to visualise the concentration profile over the whole cell. The concentration across a single plane may be more easily visualised, however, and figure 8.16 shows the concentrations at z = 1.513  10-3 cm from

Non-Analytical Convection

193

the plane of the electrode over a section of a cell 0.04 cm in height, at a centre velocity of 6.25  10-3 cms-1, simulated using the parameters above. 0 1

0.02

0.04

x /cm 0.06 0.08

C/CBulk

0.1 0.04

0

z /cm 0.02

0.04 Figure 8.16

y /cm

0 0.08

0 Concentration profile for layer at z = 1.51  10-3 cm, zr = 0.04 cm.

The concentration profile is shown coloured from blue to red, with the convection profile over the appropriate section shown in blue to green. The drop in the concentrations over the edge region where the velocity profile is also lower is obvious, and the slight extension of the diffusion layer upstream from the electrode edge may be detected. Taking single vectors of concentrations from the three dimensional profile and displaying them in conventional two dimensional plots further illustrates these effects. Figure 8.17 shows the concentration vectors at the channel edge, y = 0, and the channel centre y = 0.3 cm, at the heights above the cell base given in the legend.


z coordinate /m

1.0

Channel Centre 77.7 26.4 8.91 Channel Edge 77.7 26.4 8.91

0.8

C/CBulk

194

0.6

0.4

0.2

0.0 0.02

0.03

0.04

0.05

0.06

0.07

0.08

x coordinate /cm

Figure 8.17

Concentration vectors in x at the channel centre and edges.

The lower concentrations in both the x and z directions away from the electrode edge show that the diffusion layer is larger at the cell edge. To determine the extent of the region across which the concentration profile is depleted figure 8.18 shows vectors across the y direction at z = 0, and a range of x coordinates. Also shown are the vectors taken from an identical simulation, but using the parabolic convection profile given by equation 8.1. This check amply demonstrates that at the centre region of the channel the concentration profile does not depend at all on whether the flow profile is assumed to be parabolic or not.


Distance upstream from electrode edge / m

1.0 0.9

Parabolic Convection 245 120 58.8 28.8 14.1 6.84 3.30 Simulated Convection 245 120 58.8 28.8 14.1 6.84 3.30

0.8

C/CBulk

195

0.7 0.6 0.5 0.4 0.3 0.2 0.00

0.05

0.10

0.15

0.20

0.25

0.30

y coordinate /cm

Figure 8.18

Concentration vectors in y at the channel centre and edges.

It is clear that the central region in the simulations using a real velocity profile behaves exactly like that in a simulation using a parabolic approximation to the flow profile. The size of the depleted edge region may be seen to grow only up to around 0.06 cm from the cell edge. By repeating this analysis on a layer slightly above the plane of the electrode, the evolution of the edge region over the electrode may be examined. Figure 8.19 shows the concentration vectors from the simulation at 77.7 m from the cell base.


x coordinate /cm

1.0

0 0.0255 0.0380 0.0441 0.0471 0.0486 0.0500 0.0520 0.0539 0.0576 0.0646 0.0780 0.104 0.154 0.250

0.9 0.8 0.7

C/CBulk

196

0.6 0.5 0.4 0.3 0.2 0.1

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

y coordinate /cm

Figure 8.19

Concentration vectors in y for at z = 77.7 m.

The depleted region can be seen to grow only to 0.08 cm from the cell wall, even at the back edge of the electrode. This is exactly the region over which the convection profile drops away from the parabolic case given by equation 8.1. Examining the response for the same cell height, 0.04 cm, but using a centre velocity of 62.5 cms-1, a concentration profile for the layer 0.968 m may be generated as shown in figure 8.20. The parameters used were identical to those given for figure 8.16, except for xr = 0.01 cm, not 0.05 cm as before.


197

0 0.004 0.008

1

x /cm 0.012 0.016

C/CBulk

0.02 0.04

0

z /cm 0.02

0 0.08 0.04 Figure 8.20

0 Concentration profile for layer at z = 0.968 m, zr = 0.04 cm.

y /cm

It may be observed that the depleted edge region is much smaller in the upstream direction, noting that the x scale of this figure is a factor of 5 smaller than that of figure 8.16. This is shown by a plot of the concentration vectors in the x direction, given in figure 8.21.

z coordinate /m

1.0

Channel Centre 1.71 0.968 0.282 Channel Edge 1.71 0.968 0.282

C/CBulk

0.8

0.6

0.4

0.2

0.0

0.008

0.009

0.010

0.011

0.012

x coordinate /cm

Figure 8.21


The very high velocity used maintains the bulk concentration up until very close to the electrode in the x direction, and down to much closer to the electrode in the z direction. Examination of the output for this velocity shows that the diffusion layer extends only to 0.0078 cm from the electrode in the z direction at the limit of the simulated region in x. It may also be


198

noted that the gradient of the concentration profile is much steeper in the y direction, illustrated further by a plot, in figure 8.22, of the concentration vectors in the y direction for the layer at z = 0.968 m. x coordinate /cm 0 0.0089 0.00937 0.00964 0.0098 0.00989 0.00994 0.00997 0.00999 0.01 0.01001 0.01003 0.01007 0.0101 0.0103 0.0105 0.0111 0.0120 0.0139 0.0176 0.0246 0.0380 0.0640 0. 114 0.21

1.0

C/CBulk

0.8

0.6

0.4

0.2

0.0 0.000

0.005

0.010

0.02

0.03

0.04

0.05

0.06

0.07

0.08

y coordinate /cm

Figure 8.22

Concentration vectors in y at z = 0.968 m.

Noting the expansion of the x scale below y = 0.01 cm, the gradient of the concentration profile in the y direction is much steeper than for the slower flow rate examined in figure 8.16. The effects of increasing the height of the cell are shown in figure 8.23, which shows the concentrations in the plane 52.2 m above the electrode for a cell 0.01 cm in height. The parameters used were identical to those given for figure 8.16, except for xr = 0.08 cm.


0

0.04 0.08

1

0.12

199

x /cm 0.16 0.2

C/CBulk

0.1

0

z /cm 0.05

0 0.1 Figure 8.23

y /cm

0.2

0 Concentration profile for layer at z = 52.2 m, zr = 0.1 cm.

This appears very similar to figure 8.16, but the y and z scales here are a factor of 2.5 larger. It may be noted therefore that the edge regions over which the concentration profile is depleted are larger by the same factor, which is as expected, given that the size of the edge region for the convection profile is a function of the aspect ratio of the cell. The low gradient in the y direction is also apparent. Plotting the concentration vectors in the x direction, shown in figure 8.24, gives very similar results to those for the 0.04 cm high cell, scaled by the same factor of approximately 2.5. The diffusion layer extends further upstream in the x direction, and vertically away from the electrode.


z coordinate /m

1.0

Channel Centre 180 52.2 15.1 Channel Edge 180 52.2 15.1

0.8

C/CBulk

200

0.6

0.4

0.2

0.0 0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

x coordinate /cm

Figure 8.24


Using a 0.1cm high cell at high velocities yields concentrations profiles like that shown in figure 8.25, where the layer 1.22 m above the electrode is shown for a 0.1cm high cell with a centre velocity of 62.5 cms-1. The parameters used were identical to those for figure 8.16, except for xr = 0.01 cm. 0

0.004 0.008

1

x /cm 0.012 0.016

C/CBulk

0.02 0.1

0

z /cm 0.05

0 0.1

y /cm

0.2

0 Figure 8.25

Concentration profile for layer at z = 1.22 m, zr = 0.1 cm.

This is similar to that for the 0.04 cm high cell at the same centre velocity, except that the edge regions for both the convection and concentration profiles are increased in proportion to the cell height.


201

Steady State Currents In analysing the current responses of a full width electrode for the different cell heights and centre velocities used, the influence of the edge region on the current response may be assessed by dividing the currents obtained using a simulated, realistic convection profile, IR, by those obtained using a parabolic profile, IP. This gave the results shown in figure 8.26. 0.990

Cell Height /cm 0.04 0.06 0.08 0.1

0.985

IR / IP

0.980 0.975 0.970 0.965 0.960 0.955 0.950

0.01

0.1

1 -1

10

u0 /cms

Figure 8.26

Concentration profile for layer at z = 1.22 m, zr = 0.1 cm.

The dotted section for the 0.04 cm high cell indicates the onset of the thin layer condition, where the diffusion layer extends vertically over the whole height of the cell, due to the very low volume flow rate employed. The currents observed under these conditions fall away from those predicted by the Levich equation, and this effect adds to that observed due to the edge region for the convection profile. While it is clear that there is a small increase in the ratios of IR/IP with flow rate, the cell height appears to be the overriding parameter in determining the correspondence between the actual behaviour of a full width electrode and that predicted assuming a parabolic flow profile, invariant in the y direction. What is interesting about these results is that the deviations found in the current response are smaller than those found in the volume flow rate. For a 0.1 cm high cell, the difference in volume flow rate is 12% between the parabolic and non parabolic cases. The


202

maximum error observed in the current is only 4.5% however. A possible explanation for this observation is that as the diffusion layer grows out further into the cell in the x and z directions at these edge regions, diffusional transport in the y direction may allow mass transport from regions nearer the centre of the cell in y. In a cell where a truly parabolic diffusion profile is observed, the convection and concentration profiles are uniform in y, and with no concentration gradient, and therefore no diffusion occurs. The concentration profiles presented in figures 8.16, 8.20, 8.23, and 8.25 all show concentration gradients in the y direction, along which diffusion will act to transport material out from the centre of the cell into these depleted edge regions. In effect, the diffusional transport will mitigate the lower convective transport and act to maintain the concentration profile.

Conclusions The conclusion to be drawn from these investigations is that the use of electrodes extending across the entire width of a cell is attended by only a small error, which is only weakly dependant of the flow rate employed. As the magnitude of the error may be estimated with fair accuracy from the dimensions of the cell, it may be corrected for to within a normal experimental level of accuracy. There are electrode types for which the process of assembly is complicated by the requirement to mask off peripheral areas. The mechanism by which reduced convective transport may be masked by the appearance of lateral diffusion in the y direction is very general, and there is no reason it would not operate in a variety of different cell geometries. A particular example is the microwire analysed in chapter six. The small sizes of the wires used make the application of insulating coatings problematic, as is the accurate measurement of the exposed areas. For particularly small wires, the application of a layer of varnish over some portion may be thick enough to influence the convection profile observed. Further work may be able to establish whether the errors observed at different electrode types such as the microwire, or more conventionally a microband electrode are as low as those for a macroelectrode in a channel cell.

Chapter 9. Appendix One Dimensional Second Order Element Formulation In one dimension, linear elements are not the only choice available. Second order elements as shown in figure 2.1, with one interior node, are perfectly acceptable approximations for a one dimensional variable. The same requirements hold for the interpolation functions derived as before. It is required that 

Each interpolation function be uniquely defined by the nodal values of the element.



Each interpolation function must take a value of unity at the node with which it is associated and zero at all other nodes in the element. For the second order linear element, the three nodes define parabolic

interpolation functions. Rather than derive these in real space, the concept of a local coordinate system is helpful in simplifying the integrations necessary. A local coordinate in one dimension may be defined as



2 x  xm  a

(9.1)

where xm is the midpoint of the element, and location of the centre node, and a is the length of the element. From the above conditions, we find

Ni 



1 2

  12  2



N j

N j  1  2 Nk 



1 2

 2  12 

These functions are shown in figure 9.1.

N i   12    





 2

N k    12 

(9.2)

(9.3)

(9.4)

Appendix

204

1.0

Ni 0.5

Nj

N

Nk

0.0

-1.0

-0.5

0.0

0.5

1.0



Figure 9.1

A set of second order interpolation functions for a one dimensional simulation.

Bearing in mind that

   f    f   the matrices for the element may be x x 

constructed just as for the linear element. To illustrate the exact workings, these derivations will be shown in a number of steps which will be omitted in the following sections. The characteristic matrix may be found as

   xk N N K dx x x xi   1   2  2 12     1 2 K     2 12    4 2 1  1 1 1  2     2   2  2 

  2 N 2 N a  d a  a  2 1 1

 12     12  2  2  12   d a   12 2 

The capacitance matrix may be found as

 7 8 1  1    8 16  8 (9.5)  3a  1  8 7 

Appendix

205



  2 1   2 1    2  2 2   a  2 a K T   NN d   1    2  2 2 2  2 2 2 8 2 2 1 1   1     1   1 2  2  4 2  1 a   2 16 2  30   1 2 4  1

1











 

 2 1     1     1 2  2 2  d 2   2   1 





(9.6) The matrix for convection may also be found as

  12   1    1  1  a 1 K C   NN d    12    2  2 2 2 2 1 1  1   2     1





 2 3 1     2 2  2 2  2 3   1





  12 1      12  2  2 2   12   1



   d  



 3 4 1  1   4 0  4 12  1 4  3 (9.7) The element load vector can be found to be

1   ksl   P 4 6   1 

(9.8)

Bilinear Square Elements Grid Generation It may be recognised that the grids presented in figure 2.5 are formed by first decomposing the region into rectangular areas, each of which is then split into two triangles. If the last step is omitted, suitable rectangular grids are generated.

Appendix

Figure 9.2

Pascal’s Triangle for two dimensional polynomials.

Referring to figure 9.2, the global and local node numbering would be related by the NODE array shown:

206

Appendix

207

Element

1

2

3

4

1

1

2

12

11

2

2

3

13

12

10

11

12

22

21

Matrix Formation The isolinear square element is defined by four nodes, and so the variation of the concentration across the element will contain an extra polynomial term to those that define the concentration within the triangular element, which are just x, z and a constant. Elements could be defined including any combination of terms from Pascal’s triangle for two dimensional polynomials, shown in figure 9.3 1 x x2 x Figure 9.3

3

Constant z

Linear z2

xz 2

xz

xz

2

Quadratic z

3

Cubic

Pascal’s Triangle for two dimensional polynomials.

The choice of which extra term to use in the element is constrained by a preference for choosing the terms included in an element’s interpolation functions so as to maintain spatial iostropy. What this means is that the possible variations of the interpolation function along one axis ought to be identical those possible along the other, so there is no preferred direction for the element. This is ensured only by taking a symmetrical distribution of nodes from the set shown above. There are also some other formal requirements that must be met if convergence is to be attained as the element sizes decreasescxciii,cxciv, which are listed here: 

The interpolation function must contain a constant term, or the function may vanish as the element size tends to zero.



The interpolation function must be continuous within an element. This is automatically satisfied by choosing a simple polynomial.

Appendix 

208

The variable and derivatives of the variable of interest up to one order less than the order of the highest derivative in the weighted residual form of the governing equation must be continuous across the interelement boundaries.

The last requirement arises from the fact that the value of an expression such as

 n 1C  nC changes by only a finite jump if is continuous across a boundary between x n 1 x n two elements, but is undefined if this condition is not met. As a result of these considerations the obvious term to select as the fourth term to be used in the interpolation functions, and therefore the variation of the concentration across an element is xz, in a real coordinate system. This gives as the set of polynomial terms 1, x, z, xz. This set is called the basis function of the element. Rather than generate the interpolation functions by inspection and solving a set of simultaneous equations as for the linear one dimensional element, the basis function and nodal coordinates can be used to derive the interpolation function. The local coordinates used for the rectangular element are shown in figure 9.4.

 (-1,1)

(1,1)

4

3 

1 (-1,-1)

Figure 9.4

2 (1,-1)

Bilinear local coordinate system.

The local coordinates may be related to the real coordinates of the element by



2 x  xm  a



2  y  ym  b

(9.9)

where xm and ym are the midpoint coordinates, and a and b are the sizes of the element in the x and z directions.

Appendix

209



If the basis function is defined in local coordinates as B  1    , then the variation of the quantity of interest across the whole element will be defined by

C  a1  a 2   a3   a 4 



Defining A  a1

a2

a3

(9.10)

a4  this can be stated in vector product form as  C  BAT

(9.11)

The values of the quantity of interest at the nodes of the element may be found as

  C1   B1  C        2    B2  AT or C   B AT n n  C 3   B3      C 4   B4  

where B1  1 1

1

(9.12)

11  , which is the basis function evaluated at the nodes. This

equation may be rearranged to give

 1 AT  Bn  Cn 

(9.13)

This may be substituted into equation 9.11 to give

 1 C  BBn  Cn 

(9.14)

This expression defines the variation of the quantity of interest across a whole element in terms of the nodal values, and it may therefore be seen that the interpolation functions may be derived as

  1 N  BBn 

(9.15)

For the case of the bilinear rectangular element matrix of the basis function evaluated at the nodes, and the inverse of this matrix may be found as

1  1  1 1  1 1 1 1  1  1  1  1 1 1   and B   B  1 1 1 1  4  1  1    1  1 1  1  1 1 The interpolation functions and their derivatives are therefore given by

1 1 1  1 1 1  1  1

(9.16)

Appendix

210

N1 

1 4

1   1  

N1  

1 4

  1

N2 

1 4

1   1  

N 2  

1 4

1  

N3 

1 4

1   1  

N 3  

1 4

1  

N 3  

1 4

1  

 1  

N 4  

1 4

1   

N1   N 2  

1 4

1 4

  1

 1  

(9.17)

N4 

1 4

N 4  

1   1  

1 4

These interpolation functions are all identical in form, and the shape of Ni, rotated for clarity, is shown in figure 9.5.

1.00

0.75

N 0.50 1.0 0.25

0.5 0.0

0.00 -0.5

-0.5

0.0



Figure 9.5



0.5 1.0 -1.0

Bilinear interpolation function.

The linear variation in the directions parallel to the axis, but curvature in directions at angles to the axis can clearly be seen. Using the chain rule for differentiation and integration to convert between local and real coordinates, the matrices for the element may be found as

1  1  2  2  2 1 1  2  2 2   1  1 Da  1 2  2  1 Db  K    2  2 6b   1  2 2  2 6a   1 1     2  1 1  2 2   2  1 1

(9.18)

Appendix

 2  V x b  2 K C   12   1   1

2 2 1 1

211

1  1  2  1 1  1  1 V z a   1  2 2  2  2 12   1  2 2   2  2  2  1 1

4 2 K T   k d A  36 1  2

2 4 2 1

1 2 4 2

2 1 1  2

(9.19)

2 1 2  4

1 0  0 1        Dk s1 a 1 Dk s 2 b 1 Dk s 3 a 0 Dk s 4 b 0 k s A         P 2 0 2 1 2 1 2 0  4         0 0  1 1

(9.20)

1 1   (9.21) 1  1

where kd is the rate of decay, ks1, ks2, ks3 and ks4 are gradient boundary conditions as before, with the element sides numbered anticlockwise from the bottom face, and ks is the rate of production for sources over a whole element. A is the element area and l is the length of the side indicated. The total equation is as for 2.38.

Isoparabolic Rectangular Element Formulation By defining a rectangular element with midside and corner nodes, an element with 8 nodes, capable of defining a basis function of

 B 1 x z



x2

xz

z2

x2 z



xz 2 in real coordinates, is formed.

Grid Generation Grid geometries identical to those generated for use with the square elements may be used. The local coordinates are defined as in equation 9.9.

Appendix

7

212

6

5

 8

4 

1 Figure 9.6

2

3

An isoparabolic element.

Assemblies of isoparabolic elements involve slightly more complex relationships between global and local node numbering, with part of a typical scheme shown below, for a grid with nine elements in the x direction across the lower layer.

Figure 9.7

Global and local node numbering for an isoparabolic element.

The NODE array entries corresponding to the elements shown would be. Element

1

2

3

4

5

6

7

8

1

1

2

3

21

32

31

30

20

2

3

4

5

22

34

33

32

21

10

30

31

32

50

61

60

59

49

It is clear that the number of nodes in the grids generated using iosparabolic elements is higher than needed for the same type of discretisation using simpler square elements. The total number of nodes in a mesh of isoparabolic elements is given by

n z 3n x  2   2n x  1 where nx and nz are the numbers of elements in the x and z

Appendix

213

directions. For a simple linear square grid the total number of nodes is n z  1n x  1 , indicating that for large grids, roughly three times as many nodes are needed for the same number of elements. The possible gains in accuracy which may be set against this increased computational expense will be discussed.

Matrix Formation The interpolation functions appropriate to this element can be found using equation 9.15 to be:

Nn 

1   n 1   n  n    n   1

1 4

n  1, 3, 5, 7

Nn 

1 2

1   1   

n  2, 6

Nn 

1 2

1   n 1  2 

n  4, 8

2

n

where  n is the value of the coordinate of node n. These interpolation functions are shown in figures 9.8 and 9.9.

1.00 0.75

N

0.50

1.0

0.25 0.5 0.00 0.0

-0.25



-0.5



-0.5

0.0 0.5

1.0 -1.0

Figure 9.8

A corner interpolation function for an isoparabolic element.

Appendix

214

1.00 0.75

N 0.50 0.25 0.00 0.5



1.0 0.5

0.0 0.0 -0.5

-0.5



-1.0 -1.0

Figure 9.9

A midside interpolation function for an isoparabolic element.

It can be observed that as for the second order linear element, there are two types of interpolation function, and it must also be noted that the corner interpolation functions take positive values over only one eighth of the element, and actually make a negative contribution to the element load vector when evaluated over the whole element. The matrices for this element are:

Appendix

52 - 80  28  Db - 6 K   90a 23  - 40 17  6 52 6  17  Da - 40  90b 23  - 6 28  - 80

215

-80 160

28 -80

-6 0

23 -40

-40 80

17 -40

-80 0

52 6

6 48

17 6

-40 0

23 -6

-40 80

17 -40

6 0

52 -80

-80 160

28 -80

-40 0

23 -6

-6 -48

28 -6

-80 0

52 6

6  0  -6   -48 -6   0  6   48 

6 48 6 0 -6 -48 -6 0

17 6 52 -80 28 -6 23 -40

-40 0 -80 160 -80 0 -40 80

23 -6 28 -80 52 6 17 -40

-6 -48 -6 0 6 48 6 0

28 -6 23 -40 17 6 52 -80

-80  0  -40   80  -40   0  -80   160

(9.22)

- 12 - 20  8  V x b 14 K C    180 3  0 3  - 26 - 12 - 26  3  V z a 0  180 3  14 8  - 20

20 0

-8 20

-14 40

-3 0

0 0

3 0

-20 -40

12 26

-14 48

-3 26

0 -40

3 14

0 0

-3 0

-14 40

12 20

-20 0

8 -20

0 40

-3 -14

-14 48

-8 -14

20 40

-12 -26

14  -40 14   -48 14   -40 14   -48

14 -48 14 -40 14 -48 14 -40

3 -26 -12 -20 8 14 3 0

0 40 20 0 -20 -40 0 0

-3 -14 -8 20 12 26 -3 0

-14 48 -14 40 -14 48 -14 40

-8 -14 -3 0 -3 26 12 20

20  40  0   0  0   -40 -20  0 

(9.23)

Appendix

6 - 6  2  k d A - 8 K T   180 3  - 8 2  - 6

216

-6

2

-8

3

-8

2

32

-6

20

-8

16

-8

-6 20

6 -6

-6 32

2 -6

-8 20

3 -8

-8

2

-6

6

-6

2

16

-8

20

-6

32

-6

-8

3

-8

2

-6

6

20

-8

16

-8

20

-6

-6  20 -8   16  -8   20 -6   32 

(9.24)

1  0  0  1   1  4 0  0  0  4           1  1  0  0   1          Dk s1 a 0 Dk s 2 b 4 Dk s 3 a 0 Dk s 4 b 0 k s A  4  P     (9.25) 6 0  6 1  6 1  6 0  8  1           0  0   4 0  4 0  0  1  1   1           0 0 0 4  4  where the terms are as previously defined, for the bilinear square element. It may be evident that the evaluation of these matrices by hand is an arduous task, and the calculation of K  and K C  in this instance required over a day. It is no longer necessary for such integrations to be problematic however due to the availability of software capable of performing all the symbolic operations necessary. Mathcad 7 was used to calculate all the matrices derived from this point on, with random matrix positions checked by hand. The correct functioning of programs using the matrices generated, and adherence to the remarks made on the row and column sums of the matrix represent the best check on the matrices generated.

Linear Tetrahedral Element Formulation As well as using cubic elements with nodes either at the corners only, or at corner and mid-side positions, it is possible to subdivide a cube into a set of five tetrahedrons. This is shown below with the central tetrahedron highlighted in red. It may be recognised that this subdivision introduces no new nodes.

Appendix

217

8

7

5 6 4 3 1 Figure 9.10

2

A cube subdivided into tetrahedral subelements.

The local node numbering scheme for a linear cubic element are shown. The five tetrahedrons created may be defined by connecting the nodes of the cube and relabeling them i, j, k or l as given in the table below. Note that the nodes are labelled so that looking along the edge from node l, the other nodes form an anticlockwise series in the order i, j and k. The columns indicated by i’, j’, k’ and l’ contain the tetrahedrons that would be formed if the division of the cube were done across each face in the only other possible way. Tetrahedron

i

j

k

l

i’

j’

k’

l’

1

2

3

1

6

1

2

4

5

2

3

4

1

8

2

3

4

7

3

1

6

8

5

4

5

7

8

4

3

6

7

8

2

5

6

7

5

3

1

6

8

2

4

5

7

When subdividing a set of cubes into tetrahedrons, it is necessary to ensure that each face of a cube is placed adjacent to another face which is divided between the same two corners. Neglecting this stipulation violates the continuity between elements which is a requirement of the finite element technique. This means that each cube must be surrounded by neighbours divided in the opposite direction. A simple way to accomplish this for a regular grid of cubes arranged in layers numbered sequentially in x, y and z, is to

Appendix

218

determine if the sum of the x, y and z indices (the layer from the lower numbered edge in each direction) is odd or even and divide each set of cubes in one fashion or other. An example is shown in figure 9.11, with the cubes whose indices give even sums indicated by darker shading.

Figure 9.11

An array of cubes divided into tetrahedrons in alternating directions.

Linear Tetrahedron Matrix Formation The local coordinate system for a tetrahedron is defined in an analogous way to that of the linear triangle. The interpolation function of a point p with respect to a node is equal to the natural coordinate of the point, which is defined as the volume of the tetrahedron formed by the point p and the other three vertices of the tetrahedron, as shown in figure 9.12.

Figure 9.12

A tetrahedral interpolation function.

As before for the triangular element,

Appendix

1   1  x  x   i  y   yi     z   z i

219

1

1

xj

xk

yj

yk

zj

zk

 Ni  xl   N j    yl   N k    z l   N l 

(9.26)

This is then inverted to obtain

 Ni   ai N    j   1 a j  N k  6V a k     Nl   al

bi

ci

bj

cj

bk

ck

bl

cl

d i  1  d j   x    d k  y   dl  z 

(9.27)

where

V 

1 xi

yi

zi

1 1 xj 6 1 xk 1 xl

yj

zj

yk

zk

yl

zl

(9.28)

The values of the terms in 9.27 are the cofactors of the term in the same position in equation 9.28. These are therefore

xj

yj

zj

ai  x k

yk

zk

xl

yl

zl

1 yj bi   1 y k 1 yl

zj zk zl

xj ci   x k xl

1 zj 1 zk 1 zl

xj

yj

1

d i   xk

yk

1

xl

yl

1

(9.29) The remaining rows can therefore be found by cycling through the terms i, j k and l, and multiplying by –1 with each permutation. The matrices of interest may be found to be

 bi2  ci2  di2  D bib j  ci c j  di d j K   36V bibk  ci ck  di d k   bibl  ci cl  di d l (9.30)

bib j  ci c j  di d j b 2j  c 2j  d 2j

bibk  ci ck  d i d k b j bk  c j ck  d j d k

b j bk  c j ck  d j d k b j bl  c j cl  d j dl

bk2  ck2  d k2 bk bl  ck cl  d k dl

bibl  ci cl  di dl   b j bl  c j cl  d j dl  bk bl  ck cl  d k d l   bl2  cl2  dl2 

Appendix

bi  V x bi K C    24 bi  bi

bj

bk

bj

bk

bj

bk

bj

bk

bl  c i  bl  V y ci  bl  24 ci   bl  ci

220

cj

ck

cj

ck

cj

ck

cj

ck

cl  d i  cl  V z d i  cl  24 d i   cl  d i

dj

dk

dj

dk

dj

dk

dj

dk

dl  d l  dl   d l 

(9.31)

2  k d V 1 K T   20 1  1

1 2 1 1

1 1 2 1

1 1 1  2

(9.32)

1 1 1  Dk s ijk Aijk 1 Dk s ijl Aijl 1 Dk s ikl Aikl 0 Dk s jkl A jkl       P     1 1 0 3 3 3 3       0  1 1

0  1    k sV 1 4   1

1 1  1  1

(9.33) The load vector contains terms for gradient boundary conditions on each face of the tetrahedral element, labelled by the corner three nodes. The area of each face may be given by

A jkl 

1 4

l

 l kl  lli  lij  l jk  lil lij  l jk  lil lij  l jk  lil 2 where ljk is the 1

jk

length of the edge from node j to k, given by



l jk  x j  xk   y j  y k   z j  z k  2

2

.

1 2 2

20 Noded Hexahedral Element Formulation The grid numbering used for the 20 noded cubic element is shown below.

Appendix

Figure 9.13

221

Local node numbering for a 20 noded cubic element.

Adding midside nodes to the trilinear hexahedral element gives 20 nodes and so admits the basis function

  1 B 2 x y

x xy 2

y 2

y z

z yz 2

x2 xz 2

xy x2 z

y2 yz xyz x 2 yz

z2 xy 2 z

xz   xyz 2 

(9.34) This basis set contains a symmetrical set of terms, leaving out the three cubic terms which are just cubes of one coordinate, and including instead the central quartic terms. This is still seven terms less than the set required to allow a parabolic variation along each edge. Using equation 9.15 the interpolation functions can be found to be

Nn 

1 8

1   n 1   n 1   n   n    n    n   2



n  1, 3, 5, 7, 13, 15, 17, 19



N n  18 1   2 1  n 1   n   n  2, 6, 14, 18 Nn 

1 8

1   n 1  2 1   n  

n  4, 8, 16, 20

Nn 

1 8

1   n 1  n 1   2 

n  9, 10, 11, 12

These interpolation functions may be visualised in a similar manner to the linear interpolation function in figure 2.15. The matrices of interest may be found as

(9.35)

Appendix 98 - 160  62  18 34  - 80 46  - 18 - 18  2 Dbc 18 K    135a 24 - 24  46 - 80  34 24  11  - 40 29  - 24

98 - 18  46  - 80 34  18 62  - 160 - 18  2 Dac - 24   135b 24 18  46 - 24  29 - 40  11  24 34  - 80 98 - 160  62  18 34  - 80 46  - 18 - 18  2 Dab 18   135c 24 - 24  46 - 80  34 24  11  - 40 29  - 24

222

-160 320 -160 0 -80

62 -160 98 -18 46

18 0 -18 96 -18

34 -80 46 -18 98

-80 160 -80 0 -160

46 -80 34 18 62

-18 0 18 -96 18

-18 0 18 -60 24

18 0 -18 60 -24

24 0 -24 60 -18

-24 0 24 -60 18

46 -80 34 24 11

-80 160 -80 0 -40

34 -80 46 -24 29

24 0 -24 48 -24

11 -40 29 -24 46

-40 80 -40 0 -80

29 -40 11 24 34

160 -80 0 0 0

-80 34 18 18 -18

0 18 -96 -60 60

-160 62 18 24 -24

320 -160 0 0 0

-160 98 -18 -24 24

0 -18 96 60 -60

0 -24 60 96 -96

0 24 -60 -96 96

0 18 -60 -48 48

0 -18 60 48 -48

-40 29 -24 -18 18

80 -40 0 0 0

-40 11 24 18 -18

0 24 -48 -60 60

-80 34 24 24 -24

160 -80 0 0 0

-80 46 -24 -24 24

0 0 -80 160 -80

-24 24 34 -80 46

60 -60 24 0 -24

-18 18 11 -40 29

0 0 -40 80 -40

18 -18 29 -40 11

-60 60 -24 0 24

-48 48 -18 0 18

48 -48 18 0 -18

96 -96 24 0 -24

-96 96 -24 0 24

24 -24 98 -160 62

0 0 -160 320 -160

-24 24 62 -160 98

60 -60 18 0 -18

-18 18 34 -80 46

0 0 -80 160 -80

18 -18 46 -80 34

0 -40 80 -40 0

-24 29 -40 11 24

48 -24 0 24 -48

-24 46 -80 34 24

0 -80 160 -80 0

24 34 -80 46 -24

-48 24 0 -24 48

-60 24 0 -24 60

60 -24 0 24 -60

60 -18 0 18 -60

-60 18 0 -18 60

18 34 -80 46 -18

0 -80 160 -80 0

-18 46 -80 34 18

96 -18 0 18 -96

-18 98 -160 62 18

0 -160 320 -160 0

18 62 -160 98 -18

-18 96 -18 0 18

46 -18 98 -160 62

-80 0 -160 320 -160

34 18 62 -160 98

18 -96 18 0 -18

62 18 34 -80 46

-160 0 -80 160 -80

-18 60 -24 0 24

-24 60 -18 0 18

24 -60 18 0 -18

18 -60 24 0 -24

46 -24 29 -40 11

-24 48 -24 0 24

29 -24 46 -80 34

-40 0 -80 160 -80

11 24 34 -80 46

24 -48 24 0 -24

34 24 11 -40 29

-96 18 0 60 60

18 34 -80 -24 -18

0 -80 160 0 0

-18 46 -80 24 18

96 -18 0 -60 -60

-18 98 -160 18 24

0 -160 320 0 0

-60 18 0 96 48

-60 24 0 48 96

60 -24 0 -48 -96

60 -18 0 -96 -48

24 34 -80 -18 -24

-48 24 0 60 60

24 11 -40 -24 -18

0 -40 80 0 0

-24 29 -40 24 18

48 -24 0 -60 -60

-24 46 -80 18 24

-60 -60 -24 48 -24

18 24 29 -24 46

0 0 -40 0 -80

-18 -24 11 24 34

60 60 24 -48 24

-24 -18 34 24 11

0 0 -80 0 -40

-48 -96 -18 60 -24

-96 -48 -24 60 -18

96 48 24 -60 18

48 96 18 -60 24

24 18 98 -18 46

-60 -60 -18 96 -18

18 24 46 -18 98

0 0 -80 0 -160

-18 -24 34 18 62

60 60 18 -96 18

-24 -18 62 18 34

0 24 -48 24 0

-80 34 24 11 -40

160 -80 0 -40 80

-80 46 -24 29 -40

0 -24 48 -24 0

-40 29 -24 46 -80

80 -40 0 -80 160

0 24 -60 18 0

0 18 -60 24 0

0 -18 60 -24 0

0 -24 60 -18 0

-80 34 18 62 -160

0 18 -96 18 0

-160 62 18 34 -80

320 -160 0 -80 160

-160 98 -18 46 -80

0 -18 96 -18 0

-80 46 -18 98 -160

-160 320 -160 0 -80

62 -160 98 -18 46

18 0 -18 96 -18

34 -80 46 -18 98

-80 160 -80 0 -160

46 -80 34 18 62

-18 0 18 -96 18

-18 0 18 -60 24

18 0 -18 60 -24

24 0 -24 60 -18

-24 0 24 -60 18

46 -80 34 24 11

-80 160 -80 0 -40

34 -80 46 -24 29

24 0 -24 48 -24

11 -40 29 -24 46

-40 80 -40 0 -80

29 -40 11 24 34

160 -80 0 0 0

-80 34 18 18 -18

0 18 -96 -60 60

-160 62 18 24 -24

320 -160 0 0 0

-160 98 -18 -24 24

0 -18 96 60 -60

0 -24 60 96 -96

0 24 -60 -96 96

0 18 -60 -48 48

0 -18 60 48 -48

-40 29 -24 -18 18

80 -40 0 0 0

-40 11 24 18 -18

0 24 -48 -60 60

-80 34 24 24 -24

160 -80 0 0 0

-80 46 -24 -24 24

0 0 -80 160 -80

-24 24 34 -80 46

60 -60 24 0 -24

-18 18 11 -40 29

0 0 -40 80 -40

18 -18 29 -40 11

-60 60 -24 0 24

-48 48 -18 0 18

48 -48 18 0 -18

96 -96 24 0 -24

-96 96 -24 0 24

24 -24 98 -160 62

0 0 -160 320 -160

-24 24 62 -160 98

60 -60 18 0 -18

-18 18 34 -80 46

0 0 -80 160 -80

18 -18 46 -80 34

0 -40 80 -40 0

-24 29 -40 11 24

48 -24 0 24 -48

-24 46 -80 34 24

0 -80 160 -80 0

24 34 -80 46 -24

-48 24 0 -24 48

-60 24 0 -24 60

60 -24 0 24 -60

60 -18 0 18 -60

-60 18 0 -18 60

18 34 -80 46 -18

0 -80 160 -80 0

-18 46 -80 34 18

96 -18 0 18 -96

-18 98 -160 62 18

0 -160 320 -160 0

18 62 -160 98 -18

(9.36)

-24 0  24   -48 24   0  -24  48  60   -60  -60 60   -18 0   18  -96  18   0  -18  96 

        0  -80   160  0   0   0  0   -160 0   -80  160   -80   0  -160  320  -80 0 -40 80 -40

-24 0  24   -48 24   0  -24  48  60   -60  -60 60   -18 0   18  -96  18   0  -18  96 

Appendix - 18 0  - 18  58 - 26  20 - 6  - 22 - 22  V x bc 58 K C    135 44 4  - 6 20  - 26 44  - 29  20 - 9  4

223

0 0 0 -80 20

18 0 18 22 6

-58 80 -58 96 -58

26 -20 6 22 18

-20 0 20 -80 0

-6 20 -26 58 -18

58 -80 58 -96 58

58 -80 58 -60 44

-58 80 -58 60 -44

-44 40 -44 60 -58

44 -40 44 -60 58

-6 20 -26 44 -29

-20 0 20 -40 20

26 -20 6 -4 9

-44 40 -44 48 -44

29 -20 9 -4 6

-20 0 20 -40 20

-9 20 -29 44 -26

0 -20 80 80 -80

-20 26 -58 -58 22

80 -58 96 60 60

0 18 -58 -44 -4

0 0 80 40 -40

0 -18 -22 4 44

-80 58 -96 -60 -60

-40 44 -60 -96 -96

40 -44 60 96 96

80 -58 60 48 48

-80 58 -60 -48 -48

20 -9 4 -22 58

0 -20 40 80 -80

-20 29 -44 -58 22

40 -44 48 60 60

-20 26 -44 -44 -4

0 -20 40 40 -40

20 -6 4 4 44

-40 40 -20 0 20

-4 -44 26 -20 6

60 60 -44 40 -44

22 -58 29 -20 9

-80 80 -20 0 20

58 -22 -9 20 -29

-60 -60 44 -40 44

-48 -48 58 -80 58

48 48 -58 80 -58

96 96 -44 40 -44

-96 -96 44 -40 44

44 4 -18 0 -18

-40 40 0 0 0

-4 -44 18 0 18

60 60 -58 80 -58

22 -58 26 -20 6

-80 80 -20 0 20

58 -22 -6 20 -26

-40 20 0 -20 40

-4 9 -20 29 -44

48 -44 40 -44 48

-4 6 -20 26 -44

-40 20 0 -20 40

44 -26 20 -6 4

-48 44 -40 44 -48

-60 44 -40 44 -60

60 -44 40 -44 60

60 -58 80 -58 60

-60 58 -80 58 -60

58 -26 20 -6 -22

-80 20 0 -20 80

22 6 -20 26 -58

96 -58 80 -58 96

22 18 0 18 -58

-80 0 0 0 80

58 -18 0 -18 -22

- 18 - 22  - 6  20 - 26  58 - 18  0 - 22  V y ac 4   135 44 58  - 6 4  - 9 20  - 29  44 - 26  20

58 -96 58 -80 58

-6 -22 -18 0 -18

-20 80 0 0 0

26 -58 18 0 18

-58 96 -58 80 -58

18 -58 26 -20 6

0 80 -20 0 20

58 -60 44 -40 44

44 -60 58 -80 58

-44 60 -58 80 -58

-58 60 -44 40 -44

-6 4 -9 20 -29

44 -48 44 -40 44

-9 4 -6 20 -26

-20 40 -20 0 20

29 -44 26 -20 6

-44 48 -44 40 -44

26 -44 29 -20 9

-96 58 -80 -60 -60

58 -26 20 4 -22

-80 20 0 40 80

22 6 -20 -44 -58

96 -58 80 60 60

22 18 0 -58 -44

-80 0 0 80 40

-60 58 -80 -96 -48

-60 44 -40 -48 -96

60 -44 40 48 96

60 -58 80 96 48

44 -26 20 -22 4

-48 44 -40 -60 -60

44 -29 20 4 -22

-40 20 0 40 80

-4 9 -20 -44 -58

48 -44 40 60 60

-4 6 -20 -58 -44

-60 -60 44 -48 44

58 44 -9 4 -6

-80 -40 -20 40 -20

22 -4 29 -44 26

60 60 -44 48 -44

-4 22 26 -44 29

-40 -80 -20 40 -20

-48 -96 58 -60 44

-96 -48 44 -60 58

96 48 -44 60 -58

48 96 -58 60 -44

44 58 -18 -22 -6

-60 -60 58 -96 58

58 44 -6 -22 -18

-80 -40 -20 80 0

22 -4 26 -58 18

60 60 -58 96 -58

-4 22 18 -58 26

-40 44 -48 44 -40

20 -26 44 -29 20

0 20 -40 20 0

-20 6 -4 9 -20

40 -44 48 -44 40

-20 9 -4 6 -20

0 20 -40 20 0

-40 44 -60 58 -80

-80 58 -60 44 -40

80 -58 60 -44 40

40 -44 60 -58 80

20 -26 58 -18 0

-80 58 -96 58 -80

0 -18 58 -26 20

0 0 -80 20 0

0 18 22 6 -20

80 -58 96 -58 80

-20 6 22 18 0

- 18 - 22  - 6  4 - 9  4 - 6  - 22 0  V z bc 20   135 20 20  - 18 58  - 26 44  - 29  44 - 26  58

58 -96 58 -60 44

-6 -22 -18 -22 -6

44 -60 58 -96 58

-9 4 -6 -22 -18

44 -48 44 -60 58

-6 4 -9 4 -6

58 -60 44 -48 44

0 80 -20 40 -20

-20 80 0 80 -20

-20 40 -20 80 0

-20 40 -20 40 -20

18 -58 26 -44 29

-58 96 -58 60 -44

26 -58 18 -58 26

-44 60 -58 96 -58

29 -44 26 -58 18

-44 48 -44 60 -58

26 -44 29 -44 26

-48 44 -60 -80 -80

4 -9 4 20 0

-60 44 -48 -40 -80

-22 -6 4 20 20

-96 58 -60 -40 -40

-22 -18 -22 20 20

-60 58 -96 -80 -40

40 -20 80 0 0

40 -20 40 0 0

80 -20 40 0 0

80 0 80 0 0

-44 26 -58 0 -20

48 -44 60 80 80

-44 29 -44 -20 0

60 -44 48 40 80

-58 26 -44 -20 -20

96 -58 60 40 40

-58 18 -58 -20 -20

-40 -40 58 -96 58

20 20 -26 58 -18

-80 -40 44 -60 58

0 20 -29 44 -26

-80 -80 44 -48 44

20 0 -26 44 -29

-40 -80 58 -60 44

0 0 0 -80 20

0 0 20 -80 0

0 0 20 -40 20

0 0 20 -40 20

-20 -20 18 22 6

40 40 -58 96 -58

-20 -20 6 22 18

80 40 -44 60 -58

0 -20 9 -4 6

80 80 -44 48 -44

-20 0 6 -4 9

-60 44 -48 44 -60

58 -26 44 -29 44

-96 58 -60 44 -48

58 -18 58 -26 44

-60 58 -96 58 -60

44 -26 58 -18 58

-48 44 -60 58 -96

-40 20 -40 20 -80

-80 20 -40 20 -40

-80 0 -80 20 -40

-40 20 -80 0 -80

-4 9 -4 6 22

60 -44 48 -44 60

22 6 -4 9 -4

96 -58 60 -44 48

22 18 22 6 -4

60 -58 96 -58 60

-4 6 22 18 22

44  -40 44   -48 44   -40 44   -48 -60  -60  -60 -60  58  -80  58  -96  58   -80 58   -96 -20 40  -20  0  20   -40 20   0  80   40   -40 -80  0  80   -20 0   20   -80 0   0  -58 60  -44  48  -44  60  -58  96  80   40   40  80   -58 60   -44 48   -44  60  -58  96 

(9.37)

 Dk s1 A  1c  k sV  3c  P  8  4 m  24  4 m  where A is the area of a face, and V the volume of the element. For the element load vector, the c and m subscripts indicate the values for corner or midside nodes.

(9.38)

Appendix

224

Three Dimensional Navier Stokes Formulation The three dimensional form of the Navier Stokes equations follows the same form as the two dimensional form, and may be expressed in matrix form as

 

K a   K j  0   0  Kg 

 

0 K c   K k  0 K h 



0 0

K e   K l  K i 

K b  U   K d  V 

K  W  0

(9.39)

f

  0   P 

As before, more degrees of freedom need to be associated with the velocities than the pressure, and so the trilinear rectangular interpolation function is used for the pressure and the twenty noded rectangle for the pressure. Accordingly, K a  , K c  and K e  in this context are the matrices for diffusion for a 20 noded element given in equation 9.36,

 

and K j , K k  and K l  are the convection matrices given in equation 9.37. The other matrices are given by 8 - 16  8  - 12 7  - 8 7  - 12 - 12  8bc - 12 Kb    27 - 6 - 6  7 - 8  7 - 6  5  - 4 5  - 6

-8 16 -8 12 -7

-7 8 -7 12 -8

7 -8 7 -12 8

7 -8 7 -6 5

-7 8 -7 6 -5

-5 4 -5 6 -7

8 -7 12 12 12

16 -8 12 6 6

-16 8 -12 -6 -6

-4 5 -6 -12 -12

4 -5 6 12 12

8 -7 6 6 6

6 6 -7 8 -7

12 12 -5 4 -5

-12 -12 5 -4 5

-6 -6 8 -16 8

6 6 -8 16 -8

12 12 -7 8 -7

6 -5 4 -5 6

6 -7 8 -7 6

-6 7 -8 7 -6

-12 7 -8 7 -12

12 -7 8 -7 12

12 -8 16 -8 12

        -8  7   -6  -6   -6   -12 -12  7  -8   7  -12  8   -16 8   -12 5 -4 5 -6 7

(9.40)

Appendix 8 - 12  7  - 8 7  - 12 8  - 16 - 12  8ac - 6 K d    27 - 6 - 12  7 - 6  5 - 4  5  - 6 7  - 8

7 -12 8 -16 8

-7 12 -8 16 -8

-8 12 -7 8 -7

7 -6 5 -4 5

5 -6 7 -8 7

-5 6 -7 8 -7

-12 7 -8 -6 -12

12 -7 8 6 12

12 -8 16 12 6

-6 7 -8 -12 -6

-6 5 -4 -6 -12

6 -5 4 6 12

-12 -6 5 -6 7

12 6 -5 6 -7

6 12 -7 6 -5

-6 -12 8 -12 7

-12 -6 7 -12 8

12 6 -7 12 -8

-8 7 -6 5 -4

8 -7 6 -5 4

4 -5 6 -7 8

-8 7 -12 8 -16

-16 8 -12 7 -8

16 -8 12 -7 8

8 - 12  7  - 6 5  - 6 7  - 12 - 16  8ab - 8   27 - 4 - 8  8 - 12  7 - 6  5  - 6 7  - 12

7 -12 8 -12 7

5 -6 7 -12 8

7 -6 5 -6 7

-8 12 -7 6 -5

-7 12 -8 12 -7

-5 6 -7 12 -8

-6 5 -6 -8 -16

-12 7 -6 -4 -8

-12 8 -12 -8 -4

6 -7 12 16 8

6 -5 6 8 16

12 -7 6 4 8

-8 -4 7 -12 8

-16 -8 5 -6 7

-8 -16 7 -6 5

4 8 -8 12 -7

8 4 -7 12 -8

16 8 -5 6 -7

-12 7 -6 5 -6

-12 8 -12 7 -6

-6 7 -12 8 -12

6 -5 6 -7 12

12 -7 6 -5 6

12 -8 12 -7 6

K  f

K  g

- 8 8  7  8ab - 1  27 - 1  7 5  1

225 -7  6  -5   4  -5   6  -7   8  12  6   6  12  -8  12  -7  8   -7   12 -8   16

(9.41)

-7  6  -5   6  -7   12 -8   12 8   4   8  16  -7  6   -5  6   -7   12 -8   12

(9.42)

16

-8

12

-7

8

-1

-12

-12

12

6

-6

-1

8

-7

6

-5

4

1

-16

8

12

1

-8

7

-12

-12

12

6

-6

7

-8

1

6

-1

-4

5

-8 8

1 -7

12 12

8 -8

-16 16

8 -8

-12 -12

-6 -6

6 6

12 12

-12 -12

5 1

-4 4

-1 -5

6 6

1 -7

-8 8

7 -1

8

-7

6

-5

4

1

-6

-12

12

6

-6

-8

16

-8

12

-7

8

-1

-8

1

6

-1

-4

5

-6

-12

12

6

-6

8

-16

8

12

1

-8

7

-4

-1

6

1

-8

7

-6

-6

6

12

-12

7

-8

1

12

8

-16

8

4

-5

6

-7

8

-1

-6

-6

6

12

-12

-1

8

-7

12

-8

16

-8

-6  -6  -6   -6  -12  -12 -12  -12

(9.43)

Appendix - 8 - 1  7  8ac 8 K h   27 - 1  1 5  7

226

-12

-1

8

-7

12

-8

16

-12

-6

6

12

-1

-6

1

4

-5

6

-7

-12

-8

16

-8

12

-7

8

-6

-12

12

6

1

-6

-1

8

-7

6

-5

-12 -12

8 7

-16 -8

8 1

12 12

1 8

-8 -16

-6 -12

-12 -6

12 6

6 12

5 7

-6 -6

7 5

-8 -4

1 -1

6 6

-1 1

-6

1

4

-5

6

-7

8

-12

-6

6

12

-8

-12

-1

8

-7

12

-8

-6

-1

8

-7

6

-5

4

-6

-12

12

6

-1

-12

-8

16

-8

12

-7

-6

7

-8

1

6

-1

-4

-6

-12

12

6

7

-12

8

-16

8

12

1

-6

5

-4

-1

6

1

-8

-12

-6

6

12

8

-12

7

-8

1

12

8

8  4  -4   -8  16   8  -8   -16

(9.44) - 8 - 1  1  -1 K i   8bc  27 8  7 5  7

-12

-1

-6

1

-6

-1

-12

16

8

4

8

-8

12

-7

6

-5

6

-7

-12

-8

-12

-1

-6

1

-6

8

16

8

4

-7

12

-8

12

-7

6

-5

-6 -6

-1 1

-12 -6

-8 -1

-12 -12

-1 -8

-6 -12

4 8

8 4

16 8

8 16

-5 -7

6 6

-7 -5

12 6

-8 -7

12 12

-7 -8

-12

7

-6

5

-6

7

-12

-16

-8

-4

-8

8

12

1

6

-1

6

1

-12

8

-12

7

-6

5

-6

-8

-16

-8

-4

1

12

8

12

1

6

-1

-6

7

-12

8

-12

7

-6

-4

-8

-16

-8

-1

6

1

12

8

12

1

-6

5

-6

7

-12

8

-12

-8

-4

-8

-16

1

6

-1

6

1

12

8

12 6  6   12 12  6  6   12

(9.45)

Matrix Solution Techniques The Finite Element Method may be viewed as a tool for transforming small but complex partial differential equations into large matrix equations. The accuracy and facility with which results may be obtained is therefore dependant on the efficient solution of the matrices formed. The FE method is reasonably accommodating to the programmer in that the matrices produced may - by a sensible choice of node numbering be strongly banded, and are in general possessed of a dominant leading diagonal, as may be observed from a brief inspection of the element matrices presented in this chapter. These factors aid in producing well-conditioned matrices, and the numerical solution of such matrices has been found to be without problems. The techniques which have been used are discussed in this section. The Choleski technique, although popular in some engineering FE texts, is omitted here due the requirement for a symmetrical matrix it imposes. This is violated by the inclusion of any convective effects, and therefore is of far less use to the electrochemist.

The Gaussian Technique For a set of equations as shown below we can eliminate x1 by first rearranging to find it, giving the equation shown,

Appendix

227

11 x1  12 x 2  13 x3   1n x n  1  21 x1   22 x 2   23 x3    2 n x n   2   n1 x1   n 2 x 2   n 3 x3    nn x n   n x1 

(9.46)

  1 12  x 2  13 x3   1n x n 11 11 11 11

This can be substituted back into the original equations, giving a new set from which x1 has been eliminated.

 22 x 2   23 x3    2 n x n  2 

(9.47)

  n 2 x 2   n 3 x3    nn x n  n

The new constants in these equations are given by

 ij   ij   i11 j / 11 

i   i   i11 j / 11 

(9.48)

This process is called pivoting on the variable which we wish to eliminate from the equation. It is repeated, storing the equations generated at each step, until a final

 x n  n is found, from which x n  n  nn  gives a solution for equation of the form  nn the last unknown. The previous equations generated may then be solved in reverse order to find all the unknowns. This is referred to as back substitution. Problems arise with this technique if a value upon which we would like to pivot is zero, or a has very small value. An attempt to divide by a value of zero will lead to a program error, so a check must be made for this, and if it occurs then the row must be swapped with another which has a real value in the necessary column position. If there is no possible swap which will lead to a real value in this position then the matrix must be singular, and cannot be solved. Pivoting on a very small number will lead to larger rounding errors in the solution which may propagate through the solution process.

Appendix

228

Banded Storage Methods The property of a simple mesh of elements that the maximum disparity in node number between any two nodes belonging to a common element may be predicted, allows efficiencies in the storage of the matrices generated and the execution of operations upon them. A simple example of two grids of four noded rectangular elements is shown in figure

Figure 9.14

A simple rectangular mesh.

It is clear to see that numbering across the shortest dimensions of the grid lessens the variation of global node number across an element. The effect of this on the population of the fully assembled global matrix may be illustrated.

Figure 9.15

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

a

b

b

b

b

c

c

d

d

e

e

e

e

a

b

b

b

b

c

c

d

e

e

e

e

e

a

b

b

b

c

c

c

e

e

e

e

e

e

a

b

b

b

b

c

c

d

d

e

e

e

a

b

b

b

b

c

d

d

e

e

e

a

b

b

b

c

c

c

e

e

e

a

b

b

b

b

c

c

d

d

a

b

b

b

b

c

c

d

A possible global matrix outline.

The a region is the leading diagonal of the matrix. The outline, if not the contents, of a matrix generated by any of the methods presented here will be symmetrical about this leading diagonal, and so only half the matrix has been detailed. The b region indicates the region populated by the assembly of the right hand matrix, with the c region

Appendix

229

indicating the area which might be populated by the assembly of the left hand mesh. The d and e regions indicate the possible area occupied by the right and left hand grids if the program is required to swap rows because of a zero in the pivotal position. This matrix outline may be characterised by the concept of the bandwidth, nb. This is the largest span in node number between nodes in the same element. For the left hand matrix, nb is 7, and for the right hand mesh nb is 5. The maximum bandwidth of a matrix at assembly will be nb, but if the solution procedure swaps one row with another because of a zero or near-zero value in the pivotal position, the bandwidth will rise to 2nb1. As the matrix may be non-symmetrical nb-1 spaces to the left of the leading diagonal must also be stored. The total size of the part of the matrix which may become occupied is therefore 3nb-2 wide. The matrix coordinates may be transformed using a knowledge of the bandwidth of a problem, so that rather than declaring an array nn by nn as the global characteristic matrix, where nn is the number of nodes in the whole mesh, a mesh nn by 3nb-2 may be declared. For the example given the below, the original coordinates of each point are shown before and after the transformation.

Ai, j   Ai, j  nb  i  1

2

3

4

(1,1)

(1,2)

(2,1)

(2,2)

(2,3)

(3,2)

(3,3)

(3,4)

(4,3)

(4,4) (5,4)

5

1

(9.49) 2

3

(1,1)

(1,2)

(2,1)

(2,2)

(2,3)

(3,2)

(3,3)

(3,4)

(4,5)

(4,3)

(4,4)

(4,5)

(5,5)

(5,4)

(5,5)



It has been found that transformations of the form, Ai, j   A j  nb  i, i  produce arrays which lead to faster program execution, as the FORTRAN standard places array positions in adjacent positions in memory in the order Ai, j , Ai  1, j , Ai  2, j  . As

Appendix

230

the Gaussian solution routine used involves loops over each row, nested within loops down over the rows, this form of storage allows far more efficient caching and code optimisation. This produces an array of the form. 1

(2,1)

(3,2)

(4,3)

(5,4) (5,5)

2

(1,1)

(2,2)

(3,3)

(4,4)

3

(1,2)

(2,3)

(3,4)

(4,5)

Sparse Solution Methods For larger problems, the banded structure of the matrices created may not afford sufficient efficiencies in execution time and memory requirements. It has been found that routines to store a matrix in a form where only non-zero positions are stored may be written with relatively little deviations in structure from a standard Gaussian solver. The routine developed stores the value of the each position, and the coordinate of the value. This routine has been found to run at roughly one third of the speed of the banded Gaussian scheme, but allows problems of over triple the size to be attempted.

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