Bouzek - Modelling and Design of the Electrochemical Processes and Reactors

8/3/2019 Bouzek - Modelling and Design of the Electrochemical Processes and Reactors

1/69

Modelling and Design of the Electrochemical

Processes and Reactors

Karel Bouzek, Roman Kodm

Department of Inorganic Technology,Institute of Chemical Technology Prague


2/69

Mathematical modelling and process design

Role of the mathematical modelling in the electrochemical engineering

pedagogical

deep understanding of the process

scientific

economical

identification of the rate determining steps

learning the way of sequential thinking

analysis of the problem on the local scale based on the global experimental data

evaluation of not directly accessible parameters

verification of the theories developed

process scale-up

process optimisation

identification of the possible bottlenecks

costs reduction


3/69


4/69


Material balance

modes of operation

batch (discontinuous) processes

basic characteristics

continuous processes

processes in a closed systems (no mass exchange with the surroundings)

no steady state is reached before the reaction equilibrium

suitable mainly for the small scale processestypical design - stirred tank reactor

continuously stirred tank reactor (CSTR)

plug-flow reactor (PFR)

Batch processes

description by the dynamic model only

Fn

IVRJ

d

dcV

i

eRii

iR

*s(!X


5/69


Batch processes

material balance

high mixing rate spatially uniform concentration

BneA p Fn

I

d

dcV

A

eAR

*!

X

Fn

aj

d

dc

A

eeA *!X R

ee

V

Aa !

mass transfer limited reaction Amlim ckj !

galvanostatic process at j < jlim X!Fn

jacc

A

e0,AA

concentration decay of component A in time

galvanostatic process at j ujlim

X! em0,AA akexpcc

X!!!*em

A0,AmAAmlim

e akexpj

Fnck

j

Fnck

j

j


6/69


Continuous process - CSTR

material balance

high mixing rate spatially uniform concentration

equal inlet and outlet flow rates independent on time constant reactor volume and mean residence time

BneA p

ARI,AO,A RVccV !

Fn

IccV

A

eI,AO,A

*!

Fnjac

FVnIcc

A

eI,A

RA

I,AO,A !!galvanostatic process at j < jlim



j

Fnck

j

j AO,Amlime !!*

Fn

jac

FVn

Icc

A

limeI,A

RA

limI,AO,A !!


7/69


Continuous process - PFR

material balance

negligible axial electrolyte mixing

negligible concentration gradient perpendicular to the flow direction

BneA p A

AR

dx

dcv !

Fn

jb

dx

dcV

A

eA *!

V

bx

Fn

jcc

A

I,AA !galvanostatic process at j < jlim



!!* V

bxk

expj

Fnck

j

j mAI,Amlime

!V

bxkexpcc mI,AA

b electrode width


8/69

Steady state models

Background of the electrochemical reactors modelling

basic parameters calculated

local values of the Galvani potential

local current density values

division of the mathematical models

according to the level of simplification primary current density distributionsecondary current density distribution

tertiary current density distribution

according to the number of dimensions considered one dimensionaltwo dimensional

three dimensional

according to the mathematical methods used analyticalnumerical

way of the system description

differential or partial differential equations


9/69

Steady state models

Level of simplification

primary current density distribution

infinitely fast reaction kinetics

infinitely fast mass transfer kinetics

tertiary current density distribution

mass transfer kinetics and electrolyte hydrodynamics considered

additional reduction of the local extremes

secondary current density distribution

reaction kinetics considered

only influencing factors geometry of the system

electrolyte conductivity

more regular current density distribution

sufficient approximation for the majority of the industrially relevant systems

extremely complicated used only in a strictly limited number of cases


10/69

Steady state models

Number of dimensions considered

selection criteria

homogeneity of the system

significance of the local irregularities

consequences of the increase in the number of dimensions

one-dimensional model described by the differential equations

symmetry of the system

more dimensions requires partial differential equations

significantly more complicated mathematics

geometrically increasing hardware demands

three dimensional models

simplified models

mainly focus on the critical element of the complex system

important mainly in the tertiary current distribution models


11/69

Steady state models

Mathematical methods used

analytical solution of the model equations

most accurate way

general validity of the equations derived (for the given system)

numerical mathematics

able to describe complicated geometries and complex systems

available for the extremely simple configurations only

less significant simplification assumptions

results valid only for the particular system solved

question of results accuracy

methods of numerical mathematics used

strongly dependent on the dimensions number

one dimensional case - classical integration methods (Runge-Kutta, collocation, shooting, )

more dimensional tasks rapid development with improving hardware

strongly limited applicability in the industrially relevant systems

FDM

FEM

BEM


12/69

Steady state models

Basic model equations for electrochemical systems

equation of the mass and charge transfer in the electrolyte solution

current density value

electrolyte solution with no concentration gradients

iiiiii cvcucDJTT

N!Nernst-Planck equation

iii JFzjTT

! !i

iiJzFjTT

application of the Faraday law

N! i iii iiii iii czvFcuzFcDzFjTT

electroneutrality condition0cz

i

ii !

N!i

iii cuzFjT

NO!jT

!Oi

iii cuzF

electrolyte conductivity definition


13/69

Steady state models


electrolyte solution with concentration gradients

current less system

OO!N i iiiczD

FjT

O!N i iiiczD

F

liquid junction potential

!Ni

i

i

0

i clndz

t

F

RTd

one dimensional case


14/69

Steady state models


mass balance in the electrolyte volume

introducing Nernts-Planck equation after multiplication byziF

iii Jc

G!Xx

x

0zFJzFi

ii

i

ii !G

0jJ

zF i ii !! charge conservation

0zi

ii !Gelectroneutrality condition

introducing expression forj

0czDczuzuci

i

2

ii

i

iii

i

2

iii !NN modified Laplace equation

02 !NLaplace equation

no concentration gradients


15/69

Steady state models

Boundary conditions

significant variability according to the particular conditions

arbitrary definition

in agreement with general types of boundary conditions

constant value, i.e. Galvani potential

constant flux, i.e. current density

Boundary condition constant potential value

requirements: no influence of the current loadconstant composition, i.e. constant properties

typical choices: electrode current leads potentialselectrode body potential

arbitrary values typically used: cathode potential equal to zeroanode potential equal to the cell voltage

special case electrode / electrolyte interface


16/69

Steady state models

Boundary condition constant flux

typically the cell walls and electrolyte surface (no flux)

simplification considered: primary vs. secondary (tertiary) current density distribution

in special cases flux continuity used

Electrode / electrolyte interface

linear

Tafel

Butler-Volmer

potential rate determining steps in the electrode reaction kineticskinetics of the mass transfer to the electrode

adsorption

charge transfer kinetics

desorption of the products

kinetics of the product transfer from the electrode

possible homogeneous reactions

mass transfer kinetics subject of the individual lecture

types of the charge transfer kinetics used:


17/69

Steady state models

Linear kinetics

historical question

nowadays overcome

Tafel kinetics

considers just one part of the polarisation curvejlnba !L

!L

0j

jlnb ejj 0u

computational demands

physical models

suitable for the systems far from equilibrium

simple kinetics evaluation from the model results

linearisation of the low current densities part -

- minimisation of the divergence danger

0j

j

e

b!L ejj 0e


18/69

Steady state models

Butler - Volmer kinetics

general description of the charge transfer kinetics

more complicated kinetic evaluation (requires additional numerical procedure)

-

LE

-

LE!R

T

zFexp

c

c

R

T

zF1exp

c

cjj C0

ox

s

oxC0

red

s

red0

mass transfer limited kinetics - concentration polarisation

-

LE

-

LE

!

RT

zFexp

j

11

RT

zF1exp

j

11jj C

Clim,

C

Alim,

0

charge transfer limited kinetics

-

LE

-

LE!RT

zFexp

RT

zF1expjj CC0


19/69

Steady state models

Classical approach finite differences

first detail models of the electrochemical reactors

transformation of the partial differential equation to the set of linear equations

(((!!3

xy!2

xy!1xyyy

3

'''

i

2

''

i

'

ii1i

symmetrical formulas for the first and second derivative

(

(!

!3

xy2

!1

xy2yy

3

'''

i

'

i1i1i

Taylor's expansion used for linearisation

(

(

(!

!3

xy

!2

xy

!1

xyyy

3

'''

i

2

''

i

'

ii1i

(

(!

!4

x2y

!2

x2yyy2y

4

'v

i

2

''

i1ii1i

21i1i'i xx2

yyy (H

(

!

22

1ii1i''

i x

x

yy2yy (H

(

!


20/69

Steady state models


asymmetrical formulas boundary conditions

((! !3

x2y

!1

xy2y3yy4

3

'''

i

'

ii2i1i ((!

!3

xy4

!1

xy2y3yy4

3

'''

i

'

ii2i1i

22i1ii'i xx2

yy4y3y (H

(

! 22i1ii'i xx2

yy4y3y (H

(

!

(

(

(!

!3

x2y

!2

x2y

!1

x2yyy

3

'''

i

2

''

i

'

ii2i


21/69

Steady state models


method of replacement of the partial derivatives in the Laplace equation

1i,

j,1i

j,2

1

i

j,1i1i,xx

j

(

NN

O!

j,iN j,1iN j,1iN

1j,i N

1j,i N

principle of the Laplace equationdivergence of the flux equal to zero

i,

j,2

1i

j,i

j,ii,xx

j

(

NN

O!

1i,xi,x jj !

1i,j,ii,j,1i

j,i1i,j,ij,1ii,j,1i

j,2

1i xx

xx

(O(O

N(ON(O!N

? Aj,1ij,ij,1i

1i,

j,i

i,1i,x xx

1j

NN

O

(

O

(!

etc.


22/69

Steady state models


flux divergence in the final differences

? A ? A 0xjjyjj ij,yj,yji,xi,x !((

0CCCCC j,i5,j,i1j,i4,j,i1j,i3,j,ij,1i2,j,ij,1i1,j,i !NNNNN

j,1i

1i

j,i

i

j

1,j,i xx

yC

O

(

O

((

!

j,1i

1i

j,i

i

j

2,j,i xx

yC

O

(

O

((

!

1j,i

1j

j,i

j

i

3,j,i yy

xC

O

(

O

((

!

1j,i

1j

j,i

j

i

4,j,i yy

xC

O

(

O

((

!

4,j,i3,j,i2,j,i1,j,i5,j,i CCCCC !


23/69

potential/V

X

Y

Finite differences example

Current density simulation in the parallel plate cell

current density distribution in the zinc electrowinning cell

K. Bouzek, K. Borve, O.A. Lorentsen, K. Osmudsen, I. Rouar, J. Thonstad, J. Electrochem. Soc. 142 (1995) 64.


24/69

X

Y

1.01.0

1.0

1.0

0.9

0.90.9

0.9 0.9

0.90.9

1.01.0

0.90.9

1.0

1.0

1.0

1.0

1.11.1

1.11.1

1.11.1

1.11.1

1.2 1.2

1.21.2

1.2 1.2

1.21.2

1.3

1.3

1.3

1.3

1.3

1.3

K. Bouzek, K. Borve, O.A. Lorentsen, K. Osmudsen, I. Rouar, J. Thonstad, J. Electrochem. Soc. 142 (1995) 64.

Finite differences example

Current density simulation in the parallel plate cell

current density distribution in the zinc electrowinning cell


25/69

Steady state models

FDM applicable for solving of any type of partial differential equation

complications may be expected in the case of complex boundary conditions form

slow convergence


problems often related to the anistropic media

difficulties by solving systems with the irregular geometries

alternative approaches searched


26/69

Steady state models

original approach based on the calculus of variations

finite elements method

Recent approach finite element method (FE

M)

doesnt solve the equation directly (approximation method)

searching for the function giving extreme by replacing the differential equation

such function approximated by a sum of the basis functions (unknown coefficients)

coefficients determined by solving a system of linear algebraic equationsdrawbacks largely depends on the choice of the basis functions

cannot be satisfied for too complicated geometries

the function is not searched for the whole domain integrated

domain is divided into the a number of subdomains

in each subdomain solution approximated by a simple functionGalerkins method of weighted residuals, i.e.parameters of the basis function

modifications maybe derived by the choice of the weighting functions

necessary condition is that the combination of basis functions fulfil boundary conditions


27/69

Steady state models

application onto the solution of the Laplace equation

simplification for the one-dimensional case

Recent approach finite element method (FE

M)

approximate expression takes following form

approximate expression by the means of FEM

i

N

0i

iaU !

N!

i

N

0i

i u)x()x(u !

N!

FEM is optimising value ofui in such a way, that I is close to 0

I! )x(u2


28/69

Steady state models

simplest linear basis function

Recent approach finite element method (FEM)

!0

)( 01

1

0xx

xx

xN)( 10 xxx ee

!

0

0

)(

1

1

1

1

ii

i

ii

i

i

xx

xx

xx

xx

xN

)( 10 ee ixxx)(1 N

xxx ee

)( 1 ii xxx ee

)( 1ee ii xxx

)( 1 Ni xxx ee

!

1

1

0

)(

NN

NN

xx

xxxN)( 10 ee Nxxx

)( 1 NN xxx ee

x0

x1 xixi-1 xi+1 xNxN-1

Ni(x)

x

1

0


29/69

Examples of the FEM applications

current density distribution in the channel with parallel plate electrodes

Parametric study of the narrow gap cell


30/69

Curved boundary


presence of the gas bubble in the interelectrode space


31/69

evaluation of the process efficiency

Optimisation of the direct electrochemical water disinfection cell


10 cm

5 cm

5 cm0.5 cm 0.3 cm

2.4 cm

AA C C

x

y


32/69



U = 6.04 V

Javer. = 50 A m-2

J = 4.92 %

anode

cathode

anode

W = 667 QS cm-1

evaluation of the process efficiency plate electrodes


33/69



U = 6.04 V

Javer. = 50 A m-2

J = 4.92 %

W = 667 QS cm-1

evaluation of the process efficiency plate electrodes


34/69



evaluation of the process efficiency expanded mesh electrodes

3 mm

5 mm

1.4 mm

1.5 mm

cathode

anode

x

z

y


35/69



evaluation of the process efficiency expanded mesh electrodes

U = 4.76 V

Javer. = 42.4 A m-2

J = 4.12 %W = 667 QS cm-1


36/69

Alternative to FEM

handling flux densities

Finite volumes method

principle of the method

( i , j )( i-1, j ) ( i+1, j )

( i , j-1 )

( i , j+1 )( i-1, j+1 ) ( i+1, j+1 )

( i+1, j-1 )( i-1, j-1 )

solution of partial differential eqs.

based on the PDE integration

over the volume surrounding

controlled grid point

controlled domain covered

by the controlled volumes

integration leads formally to

equation identical with FDE

NN

WNN

W

hh

hh

j,1ij,ij,ij,1i

0hh

hh

1j,ij,ij,i1j,i !NN

WNN

W


37/69

Application to the bipolar electrode function simulation


model system under study

Bipolar Pt electrode

Electrolyte

Terminal Cathode Terminal Anode

780

14

55

x

r

Cylindrical coordinate system

r - radiusx position

1.5

[mm]


38/69




I= 40 mA; U=4 V


39/69




I= 40 mA; U=4 V


40/69



comparison of the model and experimental results


41/69


42/69

Potential and current density distribution in three dimensional electrode

simplified sketch of the cell construction

1

3

4

5

56

8

7

2

1 electrolyte inlet

2 particle electrode3 channels connecting

4 cathode feeder

5 anode feeder

individual drums

6 anode7 separator8 electrolyte outlet

Tertiary current density distribution


43/69


simplified flow patterns inside the cell - electrolyte



44/69


simplified flow patterns inside the cell electric current



45/69


46/69


basic equations describing the system definition of j el


electrode reactions considered:

cathode

overall electrode reaction current density:

anode

Cu2+ + 2e- Cu

2 H+ + 2e- H2

2 H2O 4 H+ + O2

22 OHCueljjjj !!

resistivity to the charge transfer: electrode potential


47/69


basic equations describing the system electrode reaction kinetics


L

E

-

L

E

L

E

!

x,Cu

Cuc,Cu

x,Culim,

x,Cu,0

x,Cu

Cuc,Cu

x,Cu

Cuc,Cu

x,Cu,0

x,Cu

RT

Fzexp

j

j1

RT

Fz1exp

RT

Fzexpj

j

L

E!

x,H

Hc,H

H,0x,H RT

Fzexpjj

aa jln..E ! 10550411


48/69




polarisation curves

zFkAcj x,Cux,Culim, ! Fvz

Aj

dx

dc

Cu

x,CuCu !

smE NN!

reversible potentials: Nernst equation

!i

x,i

0

rx,r clnzF

RTEE


49/69




mass transfer coefficient evaluation

? A_ apr6r

r3

1

3

1

p ReRe1004.1125exp1Re2498

Re8.52ScRe

09.1Sh

I

!

significant complication evaluation of the linear electrolyte flow rate


50/69

-15

-10

-5

0

0.000.05

0.100.15

0.200.25

10

15

20

j/Am

-2

position/m

currentload/A

electrolyteflow

1st

drum

6th

drum


selected results influence of the current load


V = 4.510-5 m3 s-1cCu0 = 7.87 mol m-3

cH0 = 100 mol m-3

k = 4.1910-6 m s-1

particle diameter 0.002 m[ = 0.047 Hz


51/69


selected results influence of the current load


5.5

6.0

6.5

7.0

7.5

8.0

0.000.05

0.100.15

0.200.25

10

15

20

cCu2+

/molm-3

position/m

currentload/A

electrolyteflow

1

st

drum6th drumdrum number

0 1 2 3 4 5 6 7

efficiency/%

0.4

0.6

0.8

1.0

I = 6 A I = 20 A


52/69


selected results cell with different particle sizes


I = 15 A

cCu0 = 7.87 mol m-3 cH

0 = 100 mol m-3

k = 1.6410-6 m s-1

V = 4.2510-5 m3 s-1

cCu0 = 4.00 mol m-3

position / m

0.00 0.05 0.10 0.15 0.20 0.25

j/Am-2

-70

-60

-50

-20

-10

0

1st

drum 6th

drum

electrolyte flow

1.0 mm1.5 mm2.0 mm2.5 mm3.0 mm3.5 mm

dp


53/69


selected results cell with different particle sizes


drum number

0 1 2 3 4 5 6 7

cu

rrentload/A

0

1

2

3

4

drum number

0 1 2 3 4 5 6 7

efficiency/%

0.4

0.6

0.8

1.0


54/69

Charge flux across the ion selective membrane

Principle of the ion selective membrane function

role of the membrane structure

simplified model: parallel cylindrical pores with an electrical

used typically for the purposes

theory of the membrane selectivity:Donnan potential and exclusion

N N N N

121,2Don NN!N( 343,4Don NN!N(

21~~

Q!Q 21~~

Q!Q

22,

0

11,

0

FzalnRTFzalnRT NQ!NQ

NQ!NQ FzalnRTFzalnRT 2,0

11,

0

!

!NN!N(

z

1

2,

1,

121,2Dona

aln

F

RT

charge located on the walls

of mathematical description

!

z

1

2,

1,

a

aln

F

RT

iz

1

2,i

1,iz

1

2,

1,z

1

2,

1,

a

a

a

a

a

a

!

!

0czczczMM2,2, !

Donnan distribution coefficient electroneutrality condition


55/69


Principle of the ion selective membrane function

role of the membrane structure

advanced structure models perfluorinated sufonated materials

dry membrane, 0 vol.% swollen membrane, 0-20 vol.% percolation, 20-40 vol.%

structure inversion, 40-60 vol.% connected network, 60-80 vol.% colloidal dispersion, 80-100 vol.%

K.A. Mauritz, R.B. Moore, Chemical Reviews 104 (2004) 4535


56/69


Basic equations for ideal behavior

Nernst-Planck equation is used to describe ion transport

material balance:

vNii

i

iiiicFz

RT

DccD

! N

ii

i divc

N!x

xJ

X

stationary state: 0!x

x

Xi

c

no chemical reaction: 0!J

electroosmotic flux: pFczk,Mk,M

! NL

Rv


57/69


Treatment of non-idealities

flux of ion i inside the membrane

solvent flux considered as well (including solution density variation)

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

, ,( )

ln1

ln

j j

j j j j j j j

j jj

s s s s s s s s si i

i i i i i M s M ss

i

D D c c z F c z c F p

RTc

K RN N

O

x! x

iN

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( )

ln1

ln

j j

j j j j j j

j

s s

s s s s s si i

i i i i is

i i i ii

D D c c z F c

RTc

KN

x! x

iN v

after rearrangement and derivation

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )20 j j j j j j j j j j

s s s s s s s s s s

i i i i i i tot tot

i i

F F z D c z D c c c

RT RT N N! v v

membrane


58/69



expression for electroosmotic flux inside the membrane remains unchanged

equation for the density dependence on the composition of the solution

current density

( ) ( ) , ,j js sf T pV ! c

electroneutrality equation

(Schlgels equation)

solution flux divergence derivative of Schlgels equation (pressure profile)

0czczjj

ions

j

s,Ms,M

N

i

)s(ii !

!ionsN

i

iizF Nj

membrane


59/69



membrane electrolyte solution interface (molar flux continuity):

membrane electrolyte solution interface (Donnan equilibrium):

iz

1

k

i

1k

ik1k)k,1k(

Dona

aln

F

RT

-

!NN!N(

0!!!

jsklklx

ix

i NN H

interface


60/69


Activity coefficients evaluation

cation

K.S. Pitzer: Activity Coefficients

where

2M M a Ma Ma c Mc a Mca a M c a caa Maaa c a a c aa

z F m 2B ZC m 2 m m m z m m C, ,,

ln( )

K ! * ] ]

2X X c cX cX a Xa c cXa a X c a caa cc Xc a a c c aa

z F m 2B ZC m 2 m m m z m m C, ,,

ln( )

K ! * ] ]

c a ca c ac cc a aa

c a c ac a

F f m m B m m m m , , , ,, ,

' ' 'K

! * *

1 21 2

1 2

I 2f A 1 bI

1 bI bln( )

K

*

! -

0 1 21 2 1 2MX MX MX 1 MX 2B g I g I! F F E F E in Electrolyte Solutions.CRC Press, London 2000.

anion


61/69


Methods used to solve the equations

boundary problem Algebraic-Ordinary Differential Equations (A-ODEs)

solution - shooting method

Boundary conditions - system non-linear equationsp modified Newton-Raphson method

1] P. N. Brown, A. C. Hindmarsh, and L. R. Petzold, Consistent Initial Condition Calculation

for Differential-Algebraic Systems, LLNL Report UCRL-JC-122175, August 1995

2] IMSL Numerical Library, 1994

Donnan potential at boundary interfaces sj-li sj-sj+1- non-linear equation

p(DZREAL2)

calculation of consistent initial conditions-system of linear equationsp(DLSLRG2)

integration of system A-ODEs for individual parts - initial problempimplicit

method based on the BDF Gears formulas (DDASPK1

)


62/69


input parameters

anolyte: 5 kmol NaCl m-3, catholyte:

0

10000

20000

0

500

1000

1500

20002500

0.00.2

0.40.6

cNa+

/m

ol

m-3

j/A

m-2

coordinate/mm

currentflowdirection

pH = 2 13 kmol NaOH m-3

simplified model

Results influence of the current load


63/69



input parameters

anolyte: 5 kmol NaCl m-3, catholyte:

pH = 2 13 kmol NaOH m-3

simplified model


64/69



input parameters

anolyte: 5 kmol NaCl m-3,

catholyte:

pH = 2

13 kmol NaOH m-3

j / A m-2

0 500 1000 1500 2000 2500

J/m

molm-2s-

1

-20

-10

0

10

20

OH-

Na+

Cl-

Na+ selectivity 52 % at 1500 A m-2

Na+ selectivity 75 % at 1500 A m-2


65/69

Dynamic models

Model of the cathodically protected pipelines in a soil

aim of the study

theory of the cathodic protection in a soil

proposed alternative theory

difficult experimental evaluation

mathematical model offers simple qualitative alternative to the experiment

0.01 m

insulation insulationdamage

boundary of the domain

x

y


66/69

Dynamic models

cathode reaction considered


p OH4e4OH2O 22

simplifying assumptions

homogeneous environmentno reaction with CO2constant oxygen flux to the cathode surface

homogeneous potential distribution on the cathode (damage) surface

water electrolysis consumes negligible portion of the current

22 HOH2e2OH2 p

model equations

0)( !NW

)cBzc(ADJ iiiii N!

2K16

15K21

K5.01A

!

II

!1

KRT

FB !

i

m

i

22

cDzART

Fi!W


67/69

Dynamic models


c0Na+ = 10 mol m

-3

c0

Cl- = 10 mol m-3

c0OH- = 0 mol m-3

selected initial conditions


68/69

Dynamic models


c0Na+ = 10 mol m

-3

c0

Cl- = 10 mol m-3

c0OH- = 0 mol m-3

selected initial conditions


69/69

Conclusion

mathematical modelling provides powerful tool in understanding and optimising

rapid development of commercial software allows faster and more efficient work

understanding of the mathematical methods still essential

two main limits exists

electrochemical as well as chemical processes

hardware limitations

reliable input data

Documents

Bouzek - Modelling and Design of the Electrochemical Processes and Reactors