Modelling and Design of the Electrochemical Processes and Reactors Karel Bouzek, Roman Kodým Department of Inorganic Technology, Institute of Chemical

Modelling and Design of the ElectrochemicalProcesses and Reactors

Karel Bouzek, Roman Kodým

Department of Inorganic Technology, Institute of Chemical Technology Prague

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 stepslearning the way of sequential thinking

analysis of the problem on the local scale based on the global experimental dataevaluation of not directly accessible parametersverification of the theories developed

process scale-upprocess optimisationidentification of the possible bottleneckscosts reduction


Levels of the mathematical modelling

material balance

based on the macroscopic kinetics data

steady state models

dynamic models

based on the two elementary modes of operationassessment of the system limits under given assumptions

detail description of the space distribution of the process parameters in steady stateallows evaluation of the construction and operational parameters suitabilitysufficient for the majority of cases

description of the non-stationary processesprocess start-up and terminationdiscontinuous processesmost sophisticated problems


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 equilibriumsuitable 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

FnI

VRJddc

Vi

eRii

iR


Batch processes

material balance

high mixing rate – spatially uniform concentration

BneA Fn

I

d

dcV

A

eAR

Fn

a j

d

dc

A

eeA

R

ee V

Aa

mass transfer limited reaction Amlim ckj

galvanostatic process at j < jlim Fn

jacc

A

e0,AA

concentration decay of component A in time

galvanostatic process at j jlim

em0,AA akexpcc

emA0,AmAAmlim

e akexpj

Fnck

j

Fnck

j

j


Continuous process - CSTR

material balance

high mixing rate – spatially uniform concentrationequal inlet and outlet flow rates independent on time –

– constant reactor volume and mean residence time

BneA ARI,AO,A RVccV

Fn

IccV

A

eI,AO,A

Fn

jac

FVn

Icc

A

eI,A

RAI,AO,A galvanostatic process at j < jlim



j

Fnck

j

j AO,Amlime

Fn

jac

FVn

Icc

A

limeI,A

RA

limI,AO,A


Continuous process - PFR

material balance

negligible axial electrolyte mixingnegligible concentration gradient perpendicular to the flow direction

BneA

AA R

dx

dcv

Fn

jb

dx

dcV

A

eA

V

bx

Fn

jcc

AI,AA galvanostatic process at j < jlim



V

bxkexp

j

Fnck

j

j mAI,Amlime

V

bxkexpcc m

I,AA

b – electrode width

Steady state models

Background of the electrochemical reactors modelling

basic parameters calculated

local values of the Galvani potentiallocal current density values

division of the mathematical models

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

according to the number of dimensions considered one dimensionaltwo dimensionalthree dimensional

according to the mathematical methods used analyticalnumerical

way of the system description

differential or partial differential equations

Steady state models

Level of simplification

primary current density distribution

infinitely fast reaction kineticsinfinitely fast mass transfer kinetics

tertiary current density distribution

mass transfer kinetics and electrolyte hydrodynamics consideredadditional reduction of the local extremes

secondary current density distribution

reaction kinetics considered

only influencing factors geometry of the systemelectrolyte conductivity

more regular current density distributionsufficient approximation for the majority of the industrially relevant systems

extremely complicated – used only in a strictly limited number of cases

Steady state models

Number of dimensions considered

selection criteria

homogeneity of the systemsignificance 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 equationssignificantly more complicated mathematicsgeometrically increasing hardware demands

three dimensional models

simplified modelsmainly focus on the critical element of the complex systemimportant mainly in the tertiary current distribution models

Steady state models

Mathematical methods used

analytical solution of the model equations

most accurate waygeneral 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 assumptionsresults valid only for the particular system solvedquestion of results accuracy

methods of numerical mathematics used

strongly dependent on the dimensions numberone 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

FDMFEM

BEM

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 cvcucDJ

Nernst-Planck equation

iii JFzj

i

iiJzFj

application of the Faraday law

i

iii

iiii

iii czvFcuzFcDzFj

electroneutrality condition

0czi

ii

i

iii cuzFj

j

i

iii cuzF

electrolyte conductivity definition

Steady state models


electrolyte solution with concentration gradients

current less system

i

iii czDFj

i

iii czDF

liquid junction potential

i

ii

0i clnd

z

t

F

RTd

one dimensional case

Steady state models


mass balance in the electrolyte volume

introducing Nernts-Planck equation after multiplication by ziF

iii J

c

0zFJzFi

iii

ii

0jJzFi

ii charge conservation

0zi

ii electroneutrality condition

introducing expression for j

0czDczuzuci

i2

iii

iiii

2iii

modified Laplace equation

02 Laplace equation

no concentration gradients

Steady state models

Boundary conditions

significant variability according to the particular conditions

arbitrary definition

in agreement with general types of boundary conditionsconstant value, i.e. Galvani potentialconstant 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

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

linearTafelButler-Volmer

potential rate determining steps in the electrode reaction kineticskinetics of the mass transfer to the electrodeadsorptioncharge transfer kineticsdesorption of the productskinetics of the product transfer from the electrodepossible homogeneous reactions

mass transfer kinetics subject of the individual lecture

types of the charge transfer kinetics used:

Steady state models

Linear kinetics

historical question

nowadays overcome

Tafel kinetics

considers just one part of the polarisation curvejlnba

0j

jlnb ejj 0

computational demandsphysical models

suitable for the systems far from equilibriumsimple kinetics evaluation from the model resultslinearisation of the low current densities part -

- minimisation of the divergence danger

0j

j

e

b ejj 0

Steady state models

Butler - Volmer kinetics

general description of the charge transfer kinetics

more complicated kinetic evaluation (requires additional numerical procedure)

RT

zFexp

c

c

RT

zF1exp

c

cjj C0

ox

sox

C0red

sred

0

mass transfer limited kinetics - concentration polarisation

RT

zFexp

j

11

RT

zF1exp

j

11jj C

Clim,C

Alim,0

charge transfer limited kinetics

RT

zFexp

RT

zF1expjj CC0

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

!1

xyyy

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 x

x2

yyy

2

21ii1i''

i xx

yy2yy

Steady state models


asymmetrical formulas – boundary conditions

!3

x2y

!1

xy2y3yy4

3'''

i'ii2i1i

!3

xy4

!1

xy2y3yy4

3'''

i'ii2i1i

22i1ii'i x

x2

yy4y3y

22i1ii'i x

x2

yy4y3y

!3

x2y

!2

x2y

!1

x2yyy

3'''

i

2''i

'ii2i

Steady state models


method of replacement of the partial derivatives in the Laplace equation

1i,

j,1ij,

2

1i

j,1i1i,x xj

j,i j,1i j,1i

1j,i

1j,i

principle of the Laplace equationdivergence of the flux equal to zero

i,

j,2

1i

j,i

j,ii,x xj

1i,xi,x jj

1i,j,ii,j,1i

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

j,2

1i xx

xx

j,1ij,i

j,1i

1i,

j,i

i,1i,x xx

1j

etc.

Steady state models


flux divergence in the final differences

0xjjyjj ij,yj,yji,xi,x

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

j,1i

1i

j,i

i

j1,j,i xx

yC

j,1i

1i

j,i

i

j2,j,i xx

yC

1j,i

1j

j,i

j

i3,j,i yy

xC

1j,i

1j

j,i

j

i4,j,i yy

xC

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

pote

ntia

l / 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. Roušar, J. Thonstad, J. Electrochem. Soc. 142 (1995) 64.

X

Y

1.0 1.0

1.0

1.0

0.90.90.9

0.9 0.9

0.90.9

1.01.0

0.90.9

1.0

1.0

1.01.0

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. Roušar, 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

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

Steady state models

original approach based on the calculus of variations

finite elements method

Recent approach – finite element method (FEM)

doesn’t solve the equation directly (approximation method)searching for the function giving extreme by replacing the differential equationsuch function approximated by a sum of the basis functions (unknown coefficients)

coefficients determined by solving a system of linear algebraic equations

drawbacks largely depends on the choice of the basis functionscannot be satisfied for too complicated geometries

the function is not searched for the whole domain integrateddomain is divided into the a number of subdomainsin each subdomain solution approximated by a simple functionGalerkin’s method of weighted residuals, i.e. parameters of the basis functionmodifications maybe derived by the choice of the weighting functionsnecessary condition is that the combination of basis functions fulfil boundary conditions

Steady state models

application onto the solution of the Laplace equation

simplification for the one-dimensional case


approximate expression takes following form

approximate expression by the means of FEM

i

N

0iiaU

i

N

0ii u)x()x(u

FEM is optimising value of ui in such a way, that is close to 0

)x(u2

Steady state models

simplest linear basis function


0

)( 01

1

0xx

xx

x)( 10 xxx

0

0

)(

1

1

1

1

ii

i

ii

i

i

xx

xxxx

xx

x

)( 10 ixxx)( 1 Nxxx

)( 1 ii xxx

)( 1 ii xxx

)( 1 Ni xxx

1

1

0)(

NN

NN

xx

xxx)( 10 Nxxx

)( 1 NN xxx

x0 x1 xixi-1 xi+1 xNxN-1

i(x)

x

1

0

Examples of the FEM applications

current density distribution in the channel with parallel plate electrodes

Parametric study of the narrow gap cell

Curved boundary


presence of the gas bubble in the interelectrode space

evaluation of the process efficiency

Optimisation of the direct electrochemical water disinfection cell


10 cm

5 cm

5 cm

0.5 cm 0.3 cm

2.4 cm

AA C C

x

y



U = 6.04 V

Javer. = 50 A m-2

= 4.92 %

anode

cathode

anode

= 667 S cm-1

evaluation of the process efficiency – plate electrodes



U = 6.04 V

Javer. = 50 A m-2

= 4.92 %

= 667 S cm-1

evaluation of the process efficiency – plate electrodes



evaluation of the process efficiency – expanded mesh electrodes

3 mm

5 mm1.4 mm

1.5 mm

cathode

anode

x

z

y



evaluation of the process efficiency – expanded mesh electrodes

U = 4.76 VJaver. = 42.4 A m-2

= 4.12 % = 667 S cm-1

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 integrationover the volume surroundingcontrolled grid pointcontrolled domain coveredby the controlled volumesintegration leads formally toequation identical with FDE

hh

hh

j,1ij,ij,ij,1i

0hh

hh

1j,ij,ij,i1j,i

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]




I = 40 mA; U =4 V




I = 40 mA; U =4 V



comparison of the model and experimental results

Potential and current density distribution in three dimensional electrode

Tertiary current density distribution



simplified sketch of the cell construction

1

3

4

5

568

7

2

1 – electrolyte inlet2 – particle electrode3 – channels connecting

4 – cathode feeder

5 – anode feeder

individual drums

6 – anode7 – separator8 – electrolyte outlet



simplified flow patterns inside the cell - electrolyte



simplified flow patterns inside the cell – electric current



basic equations describing the system


ex,el2

m2

m ajdx

d

ex,el2

s2

s ajdx

d

electrode phase

electrolyte phase

m – Galvani potential of the electrode phase

s – Galvani potential of the electrolyte phase

m – conductivity of the electrode phase

s – conductivity of the electrolyte phase

ae – electrode spec. surface

jel– current density correspondingto the electrode reaction

x – coordinate

Nomenclature


basic equations describing the system – definition of jel


electrode reactions considered:

cathode

overall electrode reaction current density:

anode

Cu2+ + 2e- Cu

2 H+ + 2e- H2

2 H2O 4 H+ + O2

22 OHCuel jjjj

resistivity to the charge transfer: electrode potential


basic equations describing the system – electrode reaction kinetics


x,CuCuc,Cu

x,Culim,

x,Cu,0

x,CuCuc,Cu

x,CuCuc,Cu

x,Cu,0

x,Cu

RT

Fzexp

j

j1

RT

Fz1exp

RT

Fzexpj

j

x,H

Hc,HH,0x,H RT

Fzexpjj

aa jln..E 10550411




polarisation curves

zFkAcj x,Cux,Culim, Fvz

Aj

dx

dc

Cu

x,CuCu

smE

reversible potentials: Nernst equation

i

x,i0rx,r cln

zF

RTEE




mass transfer coefficient evaluation

pr6

r

r3

1

3

1

p ReRe1004.1125exp1Re2498

Re8.52ScRe

09.1Sh

significant complication – evaluation of the linear electrolyte flow rate

-15

-10

-5

0

0.000.05

0.100.15

0.200.25

10

15

20

j / A

m-2

position / m

current load / A

electrolyte flow

1st drum6th drum


selected results – influence of the current load


V = 4.5·10-5 m3 s-1cCu0 = 7.87 mol m-3

cH0 = 100 mol m-3

k = 4.19·10-6 m s-1

particle diameter 0.002 m = 0.047 Hz


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

c Cu2+

/ m

ol m

-3

position / m

current load / A

electrolyte flow

1st drum6th drum

drum number

0 1 2 3 4 5 6 7

effi

cien

cy /

%

0.4

0.6

0.8

1.0

I = 6 A I = 20 A


selected results – cell with different particle sizes


I = 15 A

cCu0 = 7.87 mol m-3 cH

0 = 100 mol m-3

k = 1.64·10-6 m s-1

V = 4.25·10-5 m3 s-1

cCu0 = 4.00 mol m-3

position / m

0.00 0.05 0.10 0.15 0.20 0.25

j / A

m-2

-70

-60

-50

-20

-10

01st drum 6th drum

electrolyte flow

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

dp


selected results – cell with different particle sizes


drum number

0 1 2 3 4 5 6 7

curr

ent l

oad

/ A

0

1

2

3

4

drum number

0 1 2 3 4 5 6 7

effi

cien

cy /

%

0.4

0.6

0.8

1.0

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

121,2Don 343,4Don

21~~

21~~

22,0

11,0 FzalnRTFzalnRT

FzalnRTFzalnRT 2,0

11,0

z

1

2,

1,121,2Don a

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

0czczcz MM2,2,

Donnan distribution coefficient electroneutrality condition


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


Basic equations for ideal behavior

Nernst-Planck equation is used to describe ion transport

material balance:

vN iii

iiii cFzRT

DccD

iii div

cN

stationary state: 0

ic

no chemical reaction: 0

electroosmotic flux: pFcz k,Mk,M

v


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 ss s s s s s si i

i i i i i M s M ss

i

DD c c z F c z c F p

RTc

iN

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

( )

ln1

ln

j j

j j j j j j

j

s ss s s s s si i

i i i i isi i i ii

DD c c z F c

RTc

iN v

after rearrangement and derivation

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )20 j j j j j j j j j js s s s s s s s s s

i i i i i i tot toti i

F Fz D c z D c c c

RT RT v v

membrane



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 p c

electroneutrality equation

(Schlögel’s equation)

solution flux divergence – derivative of Schlögel’s equation (pressure profile)

0czczjj

ionsj

s,Ms,M

N

i

)s(ii

ionsN

iiizF Nj

membrane



membrane – electrolyte solution interface (molar flux continuity):

membrane – electrolyte solution interface (Donnan equilibrium):

iz

1

ki

1kik1k)k,1k(

Don a

aln

F

RT

0

jsklklxixi NN

interface


Activity coefficients evaluation

cation

K.S. Pitzer: Activity Coefficients

where

2M M a Ma Ma c Mc a Mca a M c a caa Maa

a c a a c aa

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

,

ln( )

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

c a a c c aa

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

,

ln( )

c a ca c ac cc a aac a c ac a

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

, ,

' ' '

1 21 2

1 2

I 2f A 1 bI

1 bI bln( )

0 1 21 2 1 2MX MX MX 1 MX 2B g I g I in Electrolyte Solutions.

CRC Press, London 2000.

anion


Methods used to solve the equations

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

solution - shooting method

Boundary conditions - system non-linear equations 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 (DZREAL2)

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

integration of system A-ODEs for individual parts - initial problemimplicit method based on the BDF Gears formulas (DDASPK1)


input parameters

anolyte: 5 kmol NaCl m-3, catholyte:

0

10000

20000

0

500

10001500

20002500

0.00.2

0.40.6

c Na+

/ m

ol m

-3

j / A

m-2

coordinate / mm

current flow direction

pH = 2 13 kmol NaOH m-3

simplified model

Results – influence of the current load



input parameters

anolyte: 5 kmol NaCl m-3, catholyte:pH = 2 13 kmol NaOH m-3

simplified model



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

mol

m-2

s-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

Dynamic models

Model of the cathodically protected pipelines in a soil

aim of the study

theory of the cathodic protection in a soilproposed alternative theorydifficult experimental evaluationmathematical model offers simple qualitative alternative to the experiment

0.01 m

4 m

insulation insulationdamage

boundary of the domain

x

y

Dynamic models

cathode reaction considered


OH4e4OH2O 22

simplifying assumptions

homogeneous environmentno reaction with CO2

constant oxygen flux to the cathode surfacehomogeneous potential distribution on the cathode (damage) surfacewater electrolysis consumes negligible portion of the current

22 HOH2e2OH2

model equations

0)(

)cBzc(ADJ iiiii

2K1615

K21

K5.01A

1K

RTF

B i

m

íi

22

cDzARTF

i

Dynamic models


c0Na+ = 10 mol m-3

c0Cl- = 10 mol m-3

c0OH- = 0 mol m-3

selected initial conditions

Dynamic models


c0Na+ = 10 mol m-3

c0Cl- = 10 mol m-3

c0OH- = 0 mol m-3

selected initial conditions

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 limitationsreliable input data

Documents

Modelling and Design of the Electrochemical Processes and Reactors Karel Bouzek, Roman Kodým Department of Inorganic Technology, Institute of Chemical