modelling and design of the electrochemical processes and reactors karel bouzek, roman kodým...
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![Page 1: Modelling and Design of the Electrochemical Processes and Reactors Karel Bouzek, Roman Kodým Department of Inorganic Technology, Institute of Chemical](https://reader034.vdocuments.us/reader034/viewer/2022051316/56649cff5503460f949d005d/html5/thumbnails/1.jpg)
Modelling and Design of the ElectrochemicalProcesses and Reactors
Karel Bouzek, Roman Kodým
Department of Inorganic Technology, Institute of Chemical Technology Prague
![Page 2: Modelling and Design of the Electrochemical Processes and Reactors Karel Bouzek, Roman Kodým Department of Inorganic Technology, Institute of Chemical](https://reader034.vdocuments.us/reader034/viewer/2022051316/56649cff5503460f949d005d/html5/thumbnails/2.jpg)
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
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Mathematical modelling and process design
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
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Mathematical modelling and process design
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
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Mathematical modelling and process design
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
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Mathematical modelling and process design
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
concentration decay of component A in time
galvanostatic process at j jlim
j
Fnck
j
j AO,Amlime
Fn
jac
FVn
Icc
A
limeI,A
RA
limI,AO,A
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Mathematical modelling and process design
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
concentration decay of component A in time
galvanostatic process at j jlim
V
bxkexp
j
Fnck
j
j mAI,Amlime
V
bxkexpcc m
I,AA
b – electrode width
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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
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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
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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
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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
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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
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Steady state models
Basic model equations for electrochemical systems
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
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Steady state models
Basic model equations for electrochemical systems
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
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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
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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:
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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
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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
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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
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Steady state models
Classical approach – finite differences
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
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Steady state models
Classical approach – finite differences
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.
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Steady state models
Classical approach – finite differences
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
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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.
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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
![Page 25: Modelling and Design of the Electrochemical Processes and Reactors Karel Bouzek, Roman Kodým Department of Inorganic Technology, Institute of Chemical](https://reader034.vdocuments.us/reader034/viewer/2022051316/56649cff5503460f949d005d/html5/thumbnails/25.jpg)
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
Classical approach – finite differences
problems often related to the anistropic media
difficulties by solving systems with the irregular geometries
alternative approaches searched
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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
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Steady state models
application onto the solution of the Laplace equation
simplification for the one-dimensional case
Recent approach – finite element method (FEM)
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
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Steady state models
simplest linear basis function
Recent approach – finite element method (FEM)
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
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Examples of the FEM applications
current density distribution in the channel with parallel plate electrodes
Parametric study of the narrow gap cell
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Curved boundary
Examples of the FEM applications
presence of the gas bubble in the interelectrode space
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evaluation of the process efficiency
Optimisation of the direct electrochemical water disinfection cell
Examples of the FEM applications
10 cm
5 cm
5 cm
0.5 cm 0.3 cm
2.4 cm
AA C C
x
y
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Optimisation of the direct electrochemical water disinfection cell
Examples of the FEM applications
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
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Optimisation of the direct electrochemical water disinfection cell
Examples of the FEM applications
U = 6.04 V
Javer. = 50 A m-2
= 4.92 %
= 667 S cm-1
evaluation of the process efficiency – plate electrodes
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Optimisation of the direct electrochemical water disinfection cell
Examples of the FEM applications
evaluation of the process efficiency – expanded mesh electrodes
3 mm
5 mm1.4 mm
1.5 mm
cathode
anode
x
z
y
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Optimisation of the direct electrochemical water disinfection cell
Examples of the FEM applications
evaluation of the process efficiency – expanded mesh electrodes
U = 4.76 VJaver. = 42.4 A m-2
= 4.12 % = 667 S cm-1
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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
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Application to the bipolar electrode function simulation
Finite volumes method
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]
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Application to the bipolar electrode function simulation
Finite volumes method
model system under study
I = 40 mA; U =4 V
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Application to the bipolar electrode function simulation
Finite volumes method
model system under study
I = 40 mA; U =4 V
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Application to the bipolar electrode function simulation
Finite volumes method
comparison of the model and experimental results
![Page 41: Modelling and Design of the Electrochemical Processes and Reactors Karel Bouzek, Roman Kodým Department of Inorganic Technology, Institute of Chemical](https://reader034.vdocuments.us/reader034/viewer/2022051316/56649cff5503460f949d005d/html5/thumbnails/41.jpg)
Potential and current density distribution in three dimensional electrode
Tertiary current density distribution
model system under study
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Potential and current density distribution in three dimensional electrode
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
Tertiary current density distribution
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Potential and current density distribution in three dimensional electrode
simplified flow patterns inside the cell - electrolyte
Tertiary current density distribution
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Potential and current density distribution in three dimensional electrode
simplified flow patterns inside the cell – electric current
Tertiary current density distribution
![Page 45: Modelling and Design of the Electrochemical Processes and Reactors Karel Bouzek, Roman Kodým Department of Inorganic Technology, Institute of Chemical](https://reader034.vdocuments.us/reader034/viewer/2022051316/56649cff5503460f949d005d/html5/thumbnails/45.jpg)
Potential and current density distribution in three dimensional electrode
basic equations describing the system
Tertiary current density distribution
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
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Potential and current density distribution in three dimensional electrode
basic equations describing the system – definition of jel
Tertiary current density distribution
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
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Potential and current density distribution in three dimensional electrode
basic equations describing the system – electrode reaction kinetics
Tertiary current density distribution
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
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Potential and current density distribution in three dimensional electrode
basic equations describing the system – electrode reaction kinetics
Tertiary current density distribution
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
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Potential and current density distribution in three dimensional electrode
basic equations describing the system – electrode reaction kinetics
Tertiary current density distribution
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
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-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
Potential and current density distribution in three dimensional electrode
selected results – influence of the current load
Tertiary current density distribution
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
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Potential and current density distribution in three dimensional electrode
selected results – influence of the current load
Tertiary current density distribution
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
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Potential and current density distribution in three dimensional electrode
selected results – cell with different particle sizes
Tertiary current density distribution
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
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Potential and current density distribution in three dimensional electrode
selected results – cell with different particle sizes
Tertiary current density distribution
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
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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
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Charge flux across the ion selective membrane
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
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Charge flux across the ion selective membrane
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
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Charge flux across the ion selective membrane
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
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Charge flux across the ion selective membrane
Treatment of non-idealities
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
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Charge flux across the ion selective membrane
Treatment of non-idealities
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
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Charge flux across the ion selective membrane
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
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Charge flux across the ion selective membrane
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)
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Charge flux across the ion selective membrane
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
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Charge flux across the ion selective membrane
Results – influence of the current load
input parameters
anolyte: 5 kmol NaCl m-3, catholyte:pH = 2 13 kmol NaOH m-3
simplified model
![Page 64: Modelling and Design of the Electrochemical Processes and Reactors Karel Bouzek, Roman Kodým Department of Inorganic Technology, Institute of Chemical](https://reader034.vdocuments.us/reader034/viewer/2022051316/56649cff5503460f949d005d/html5/thumbnails/64.jpg)
Charge flux across the ion selective membrane
Results – influence of the current load
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
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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
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Dynamic models
cathode reaction considered
Model of the cathodically protected pipelines in a soil
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
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Dynamic models
Model of the cathodically protected pipelines in a soil
c0Na+ = 10 mol m-3
c0Cl- = 10 mol m-3
c0OH- = 0 mol m-3
selected initial conditions
![Page 68: Modelling and Design of the Electrochemical Processes and Reactors Karel Bouzek, Roman Kodým Department of Inorganic Technology, Institute of Chemical](https://reader034.vdocuments.us/reader034/viewer/2022051316/56649cff5503460f949d005d/html5/thumbnails/68.jpg)
Dynamic models
Model of the cathodically protected pipelines in a soil
c0Na+ = 10 mol m-3
c0Cl- = 10 mol m-3
c0OH- = 0 mol m-3
selected initial conditions
![Page 69: Modelling and Design of the Electrochemical Processes and Reactors Karel Bouzek, Roman Kodým Department of Inorganic Technology, Institute of Chemical](https://reader034.vdocuments.us/reader034/viewer/2022051316/56649cff5503460f949d005d/html5/thumbnails/69.jpg)
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