computation of corrosion

7/28/2019 Computation of Corrosion

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COMPUTATION OF

CORROSION

Ernesto Gutierrez-Miravete

Universidad IberoAmericanaJune 1998



CONTENTS

1 INTRODUCTION

2 CORROSION

THERMODYNAMICS

3 ELECTRODE KINETICS

4 CORROSION KINETICS5 COMPLEX CORROSION

SYSTEMS



1 INTRODUCTION

1.1 ECONOMIC SIGNIFICANCE

1.2 DEFINITION OF CORROSION1.3 THERMODYNAMICS

1.4 ELECTROCHEMISTRY

1.5 KINETICS



1.1 ECONOMIC

SIGNIFICANCE• CONSEQUENCES OF CORROSION

– WASTE OF UP TO ~ 3% OF GNP

– WASTE OF NATURAL RESOURCES

– ECOLOGICAL DAMAGE

• UP TO ~ 1% OF THE WASTED GNP

COULD BE SAVED IF EXISTINGKNOWLEDGE IS APPLIED!



1.2 DEFINITION OF

CORROSION• ELECTROCHEMICAL WEAR OF

METALS EXPOSED TO REACTIVE

AQUEOUS ENVIRONMENTS

• WEAR PRODUCTS ARE STABLE

METAL COMPOUNDS

• THERE ARE MANY DIFFERENTFORMS OF CORROSION



1.3 THERMODYNAMICS

• SYSTEM; SURROUNDINGS;

EQUILIBRIUM

• STANDARD STATE

• LAWS OF THERMODYNAMICS

• CHEMICAL POTENTIAL

• EQUILIBRIUM CONSTANT

• REACTION ISOTHERM



1.4 ELECTROCHEMISTRY

• ELECTROCHEMICAL POTENTIAL

• ELECTROCHEMICAL CELLS

– OXIDATION REACTION (ANODE) – REDUCTION REACTION (CATHODE)

• HALF CELL AND SINGLE POTENTIAL

– NERNST EQUATION

• ELECTROCHEMICAL SERIES



1.5 KINETICS

• CHEMICAL REACTION RATE

• ARRHENIUS RATE EQUATION

• ACTIVATED COMPLEX• ACTIVATION ENERGY

• REACTION MECHANISMS AT AN

ELECTRODE-ELECTROLYTEINTERFACE



2 CORROSION

THERMODYNAMICS2.1 CHEMICAL SYSTEMS

2.2 ELECTROCHEMICALSYSTEMS

2.3 DIAGRAMS POTENTIAL-pH

2.4 IMMUNITY, PASSIVITY ANDCORROSION

2.5 EXAMPLES



2.1 CHEMICAL SYSTEMS I

• GIBBS FREE ENERGY

G = H - T S

• EQUILIBRIUM CRITERION

DG = 0

•CHEMICAL POTENTIAL

mi = dG/dni = mio + RT ln ai



2.1 CHEMICAL SYSTEMS II

• GENERIC CHEMICAL REACTION

S ni Mi = 0

• CRITERION FOR REACTIONEQUILIBRIUM

S ni mio + RT S ni ln ai = 0

• IN TERMS OF EQUILIBRIUM

CONSTANT

-S ni mio = RT S ni ln ai = RT ln K



2.1 CHEMICAL SYSTEMS III

• EXAMPLES

– DISSOCIATION OF WATER

H2O = H+ + OH-

K = aH+ aOH-/aH2O = aH+ aOH- = 10-14

– PRECIPITATION OF METAL HYDROXIDE

K = (aH+ )2/ aMg2+ = 10-16.95



2.2 ELECTROCHEMICAL

SYSTEMS I• GENERIC ELECTROCHEMICAL

REACTION

S ni Mi + ne = 0

• CRITERION FOR ELECTROCHEMICAL

EQUILIBRIUM

S ni mio + RT S ni ln ai - 23061 n Eo = 0

• WHERE Eo IS THE EQUILIBRIUM

POTENTIAL



2.2 ELECTROCHEMICAL

SYSTEMS II• IN TERMS OF THE EQUILIBRIUM

POTENTIAL (NERNST EQUATION)

Eo = Eoo + (2.3RT/nF)S ni log ai

• WHERE

Eoo

=

S ni mio /(23061 n)• IS THE STANDARD EQUILIBRIUM

POTENTIAL



2.2 ELECTROCHEMICAL

SYSTEMS III• EXAMPLE: THE STANDARD

HYDROGEN ELECTRODE

2 H+ + 2 e = H2

• DATA

mH+

o = mH2

o = 0 cal/mol

• THUS

Eoo = S ni mi

o /(23061 n) = 0



2.2 ELECTROCHEMICAL

SYSTEMS IV• THUS

Eo = Eoo + (2.3RT/nF)S ni log ai

= 0.0 + 0.059 log aH+ - 0.0295 log PH2

• AND FOR PH2 = 1 atm

Eo = 0.0 - 0.059 pH



2.2 ELECTROCHEMICAL

SYSTEMS V• POSSIBILITY OF

ELECTROCHEMICAL REACTIONS

– IF E = Eo >> EQUILIBRIUM (NO

REACTION)

– IF E > Eo >> OXIDATION

POSSIBLE

– IF E < Eo >> REDUCTION

POSSIBLE



2.2 ELECTROCHEMICAL

SYSTEMS VI• THE VELOCITY OF A REACTION (AS

MEASURED BY THE INTENSITY OF

THE REACTION CURRENT i ) ISEITHER ZERO OR OF THE SAME

SIGN AS ITS OVERPOTENTIAL

(E - Eo) i = 0 OR

(E - Eo) i > 0



2.3 DIAGRAMS E-pH I

• FIRST PRESENTED BY POURBAIX

• GRAPHICAL REPRESENTATIONS OF

Eo-pH RELATIONSHIPS FOR

SELECTED REDOX COUPLES

• OBTAINED FROM NERNST

EQUATIONS

• USED TO IDENTIFY REGIONS OF

PREDOMINANCE



2.3 DIAGRAMS E-pH II

• HYDROGEN EVOLUTION (LINE a)

2 H+ + 2 e = H2

E = 0.0 - 0.059 pH

• OXYGEN REDUCTION (LINE b)

O2 + 2 H2O + 4 e = 4 OH-

E = 1.228 - 0.059 pH



2.3 DIAGRAMS E-pH III

• MAIN REACTION TYPES

– REACTIONS DEPENDING ON pH ONLY

Fe3+ + H2O = Fe(OH)2+ + H+

– REACTIONS DEPENDING ON E ONLY

Fe = Fe2+ + 2 e

– REACTIONS DEPENDING ON pH AND E

Fe3+ + H2O = Fe(OH)2+ + H+ + e

– REACTIONS INDEPENDENT OF pH AND E

CO2 + H2O = H2CO3



2.3 DIAGRAMS E-pH IV

• DIAGRAM CONSTRUCTION

PROCEDURE

– DETERMINE SPECIES INVOLVED

– WRITE DOWN REACTIONS FOR ALL

SPECIES INVOLVED

– WRITE DOWN NERNST EQUATIONS

FOR ALL REACTIONS USING ai = 10-6

– FIND E AND pH VALUES FOR

INTERSECTIONS BETWEEN

REACTIONS



2.4 IMMUNITY , PASSIVITY

AND CORROSION• IMMUNITY: DOMAIN IN E-pH SPACE

WHERE THE METAL PHASE IS

THERMODYNAMICALLY STABLE• PASSIVITY: DOMAIN IN E-pH SPACE

WHERE AN INSOLUBLE FILM IS

THERMODYNAMICALLY STABLE• CORROSION: DOMAIN IN E-pH SPACE

WHERE SOLUBLE PRODUCTS ARE

THERMODYNAMICALLY STABLE



2.5 EXAMPLES I

Fe/Fe2+/Fe3+/Fe3O4/Fe2O3 SYSTEM

• REACTIONS (NUMBERS FROM ATLAS)

Fe2+ = Fe3+ + 1e ............(4)

3Fe + 4H2O = Fe3O4 + 8H+ + 8e ......(13)

2Fe3O4 + H2O = 3Fe2O3 + 2H+ + 2e ...(17)

2Fe3+ + 3H2O = Fe2O3 + 6H+ .....(20)

Fe = Fe2+ + 2e .............(23)

3Fe2+ + 4H2O = Fe3O4 + 8H+ + 2e ...(26)

2Fe2+ + 3H2O = Fe2O3 + 6H+ + 2e ...(28)



2.5 EXAMPLES II


• EQUILIBRIUM POTENTIALS/EQNS

E4 = 0.771 + 0.0591 log( Fe3+/Fe2+)

E13 = -0.085 - 0.0591 pH

E17 = 0.221 - 0.0591 pH

log(Fe3+)20

= -0.72 - 3 pH

E23 = -0.44 + 0.0295 log( Fe2+)

E26 = 0.98 - 0.2364 pH - 0.0886 log( Fe2+)

E28 = 0.728 - 0.1773 pH - 0.0591 log( Fe2+)



2.5 EXAMPLES III


• INTERSECTION 1: 20-28

• INTERSECTION 2: 17-26-28

• INTERSECTION 3: 13-23-26

• VERTICAL LINES: 20

• HORIZONTAL LINES: 23



2.5 EXAMPLES IV

• EXERCISE: CONSTRUCT POURBAIXDIAGRAMS

– Mg/Mg2+/Mg(OH)2 SYSTEM

Mg2+ + 2e = Mg

E3 = -2.363 + 0.0295 log(Mg2+)

– OTHER SYSTEM OF YOUR

CHOICE



3 ELECTRODE KINETICS

3.1 MECHANISMS OF ELECTRODE

REACTIONS

3.2 EXCHANGE CURRENT

3.3 POLARIZATION AND

OVERVOLTAGE

3.4 TYPES OF OVERVOLTAGE

3.5 EXAMPLES



3.1 MECHANISMS OF

ELECTRODE REACTIONS I• ELECTRODE PROCESSES ARE

HETEROGENOEUS REACTIONS

• THE ELECTRODE-ELECTROLYTEINTERFACE HAS COMPLEX

STRUCTURE (ELECTROCHEMICAL

DOUBLE LAYER/DIFFUSE LAYER)• THE STRUCTURE OF THE ECDL

INFLUENCES REACTION RATES



3.1 MECHANISMS OF

ELECTRODE REACTIONS II

1.- MASS TRANSFER OF REACTANTS

FROM BULK OF SOLUTION TO

ELECTRODE BOUNDARY LAYER 2.- ADSORPTION OF REACTANTS

ONTO ELECTROCHEMICAL DOUBLE

LAYER 3.- DEHYDRATION/DESOLVATION



3.1 MECHANISMS OF

ELECTRODE REACTIONS III4.- CHEMICAL CHANGES LEADING

TO INTERMEDIARIES

5.- ELECTRON TRANSFER

6.- ADSORPTION OF REACTION

PRODUCTS

7.- DESORPTION OF REACTION

PRODUCTS



3.1 MECHANISMS OF

ELECTRODE REACTIONS IV8.- SECONDARY CONVERSION OF

PRIMARY PRODUCTS

9.- MASS TRANSFER OF PRODUCTS

FROM BOUNDARY LAYER INTO

BULK OF THE SOLUTION



3.1 MECHANISMS OF

ELECTRODE REACTIONS VTHE MOST CHARACTERISTIC

FEATURE OF ELECTRODE

PROCESSES IS THE DEPENDENCEOF THE ACTIVATION ENERGY FOR

THE ELECTRON TRANSFER STEP

ON THE POTENTIAL DIFFERENCEBETWEEN ELECTRODE AND

SOLUTION.



3.2 EXCHANGE CURRENT I

THE CURRENT DENSITY i (A/m2) IS A

MEASURE OF THE RATE OF

ELECTROCHEMICAL REACTIONi = z F v

z = number of electrons/molecule

F = Faraday’s constant = 96487 Coulomb/mol

v = reaction rate (mole/ m2 s)



3.2 EXCHANGE CURRENT II

• THE EQUILIBRIUM AT A REVERSIBLE

ELECTRODE IS DYNAMIC

va = vc

ia = ic

• WHERE

va = RATE OF ANODIC (OXIDATION) PROCESS

vc = RATE OF CATHODIC (OXIDATION) PROCESS



3.2 EXCHANGE CURRENT III

EQUILIBRIUM SINGLE ELECTRODE

M = M+z + z e

ia = z F k a exp( aa z F fo/RT)

ic = z F k c aMz+ exp(-ac z F fo/RT)

inet = ia+ ic = 0

io = ia = - ic



3.2 EXCHANGE CURRENT IV

k a , k c = RATE CONSTANTS FOR

ANODIC AND CATHODIC PROCESSES

aa , ac = TRANSFER COEFFICIENTSFOR ANODIC AND CATHODIC

PROCESSES

fo = (EQUILIBRIUM) ELECTRODEPOTENTIAL

io = EXCHANGE CURRENT



3.2 EXCHANGE CURRENT V

• FROM THE CONDITION inet = 0

fo =[RT/zF(a

a+ a

c)] ln[k

ca

M

z+ / k a]

• FROM THE CONDITION io = ia = - ic

io = zF k a [k c aMz+ / k a](aa /(aa + ac))



3.2 EXCHANGE CURRENT VI

• EXAMPLES

Ag = Ag+ + e ; io = 104 A/m2

Cu = Cu2+ + 2e ; io = 5 A/m2

Zn = Zn2+ + 2e ; io = 10-2 A/m2

H2

= 2H+ + 2e (on Pt) ;io

= 102 A/m2

H2 = 2H+ + 2e (on Fe) ; io = 10-2 A/m2

H2 = 2H+ + 2e (on Pb) ; io = 10-9 A/m2




OVERVOLTAGE IIN A NON-EQUILIBRIUM EQUILIBRIUM

ELECTRODE f IS DISTINCT FROM fo

ia = z F k a exp( aa z F f/RT)= io exp( aa z F [f - fo]/RT)

ic = z F k c aMz+ exp(-ac z F f/RT)= io exp(-ac z F [f - fo]/RT)




OVERVOLTAGE II

IN A NON-EQUILIBRIUM EQUILIBRIUM

ELECTRODE THERE IS A NET FLOW OF

CURRENT, THE POTENTIAL IS f ANDh = f - fo IS THE (ACTIVATION)

OVERVOLTAGE

inet = io {exp(aa z F h/RT) -exp(-ac z F h/RT) }

BUTLER-VOLMER EQUATION




OVERVOLTAGE III• “SMALL” h APPROXIMATION

IF | h | < 0.01 V SINCE e

x

~ 1 + x

inet = io (z F h/RT)

THE LINEAR “LAW”




OVERVOLTAGE V• “LARGE” h APPROXIMATION

(CATHODIC POLARIZATION)

IF h < - 0.1 V THEN inet ~ ic

inet = io exp(-ac z F h/RT) OR

h = bc log(inet / io )

CATHODIC TAFEL EQUATION




OVERVOLTAGE VI• EXAMPLES

Ag = Ag+ + e ; ba = 0.12 V/decade

Cu = Cu2+ + 2e ; ba = 0.06 V/decade

Zn = Zn2+ + 2e ; ba = 0.06 V/decade

H2 = 2H+ + 2e (on Pt) ; bc = -0.12 V/decade

H2 = 2H+ + 2e (on Fe) ; bc = - 0.12 V/decade

H2

= 2H+ + 2e (on Pb) ; bc

= -0.12 V/decade



3.4 TYPES OF

OVERVOLTAGE I• ACTIVATION OVERVOLTAGE

• CONCENTRATION OVERVOLTAGE

• RESISTANCE OVERVOLTAGE

• REACTION OVERVOLTAGE

• NUCLEATION OVERVOLTAGE



3.4 TYPES OF

OVERVOLTAGE II• WHEN THE ELECTRON TRANSFER

PROCESS IS FAST, IONIC

CONCENTRATION GRADIENTS ATTHE ELECTRODE SURFACE GIVE

RISE TO CONCENTRATION

OVERVOLTAGEhconc = (RT/zF) ln(Csurf /Cbulk )



3.4 TYPES OF

OVERVOLTAGE III• IONIC CONCENTRATION GRADIENTS

PRODUCES CURRENT

i = - z F D (dC/dx)

= - z F D (Csurf - Cbulk )/d

D = DIFFUSION COEFFICIENT (m2/s)

d = CONCENTRATION BOUNDARY

LAYER THICKNESS (m)



3.4 TYPES OF

OVERVOLTAGE IV• IN THE LIMIT WHEN Csurf ~ 0, THIS

BECOMES THE LIMITING CURRENT

DENSITY iL

iL = z F D Cbulk /d THUS

i/iL = 1 - Csurf /Cbulk AND

hconc = (RT/zF) ln(1 - i/iL)



3.4 TYPES OF

OVERVOLTAGE V• THE BOUNDARY LAYER THICKNESS

DEPENDS ON FLUID FLOW

• IN A ROTATING DISK ELECTRODE(RDE) SYSTEM, LEVICH FOUND

d = 1.61 D1/3 n 1/6 w -1/2

n = KINEMATIC VISCOSITY (m2/s)

w = DISK ROTATION RATE (rad/s)



3.4 TYPES OF

OVERVOLTAGE VI• IF THE ELECTRICAL RESISTANCE

(R ) TO THE FLOW OF CURRENT ( I )

IN THE ELECTROLYTE BETWEENANODE AND CATHODE IS

SIGNIFICANT THERE CAN BE

(ELECTROLYTE) RESISTANCEOVERVOLTAGE

hres = I R



3.5 EXAMPLES I

• EXERCISE: SHOW THAT A NET

CURRENT DENSITY OF 1 A/m2 IS

APPROXIMATELYEQUIVALENT TO A UNIFORM

CORROSION WASTAGE RATE

OF 1 mm/year



3.5 EXAMPLES II

Fe = Fe2+ + 2e

aM

z+ = aFe

2+ = 10-6

aa = ac = 0.5

k a = 14.9 ; k a = 1.8*10-14 THUS

fo = - 0.617 V

io = 10-4 A/m2



3.5 EXAMPLES III

• EXERCISE: VERIFY THE GIVEN

EXPRESSIONS FOR fo AND io

• EXERCISE: FIND THE VALUES OF k a

AND k c FOR THE Mg = Mg2+ + 2e

SYSTEM

• EXERCISE: FIND THE VALUES OF k a

AND k c FOR OTHER SYSTEM OF

YOUR CHOICE



3.5 EXAMPLES IV

• Fe ELECTRODE

inet

= io

{e38.94h - e-38.94h }

h (V) 0.01 0.02 0.05 0.1

e38.94h 1.47 2.18 7.00 49.11

e-38.94h 0.68 0.46 0.14 0.02

inet / io 0.80 1.72 6.85 49.09



4 CORROSION KINETICS

4.1 ANODIC AND CATHODIC

REACTIONS

4.2 POLARIZATION4.3 WAGNER AND TRAUD’S

SUPERPOSITION PRINCIPLE

4.4 STERN AND GEARY’S ANALYSIS

4.5 EXAMPLES



4.1 ANODIC AND

CATHODIC REACTIONS II• AT ANODIC SITES OXIDATION

OCCURS ( ia )

• AT CATHODIC SITES REDUCTION

OCCURS ( ic )

• ALL THE ELECTRONS PRODUCED

BY THE ANODIC REACTION ARECONSUMED BY THE CATHODIC

PROCESS



4.1 ANODIC AND

CATHODIC REACTIONS III• SIMPLEST ANODIC REACTION

(SINGLE STEP)M = Mz+ + ze

• EVEN IN THIS CASE A COMPLEX

MECHANISM MAY BEINVOLVED



4.1 ANODIC AND

CATHODIC REACTIONS IV• COMPLEX ANODIC REACTION

(MULTIPLE STEP)

STEP 1

M + nH2O = [M(H2O)n]z1 + z1e



4.1 ANODIC AND

CATHODIC REACTIONS V• STEP 2a: INHIBITION

M + nH2O = [M(H2O)n]z1

+ z1e• STEP 2b: COMPLEX IONISATION

[M(H2O)n]z1 = Mz+ + n H2O + (z-z1 )e

• STEP 2c: PASSIVATION

M + (z/2)H2O = MOz/2 + zH+ + ze



4 1 ANODIC AND CATHODIC



4.1 ANODIC AND CATHODIC

REACTIONS VII• FOR NON-UNIFORM CORROSION

IS BETTER TO USE THE TOTAL

CURRENTS ( I ) RATHER THANTHE CURRENT DENSITIES ( i )

• PLOTS OF E vs Ia

AND Ic

ARE

CALLED EVANS DIAGRAMS



4.2 POLARIZATION II

• BECAUSE OF POLARIZATION THE

ELECTRODE POTENTIALS OF THE

TWO REACTIONS MOVE TOWARDS

EACH OTHER AND IF THEELECTROLYTE RESISTANCE IS LOW

fa

= Ea

= fc

= Ec

= f* = Ecorr

ia = ic = icorr

Ia = Ic = Icorr



4.2 POLARIZATION III

• A FREQUENTLY USED ASSUMPTION

IS THAT THE ANODIC AND

CATHODIC OVERVOLTAGES AREADDITIVE

hT = h + hconc + hres + hother



4.2 POLARIZATION IV

log i

E

ANODIC

CURVE

CATHODIC

CURVE

Ecorr

log icorr



4.2 POLARIZATION V

I

EANODIC

CURVE

CATHODIC

CURVE

Icorr

Ecorr

4 3 WAGNER AND TRAUD’S



4.3 WAGNER AND TRAUD S

SUPERPOSITION

PRINCIPLE I

• IN A HOMOGENOUS SURFACE

WITH MULTIPLE REACTIONSWITH CURRENT DENSITIES iia

iic

iaT = S iia = - S iic = - icT



4 4 STERN AND GEARY’S



4.4 STERN AND GEARY’S

ANALYSIS IVIF ON A FREELY CORRODING

SPECIMEN (f*), A SMALL

OVERVOLTAGE IS APPLIED, THE

POTENTIAL INCREASES BY D f AND

THE NET CURRENT DENSITY

INCREASES BY

D i = D f {(ba + |bc|)/(ba |bc|)}

4 4 STERN AND GEARY’S



4.4 STERN AND GEARY’S

ANALYSIS VE

i

icorr

Ecorr

Df

Di

slope = ba

slope = bc



4.5 EXAMPLES I

• THE EVANS SALT DROP EXPERIMENT

– 3% NaCl SOLUTION IN H2O

– POTASIUM FERRICYANIDE (BECOMES

BLUE IN PRESENCE OF Fe2+)

– PHENOLPHTALEIN (BECOMES PINK IN

PRESENCE OF ALKALI)

– FINELY ABRADED Fe SURFACE

• FORMATION OF PINK-RUST-BLUE

RINGS AFTER SOME TIME



4.5 EXAMPLES III

• EXERCISE: ASSUMING

aaa = aac = aca = acc = 1/2

FOR SINGLE METAL DISSOLUTION

COUPLED WITH A SINGLE

CATHODIC REACTION FIND AN

EXPRESSION FOR f*



4.5 EXAMPLES IV• EXERCISE: VERIFY THE GIVEN

STERN&GEARY EXPRESSION FOR

inet

• EXERCISE: USE THE GIVEN

STERN&GEARY EXPRESSION FOR

inet USE THE GIVEN TO COMPUTE

THE f - log inet POLARIZATION

CURVE FOR THE FOUR SYSTEMS IN

EXAMPLES II ABOVE



4.5 EXAMPLES V

• DETERMINE THE CORROSION

POTENTIALS AND CORROSION

CURRENT DENSITIES FOR THEFOUR SYSTEMS IN EXAMPLE II

ABOVE



4.5 EXAMPLES VI

• EXERCISE: VERIFY THE GIVEN

EXPRESSIONS FOR ba AND bc IN

THE STERN-GEARY ANALYSIS



5 COMPLEX CORROSION



5 COMPLEX CORROSION

SYSTEMS5.1 GOVERNING EQUATIONS

5.2 BOUNDARY CONDITIONS5.3 ANALYTICAL SOLUTIONS

5.4 NUMERICAL SOLUTIONS

5.5 EXAMPLES



5 1 GOVERNING



5.1 GOVERNING

EQUATIONS II• CURRENT DENSITY EQUATION

i = F S zi Ni

• MATERIAL BALANCE

dCi/dt = - div Ni

•ELECTRONEUTRALITY CONDITION

S zi Ci = 0



5 1 GOVERNING



5.1 GOVERNING

EQUATIONS IV• SIMPLIFICATIONS FOR POTENTIAL

THEORY

–

WHEN CONCENTRATION GRADIENTS,ARE NOT IMPORTANT

i = - k grad f

– WHERE k IS THE CONDUCTIVITY OF

THE SOLUTION

5 1 GOVERNING



5.1 GOVERNING

EQUATIONS V• SIMPLIFICATIONS FOR POTENTIAL

THEORY (contd)

– FROM THE MASS BALANCE EQUATION

div grad f = 0

LAPLACE’S EQUATION

5 2 BOUNDARY



5.2 BOUNDARY

CONDITIONS I• FLUX NORMAL TO ELECTRODE

SURFACE

Nin = - (ni/zF) in = f(hs,Ci)• NO-SLIP AT FLUID-SOLID

BOUNDARIES

vf = vs



5 3 ANALYTICAL



5.3 ANALYTICAL

SOLUTIONS I• ANALYTICAL SOLUTIONS OF

POTENTIAL PROBLEMS ARE

POSSIBLE IN A FEW SELECTEDCASES

• ANALYTICAL SOLUTIONS CAN BE

OBTAINED BY FOURIER SERIES OR CONFORMAL MAPPING METHODS

5.3 ANALYTICAL



5 3 N C

SOLUTIONS II• WABER SOLUTION : GALVANIC

SYSTEM AS BEFORE x1 = 0, x2 = a

AND x3 = c AND f = Ea ON 1-2 ANDf = 0 ON 2-3

f(x,y) = Ea (p a/c) + (2 Ea / p)

S (1/n) e -n p y/c sin(n p a/c) cos(n p x/c)



5 4 NUMERICAL



5.4 NUMERICAL

SOLUTIONS I• DISCRETIZED VERSIONS OF THE

MODELING EQUATIONS ARE

SOLVED USING COMPUTERS• DISCRETIZATION METHODS

– FINITE DIFFERENCES

–

CONTROL VOLUME (BOX) METHOD – FINITE ELEMENTS

– BOUNDARY ELEMENTS

5 4 NUMERICAL



5.4 NUMERICAL

SOLUTIONS II• FINITE DIFFERENCE METHOD FOR

POTENTIAL PROBLEMS

1.- SUBDIVIDE THE DOMAIN OFINTEREST INTO A LATTICE OF

MESH POINTS OR NODES

2.- REPLACE DERIVATIVES IN THELAPLACE’S EQUATION FOR f BY

FINITE DIFFERENCES



5.5 EXAMPLES II

• GALVANIC CORROSION (contd)

– ELECTROLYTE (LAPLACE’S

EQUATION)

d2f/dx2 + d2f/dy2 = 0 – BOUNDARIES 3-4, 5-1

df/dx = 0

5 5 EXAMPLES III



5.5 EXAMPLES III


– BOUNDARY 4-5

df/dy

= 0 – BOUNDARY 1-2

df/dy = iA/k

– BOUNDARY 2-3

df/dy = iB/k



5.5 EXAMPLES IV


• FINITE DIFFERENCE FORMULAE

(ELECTROLYTE; INTERIOR NODES)

(fE + fW - 2 fP )/Dx2 +

(fN + fS - 2 fP )/Dy2 = 0



5.5 EXAMPLES V

N

S

EW

P

Dx

Dy

INTERIOR NODES



5.5 EXAMPLES VI


(BOUNDARY NODES; BOUNDARY

1-2-3)

fP = [1/(2(1+l2))] {(fE + fW) l2 +

2 fN + (2Dy/k) i }WITH l = Dy/Dx



5.5 EXAMPLES VII


(BOUNDARY NODES; BOUNDARY

3-4)

fP = [1/(2(1+(1/l2)))]

{(fN + fS)(1/ l2 ) + 2 fE }



5.5 EXAMPLES VIII

BOUNDARY NODES

N

EW

-N

P



5.5 EXAMPLES IX

• SOLUTION BY SUCCESIVE OVER-

RELAXATION ITERATION FOR ALL

MESH POINTS. SOR EQUATION FOR MESH POINT i IS

fin+1 = fi

n +

(w/aii) { bi - S aij f jn+1 - S aij f j

n }

• WHERE w = RELAXATION FACTOR



5.5 EXAMPLES X

• EXERCISE: DERIVE THE FINITE

DIFFERENCE FORMULAE GIVEN

•EXERCISE: DERIVE THE S.O.R.FORMULAE GIVEN

• EXERCISE: EXPLORE THE

SENSITIVITY OF THE GALVANICSYSTEM



REFERENCES II

• Charlot, Badoz-Lambling & Tremillon,“Las Reacciones Electroquimicas”,

Toray-Mason, Barcelona, 1969

• Bockris & Ready, “ModernElectrochemistry”, 2 V, Plenum, New

York, 1977

• Pourbaix, “Atlas of ElectrochemicalEquilibria in Aqueous Solutions”,

NACE, Houston, 1974



REFERENCES III

• Gileadi, “Electrode Kinetics”, VCH, New

York, 1993

•Newman, “Electrochemical Systems”,2nd ed., Prentice-Hall, Englewood Cliffs,

1991

•

Iserles, “A First Course in NumericalAnalysis of Differential Equations”,

Cambridge UP Cambridge 1996

computation of corrosion

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