def ConcentrationsCalculator(self, Ce_MP = None, CeH2_MP = None, Ce_OB = None, CeH2_OB = None, DeltaOX = None):
gibbsValues = GibbsEnergyClass.gibbs(self.Temperature)
DeltaGFe_Fe2_MP = gibbsValues.getDeltaGFe_Fe2()
DeltaGFe2p_Fe3O4_MP = gibbsValues.getDeltaGFe2p_Fe304()
DeltaGFe2p_Fe3O4_OB = gibbsValues.getDeltaGFe2p_Fe304()
DeltaGH2_Hp_MP = gibbsValues.getDeltaGfH2_Hp()
DeltaGH2_Hp_OB = gibbsValues.getDeltaGfH2_Hp()
kHenry = gibbsValues.getH2HenryConstant()
# PHvalue = PHCalculator.equilibriumConstants(self.Temperature, self.ConcC4H9ONTotal)
# pH = PHvalue.PHCalculation()
pH = 6.88 # for now not using the PHCalculator function
CeHp = math.pow (10, -pH)
#pH = 9.58
value = Constants.Const()
n = value.n()
F = value.F()
R = value.R()
h = value.h()
kboltzman = value.kboltzman()
Beta = value.Beta()
PhiOX = value.PhiOX()
ChiOX = value.ChiOX()
RhoOX = value.RhoOX()
DFe = value.DFe()
kd_OB = value.kd()
hH2 = value.km() # for now equals to km
Temperature = value.Temperature()
ConcC4H9ONTotal = value.ConcC4H9ONTotal()
FeMolarMass = value.FeMolarMass()
H2MolarMass = value.H2MolarMass()
RhoH2O = value.RhoH2O()
CH2coolant = value.CH2coolant()
TKelvin = self.Temperature + 273.15;
if Ce_MP == None:
Ce_MP = self.Ce_MP
if CeH2_MP == None:
CeH2_MP = self.CeH2_MP
if Ce_OB == None:
Ce_OB = self.Ce_OB
if CeH2_OB == None:
CeH2_OB = self.CeH2_OB
if DeltaOX == None:
DeltaOX = self.DeltaOX
# At the oxide-Bulk interface O_B
# Corrosion reaction at the M_P interface; All the reaction are considered in the following direction: O + ne- -> R
# 2H+ +2e- + Fe(OH)2 = Fe + 2H2O
# Fe concentration should be converted to mol/L from g/cm^3 in potential equations g/cm^3 * 1000 / Molar mass Fe = mol/L
EeFe_Fe2p_MP = -1 * DeltaGFe_Fe2_MP* 1000 / (n * F) - math.log(10) * 2 * R * TKelvin / (n * F) * pH + R * TKelvin / (n * F) * math.log(Ce_MP * 1000 / FeMolarMass) # Equilibrium potential for the oxidation reaction at M_P interface
#in i0 equationsthe unit of concentration should be mol/cm^3 according to Dr. Cookthesis and Olga program ( the power 2/3 is not clear and the units does not match the A/cm^2 ) but according to Lisa Lang program Concentration in mol/lit should be multiplied by diameter of pipe/4 (cm). This would make sense in terms of units. I tried all the variations and they make insignificant changes in the results
i0Fe_Fe2p_MP = F * kboltzman * TKelvin / h * (math.exp(-1 * self.ActivationEFe_Fe2_MP / (R * TKelvin)) * math.pow(Ce_MP / FeMolarMass,0.67) * math.exp(-1 * Beta * n * F * EeFe_Fe2p_MP / (R * TKelvin)));
#Precipitation of magnetite at M_P interface Fe3O4 + 2H2O + 2H+ + 2e- = 3Fe(OH)2
EeFe2p_Fe3O4_MP = -1 * DeltaGFe2p_Fe3O4_MP * 1000 / (n * F ) - math.log(10) * 2 * R * TKelvin / (n * F) * pH - 3 * R * TKelvin / (n * F) * math.log(Ce_MP * 1000 / FeMolarMass);
#Hydrogen production 2H+ + 2e- = H2 at M_P;
EeH2_Hp_MP = -1 * DeltaGH2_Hp_MP * 1000 / (n * F ) - math.log(10) * 2 * R * TKelvin / (n * F) * pH - R * TKelvin / (n * F) * math.log((CeH2_MP * 1000 / H2MolarMass) / kHenry) # At M_P interface, CH2_MP converted to mol/l, g/cm^3 * 1000/Molar mass H2 and then to atm: mol/l/KHenry(mol/l.atm) => atm
i0H2_Hp_MP = F * kboltzman * TKelvin / h * (math.exp(-1 * self.ActivationEH2_Hp_MP / (R * TKelvin)) * math.pow(CeHp/1000, 0.67) * math.exp(-1 * Beta * n * F * EeH2_Hp_MP / (R * TKelvin)))
# Mixed Potential at the M_P interface
Ecorr = R * TKelvin /( n * F) * math.log((i0H2_Hp_MP * math.exp(Beta * n * F * EeH2_Hp_MP / (R * TKelvin)) + i0Fe_Fe2p_MP * math.exp(Beta * n * F * EeFe_Fe2p_MP / (R * TKelvin)))/(i0H2_Hp_MP * math.exp(-1 * (1 - Beta) * n * F * EeH2_Hp_MP / (R * TKelvin)) + i0Fe_Fe2p_MP * math.exp(-1 * (1 - Beta) * n * F * EeFe_Fe2p_MP / (R * TKelvin))));
# Current calculations at M_P interface
iFe_Fe2p_MP = i0Fe_Fe2p_MP * (math.exp(Beta * n * F / (R * TKelvin) * (Ecorr - EeFe_Fe2p_MP)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (Ecorr - EeFe_Fe2p_MP)));
iH2_Hp_MP = i0H2_Hp_MP * (math.exp(Beta * n * F / (R * TKelvin) * (EeH2_Hp_MP - Ecorr)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (EeH2_Hp_MP - Ecorr)));
Corrosionrate = iFe_Fe2p_MP * FeMolarMass/ (n * F)
#Adjusting Ce_MP for the Ecorr (in mol/l should be calculated back to g/cm^3)
Ce_MPNew = (math.exp((Ecorr + DeltaGFe_Fe2_MP * 1000 / (n * F ) + math.log(10) * 2 * R * TKelvin / (n * F) * pH) * (n * F) / (R * TKelvin)))* FeMolarMass / 1000
#Solubility at M-P: CMPsat assuming that the solubility at the M_P interface is summation of what is diffused at this interface and the equilibrium concentration considering the developed Ecorr
CMP_Diffused = ((0.476 * (1.101 + PhiOX) * ChiOX * DeltaOX) * Corrosionrate / (PhiOX * DFe * RhoOX * (1 - PhiOX))) * FeMolarMass / 1000
CSat_MPNew = CMP_Diffused + Ce_MPNew
#Dissolution of magnetite at O_B interface Fe3O4 + 2H2O + 2H+ + 2e- = 3 Fe(OH)2
EeFe2p_Fe3O4_OB = -1 * DeltaGFe2p_Fe3O4_OB * 1000/ (n * F) - math.log(10) * 2 * R * TKelvin / (n * F) * pH - 3 * R * TKelvin / (n * F) * math.log(Ce_OB * 1000 / FeMolarMass); # Equilibrium potential for dissolution reaction at O_B interface
i0Fe2p_Fe3O4_OB = F * kboltzman * TKelvin / h * (math.exp(-1 * self.ActivationEFe2p_Fe3O4_OB / (R * TKelvin )) * math.exp(-1 * Beta * n * F * EeFe2p_Fe3O4_OB / (R * TKelvin)));
#Hydrogen consumption at O_B interface 2H+ + 2e- = H2
#Standard DeltaGH2_Hp is zero at any temperature
EeH2_Hp_OB = -1 * DeltaGH2_Hp_OB * 1000 / (n * F ) - math.log(10) * 2 * R * TKelvin / (n * F) * pH - R * TKelvin / (n * F) * math.log((CeH2_OB * 1000 / H2MolarMass) / kHenry); # At O_B interface
i0H2_Hp_OB = F * kboltzman * TKelvin / h * math.exp(-1 * self.ActivationEH2_Hp_OB / (R * TKelvin)) * math.pow(CeHp/1000 , .67) * math.exp(-1 * Beta * n * F * EeH2_Hp_OB/(R * TKelvin));
# Mixed Potential at the O_B interface
Emixed = R * TKelvin / ( n * F) * math.log((i0H2_Hp_OB * math.exp(Beta * n * F * EeH2_Hp_OB / (R * TKelvin)) + i0Fe2p_Fe3O4_OB * math.exp(Beta * n * F * EeFe2p_Fe3O4_OB / (R * TKelvin))) / (i0H2_Hp_OB * math.exp(-1 * (1 - Beta) * n * F * EeH2_Hp_OB / (R * TKelvin)) + i0Fe2p_Fe3O4_OB * math.exp(-1 * (1 - Beta) * n * F * EeFe2p_Fe3O4_OB / (R * TKelvin))));
# Current calculations at O_B interface
iFe3O4_Fe2p_OB = i0Fe2p_Fe3O4_OB * (math.exp(Beta * n * F / (R * TKelvin) * (EeFe2p_Fe3O4_OB - Emixed)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (EeFe2p_Fe3O4_OB - Emixed)));
iH2_Hp_OB = i0H2_Hp_OB * (math.exp(Beta * n * F / (R * TKelvin) * (Emixed - EeH2_Hp_OB)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (Emixed - EeH2_Hp_OB)));
#Adjusting the concentration of iron species at O_B interface (in mol/l should be calculated back to g/cm^3)
Ce_OBNew = ( math.exp((Emixed + DeltaGFe2p_Fe3O4_OB * 1000/ (n * F) + 2 * R * TKelvin * math.log(10) * pH / (n * F)) * (-1) * n * F / (3 * R * TKelvin)))* FeMolarMass / 1000
CSat_OBNew = Ce_OBNew + CMP_Diffused
#Adjusting the dissolution rate constant of magnetite
ked_OB = kd_OB * math.exp(-1 * (1 - Beta) * n * F * (Emixed - EeFe2p_Fe3O4_OB) / (R * TKelvin))
#H2 concentration
CH2generated = 0.005659 * Corrosionrate * (7.32 - PhiOX )
#Diffusivity of H2
DH2 = 0.0000222 * TKelvin * math.exp(-12400 / (R * TKelvin))
# H2 concentration at M_P interface assuming high mass transfer to the bulk
CH2coolant = CH2coolant * 1e-6 * H2MolarMass * 101325 * RhoH2O / (8.314 * 298.15 * 1000) # cm^3/kg water *1e-6 * H2MolarMass *P/(RT) * rowater/1000 = g H2/cm^3 H2O
CeH2_MPNew = CH2coolant + (0.005659 * Corrosionrate * (7.32 - PhiOX) * (hH2 * DeltaOX * ChiOX / (RhoOX * (1 - PhiOX)) + (DH2 * PhiOX))) / (DH2 * hH2 * PhiOX)
# H2 concentration at O_B interface
CeH2_OBNew = CH2coolant + (0.005659 * Corrosionrate * (7.32 - PhiOX ) / hH2)
return CSat_MPNew, CSat_OBNew, CeH2_MPNew, CeH2_OBNew, Ce_OBNew , Ce_MPNew, ked_OB, Emixed, Ecorr
def ConcentrationsCalculator(self,
DeltaOX=None,
DMOX=None,
Ce_MP=None,
Ce_OP=None,
Ce_OB=None,
pH=None):
gibbsValues = GibbsEnergyClass.gibbs(self.Temperature)
DeltaGFe_Fe2_MP = gibbsValues.getDeltaGFe_Fe2()
DeltaGFe2p_Fe3O4_OP = 80.0 #gibbsValues.getDeltaGFe2p_Fe304() *.33 #( for one mole of Fe(OH)3 instead of 3)
DeltaGFe2p_Fe3O4_OB = 80.0 #gibbsValues.getDeltaGFe2p_Fe304()* .33
DeltaGH2_Hp_OP = gibbsValues.getDeltaGfH2_Hp()
DeltaGH2_Hp_OB = gibbsValues.getDeltaGfH2_Hp()
DeltaGH2_Hp_MP = gibbsValues.getDeltaGfH2_Hp()
kHenry = gibbsValues.getH2HenryConstant()
value = Constants.Const()
n = value.n()
F = value.F()
R = value.R()
h = value.h()
kboltzman = value.kboltzman()
Beta = value.Beta()
PhiOX = value.PhiOX()
ChiOX = value.ChiOX()
RhoOX = value.RhoOX()
DFe = value.DFe()
PhiP = value.PhiP()
ChiP = value.ChiP()
RhoP = value.RhoP()
DeltaP = value.DeltaP()
DP = value.DP()
kd_OB = value.kd()
kd_OP = value.kd()
hH2 = 0.3 #value.km() # for now equals to km
Temperature = value.Temperature()
ConcC4H9ONTotal = value.ConcC4H9ONTotal()
FeMolarMass = value.FeMolarMass()
H2MolarMass = value.H2MolarMass()
RhoH2O = value.RhoH2O()
CH2coolant = value.CH2coolant()
TKelvin = self.Temperature + 273.15
if Ce_MP == None:
Ce_MP = self.Ce_MP
if Ce_OP == None:
Ce_OP = self.Ce_OP
# if CeH2_MP == None:
# CeH2_MP = self.CeH2_MP
if Ce_OB == None:
Ce_OB = self.Ce_OB
# if CeH2_OB == None:
# CeH2_OB = self.CeH2_OB
if DeltaOX == None:
DeltaOX = self.DeltaOX
if DMOX == None:
DMOX = self.DMOX
# if CSat_MP == None:
# CSat_MP = self.CSat_MP
# if CSat_OB == None:
# CSat_OB = self.CSat_OB
if pH == None:
pH = self.pH
CeHp = math.pow(10, -pH)
#Diffusivity of H2
DH2 = 0.0000222 * TKelvin * math.exp(-12400 / (R * TKelvin))
Corrosionrate = DMOX
AP = (RhoP * (1 - PhiP) * DH2 * PhiP) / (DeltaP * ChiP)
AOX = (RhoOX * (1 - PhiOX) * DH2 * PhiOX) / (DeltaOX * ChiOX)
# H2 concentration at M_P interface assuming high mass transfer to the bulk
CH2coolant = CH2coolant * 1e-6 * H2MolarMass * 101325 * RhoH2O / (
8.314 * 298.15 * 1000
) # cm^3/kg water *1e-6 * H2MolarMass *P/(RT) * rowater/1000 = g H2/cm^3 H2O
CeH2_MP = CH2coolant + (
0.005659 * Corrosionrate * (7.32 - PhiOX) *
(1 / AOX + 1 / hH2)) + 3.58e-2 * Corrosionrate / AP
CeH2_OP = CH2coolant + (0.005659 * Corrosionrate * (7.32 - PhiOX) *
(1 / AOX + 1 / hH2))
# H2 concentration at O_B interface
CeH2_OB = CH2coolant + (0.005659 * Corrosionrate *
(7.32 - PhiOX) / hH2)
#Dissolution of magnetite at O_B interface Fe3O4 + 2H2O + 2H+ + 2e- = 3 Fe(OH)2
nFe2p_Fe3O4_OB = 2.0
EeFe2p_Fe3O4_OB = -1 * DeltaGFe2p_Fe3O4_OB * 1000 / (
nFe2p_Fe3O4_OB * F
) - math.log(10) * 2 * R * TKelvin / (
nFe2p_Fe3O4_OB * F
) * pH - 3 * R * TKelvin / (nFe2p_Fe3O4_OB * F) * math.log(
Ce_OB * 1000 / 90.0
) # Equilibrium potential for dissolution reaction at O_B interface
i0Fe2p_Fe3O4_OB = F * kboltzman * TKelvin / h * (
math.exp(-1 * self.ActivationEFe2p_Fe3O4_OB /
(R * TKelvin)) * math.pow(Ce_OB * 1000 / 90.0, 1.0) *
math.exp(-1 * Beta * nFe2p_Fe3O4_OB * F * EeFe2p_Fe3O4_OB /
(R * TKelvin)))
#Hydrogen consumption at O_B interface 2H+ + 2e- = H2
nH2_Hp_OB = 2.0
EeH2_Hp_OB = -1 * DeltaGH2_Hp_OB * 1000 / (
nH2_Hp_OB * F) - math.log(10) * 2 * R * TKelvin * pH / (
nH2_Hp_OB * F) - R * TKelvin / (nH2_Hp_OB * F) * math.log(
(CeH2_OB * 1000 / 2.0)) # At O_B interface
i0H2_Hp_OB = F * kboltzman * TKelvin / h * math.exp(
-1 * self.ActivationEH2_Hp_OB / (R * TKelvin)) * math.pow(
CeH2_OB * 1000 / H2MolarMass, 1.0) * math.exp(
-1 * Beta * nH2_Hp_OB * F * EeH2_Hp_OB / (R * TKelvin))
# Mixed Potential at the O_B interface
i0_H1 = i0H2_Hp_OB * math.exp(Beta * nH2_Hp_OB * F * EeH2_Hp_OB /
(R * TKelvin))
i0_H2 = i0H2_Hp_OB * math.exp(-1 *
(1 - Beta) * nH2_Hp_OB * F * EeH2_Hp_OB /
(R * TKelvin))
i0_Fe1 = i0Fe2p_Fe3O4_OB * math.exp(
Beta * nFe2p_Fe3O4_OB * F * EeFe2p_Fe3O4_OB / (R * TKelvin))
i0_Fe2 = i0Fe2p_Fe3O4_OB * math.exp(
-1 * (1 - Beta) * nFe2p_Fe3O4_OB * F * EeFe2p_Fe3O4_OB /
(R * TKelvin))
#just for Beta = 0.5
Emixed = R * TKelvin / (nFe2p_Fe3O4_OB * F * Beta * 2) * math.log(
(i0_H1 + i0_Fe1) / (i0_H2 + i0_Fe2))
# print EeFe2p_Fe3O4_OB,i0Fe2p_Fe3O4_OB,EeH2_Hp_OB,i0H2_Hp_OB, Emixed,Corrosionrate
#Adjusting the concentration of iron species at O_B interface (in mol/l should be calculated back to g/cm^3)
CSat_OBNew = (math.exp(
(Emixed + DeltaGFe2p_Fe3O4_OB * 1000 /
(nFe2p_Fe3O4_OB * F) + 2 * R * TKelvin * math.log(10) * pH /
(nFe2p_Fe3O4_OB * F)) * (-1) * nFe2p_Fe3O4_OB * F /
(3 * R * TKelvin))) * 90 / 1000
# print CSat_OBNew
#Adjusting the dissolution rate constant of magnetite
ked_OB = kd_OB * math.exp(-1 * (1 - Beta) * nFe2p_Fe3O4_OB * F *
(Emixed - EeFe2p_Fe3O4_OB) / (R * TKelvin))
# # Fe concentration at M_P interface
#
# CMP_Diffused = ((0.476 * (1.101 + PhiOX) * ChiOX * DeltaOX) * Corrosionrate / (PhiOX * DFe * RhoOX * (1 - PhiOX))) * FeMolarMass / 1000
#
# Ce_MP = CSat_MP - CMP_Diffused
# At O_P interface
#Precipitation of magnetite at O_P interface Fe3O4 + 2H2O + 2H+ + 2e- = 3Fe(OH)2
# R * 6 is due to the fact that the n=1/3 in these equation instead of 2
nFe2p_Fe3O4_OP = .33 #1/3
EeFe2p_Fe3O4_OP = -1 * DeltaGFe2p_Fe3O4_OP * 1000 / (2 * F) - math.log(
10) * R * TKelvin / (F) * pH - 3 * R * TKelvin / (
2 * F) * math.log(Ce_OP * 1000 / FeMolarMass)
i0Fe2p_Fe3O4_OP = F * kboltzman * TKelvin / h * (
math.exp(-1 * self.ActivationEFe2p_Fe3O4_OP /
(R * TKelvin)) * math.pow(Ce_OP * 1000 / 90.0, 1.0) *
math.exp(-1 * Beta * nFe2p_Fe3O4_OP * F * EeFe2p_Fe3O4_OP /
(R * TKelvin)))
#Hydrogen production 2H+ + 2e- = H2 at O_P;
nH2_Hp_OP = .33 #1/3
EeH2_Hp_OP = -1 * DeltaGH2_Hp_OP * 1000 / (2 * F) - math.log(
10) * R * TKelvin * pH / (F) - R * TKelvin / (2 * F) * math.log(
(CeH2_OP * 1000 / 2.0))
i0H2_Hp_OP = F * kboltzman * TKelvin / h * (
math.exp(-1 * self.ActivationEH2_Hp_OP / (R * TKelvin)) *
math.pow(CeH2_OP * 1000 / H2MolarMass, 1.0) *
math.exp(-1 * Beta * nH2_Hp_OP * F * EeH2_Hp_OP / (R * TKelvin)))
# Mixed Potential at the O_P interface
i0_H1OP = i0H2_Hp_OP * math.exp(Beta * nH2_Hp_OP * F * EeH2_Hp_OP /
(R * TKelvin))
i0_H2OP = i0H2_Hp_OP * math.exp(
-1 * (1 - Beta) * nH2_Hp_OP * F * EeH2_Hp_OP / (R * TKelvin))
i0_Fe1OP = i0Fe2p_Fe3O4_OP * math.exp(
Beta * nFe2p_Fe3O4_OP * F * EeFe2p_Fe3O4_OP / (R * TKelvin))
i0_Fe2OP = i0Fe2p_Fe3O4_OP * math.exp(
-1 * (1 - Beta) * nFe2p_Fe3O4_OP * F * EeFe2p_Fe3O4_OP /
(R * TKelvin))
#just for Beta = 0.5
EmixedOP = R * TKelvin / (nFe2p_Fe3O4_OP * F * Beta * 2) * math.log(
(i0_H1OP + i0_Fe1OP) / (i0_H2OP + i0_Fe2OP))
#Adjusting the dissolution rate constant of magnetite
ked_OP = kd_OP * math.exp(-1 * (1 - Beta) * nFe2p_Fe3O4_OP * F *
(EmixedOP - EeFe2p_Fe3O4_OP) / (R * TKelvin))
#Adjusting the concentration of iron species at O_P interface (in mol/l should be calculated back to g/cm^3)
CSat_OPNew = (math.exp(
(EmixedOP + DeltaGFe2p_Fe3O4_OP * 1000 /
(2 * F) + R * TKelvin * math.log(10) * pH / (F)) * (-1) * 2 * F /
(3 * R * TKelvin))) * 90 / 1000
Ce_OPNew = CSat_OPNew + (0.476 * (1 - PhiOX) * Corrosionrate /
ked_OP) * 90 / 1000
Ce_MPNew = (Corrosionrate * (value.ChiP() * value.DeltaP()) /
(value.PhiP() * value.DP() * value.RhoP() *
(1 - value.PhiP())) + Ce_OPNew * 1000 / 90) * 56 / 1000
# print EeFe2p_Fe3O4_OP,i0Fe2p_Fe3O4_OP,EeH2_Hp_OP,i0H2_Hp_OP, EmixedOP,Ce_MPNew,Ce_OP
# # At M-P interface
#
# EeFe_Fe2p_MP = -1 * DeltaGFe_Fe2_MP * 1000 / (n * F) - math.log(10) * 2 * R * TKelvin * pH / (n * F) + R * TKelvin / (n * F) * math.log(Ce_MP * 1000 / FeMolarMass) # Equilibrium potential for the oxidation reaction at M_P interface, concentration unit changed from g/cm^3 to mol/L
# #in i0 equationsthe unit of concentration should be mol/cm^3 according to Dr. Cook thesis and Olga program ( the power 2/3 is not clear and the units does not match the A/cm^2 ) but according to Lisa Lang program Concentration in mol/lit should be multiplied by diameter of pipe/4 (cm). This would make sense in terms of units. I tried all the variations and they make insignificant changes in the results
# i0Fe_Fe2p_MP = (F * kboltzman * TKelvin / h) * (math.exp(-1 * self.ActivationEFe_Fe2_MP / (R * TKelvin)) * math.pow(Ce_MP * 1000 / FeMolarMass,1.0) * math.exp(-1 * Beta * n * F * EeFe_Fe2p_MP / (R * TKelvin)));
#Precipitation of magnetite at M_P interface Fe3O4 + 2H2O + 2H+ + 2e- = 3Fe(OH)2
# EeFe2p_Fe3O4_MP = -1 * DeltaGFe2p_Fe3O4_MP * 1000 / (n * F ) - math.log(10) * 2 * R * TKelvin / (n * F) * pH - 3 * R * TKelvin / (n * F) * math.log(Ce_MP * 1000 / FeMolarMass);
nFe_Fe2p_MP = 2
EeFe_Fe2p_MP = -1 * DeltaGFe_Fe2_MP * 1000 / (
nFe_Fe2p_MP * F
) - math.log(10) * 2 * R * TKelvin / (
nFe_Fe2p_MP * F
) * pH + R * TKelvin / (nFe_Fe2p_MP * F) * math.log(
Ce_MPNew * 1000 / FeMolarMass
) # Equilibrium potential for the oxidation reaction at M_P interface
#in i0 equationsthe unit of concentration should be mol/cm^3 according to Dr. Cookthesis and Olga program ( the power 2/3 is not clear and the units does not match the A/cm^2 ) but according to Lisa Lang program Concentration in mol/lit should be multiplied by diameter of pipe/4 (cm). This would make sense in terms of units. I tried all the variations and they make insignificant changes in the results
i0Fe_Fe2p_MP = F * kboltzman * TKelvin / h * (
math.exp(-1 * self.ActivationEFe_Fe2_MP / (R * TKelvin)) *
math.pow(Ce_MPNew * 1000 / FeMolarMass, 1.0) *
math.exp(-1 * Beta * nFe_Fe2p_MP * F * EeFe_Fe2p_MP /
(R * TKelvin)))
#Hydrogen production 2H+ + 2e- = H2 at M_P;
nH2_Hp_MP = 2
EeH2_Hp_MP = -1 * DeltaGH2_Hp_MP * 1000 / (
nH2_Hp_MP * F
) - math.log(10) * 2 * R * TKelvin * pH / (
nH2_Hp_MP * F
) - R * TKelvin / (nH2_Hp_MP * F) * math.log(
(CeH2_MP * 1000 / 2.0)
) # At M_P interface, CH2_MP converted to mol/l, g/cm^3 * 1000/Molar mass H2 and then to atm: mol/l/KHenry(mol/l.atm) => atm
i0H2_Hp_MP = F * kboltzman * TKelvin / h * (
math.exp(-1 * self.ActivationEH2_Hp_MP / (R * TKelvin)) *
math.pow(CeH2_MP * 1000 / H2MolarMass, 1.0) *
math.exp(-1 * Beta * nH2_Hp_MP * F * EeH2_Hp_MP / (R * TKelvin)))
# having the corrosion rate, the Ecorr can be calculated by solving the following equation:
# Corrosionrate = iFe_Fe2p_MP * FeMolarMass/ (n * F)
# iFe_Fe2p_MP = i0Fe_Fe2p_MP * (math.exp(Beta * n * F / (R * TKelvin) * (Ecorr - EeFe_Fe2p_MP)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (Ecorr - EeFe_Fe2p_MP)))
# i0Fe_Fe2p_MP * (math.exp(Beta * n * F / (R * TKelvin) * (Ecorr - EeFe_Fe2p_MP)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (Ecorr - EeFe_Fe2p_MP)))* FeMolarMass/ (n * F) - Corrosionrate =0.0
A1 = Corrosionrate * (n * F) / (FeMolarMass * i0Fe_Fe2p_MP)
C1 = Beta * n * F / (R * TKelvin)
C2 = math.exp(C1 * (EeFe_Fe2p_MP))
X1 = (A1 + math.pow(math.pow(A1, 2) + 4, 0.5)) / 2
X2 = (A1 - math.pow(math.pow(A1, 2) + 4, 0.5)) / 2
# C1 = Beta * n * F / (R * TKelvin)
# C2 = (1-Beta) * n * F / (R * TKelvin)
# C3 = math.exp(C1 * (EeFe_Fe2p_MP))
# C4 = math.exp(C2 * (EeFe_Fe2p_MP))
# print C1+C2, A1*C3, C2, C3*C4
if X2 > 0:
Ecorr = math.log(X2) / C1 + EeFe_Fe2p_MP
elif X1 > 0:
Ecorr = math.log(X1) / C1 + EeFe_Fe2p_MP
else:
print('error')
if (X1 > 0 and X2 > 0):
Ecorr = math.log(X1) / C1 + EeFe_Fe2p_MP
# print A1,C1, C2, X1, X2, Ecorr, Emixed,
# print EeFe_Fe2p_MP,i0Fe_Fe2p_MP,EeH2_Hp_MP,i0H2_Hp_MP,Ecorr
# Current calculations at M_P interface
# iH2_Hp_MP = i0H2_Hp_MP * (math.exp(Beta * n * F / (R * TKelvin) * (EeH2_Hp_MP - Ecorr)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (EeH2_Hp_MP - Ecorr)))
#Adjusting Ce_MP for the Ecorr (in mol/l should be calculated back to g/cm^3)
CSat_MPNew = (math.exp(
(Ecorr - EeFe_Fe2p_MP + DeltaGFe_Fe2_MP * 1000 /
(n * F) + math.log(10) * 2 * R * TKelvin * pH / (n * F)) *
(1) * n * F / (R * TKelvin))) * FeMolarMass / 1000
# CSat_MPNew = Ce_MPNew + CMP_Diffused
#Solubility at M-P: CMPsat assuming that the solubility at the M_P interface is summation of what is diffused at this interface and the equilibrium concentration considering the developed Ecorr
# Current calculations at O_B interface
# iFe3O4_Fe2p_OB = i0Fe2p_Fe3O4_OB * (math.exp(Beta * n * F / (R * TKelvin) * (EeFe2p_Fe3O4_OB - Emixed)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (EeFe2p_Fe3O4_OB - Emixed)));
# iH2_Hp_OB = i0H2_Hp_OB * (math.exp(Beta * n * F / (R * TKelvin) * (Emixed - EeH2_Hp_OB)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (Emixed - EeH2_Hp_OB)));
return CSat_MPNew, CSat_OPNew, CSat_OBNew, ked_OB, ked_OP, Emixed, EmixedOP, Ecorr, CeH2_MP, CeH2_OP, CeH2_OB, Corrosionrate, Ce_MPNew, Ce_OPNew
def ConcentrationsCalculator(self,
Ce_MP=None,
CeH2_MP=None,
Ce_OB=None,
CeH2_OB=None,
DeltaOX=None):
gibbsValues = GibbsEnergyClass.gibbs(self.Temperature)
DeltaGFe_Fe2_MP = gibbsValues.getDeltaGFe_Fe2()
DeltaGFe2p_Fe3O4_MP = gibbsValues.getDeltaGFe2p_Fe304()
DeltaGFe2p_Fe3O4_OB = gibbsValues.getDeltaGFe2p_Fe304()
DeltaGH2_Hp_MP = gibbsValues.getDeltaGfH2_Hp()
DeltaGH2_Hp_OB = gibbsValues.getDeltaGfH2_Hp()
kHenry = gibbsValues.getH2HenryConstant()
# PHvalue = PHCalculator.equilibriumConstants(self.Temperature, self.ConcC4H9ONTotal)
# pH = PHvalue.PHCalculation()
pH = 6.88 # for now not using the PHCalculator function
CeHp = math.pow(10, -pH)
#pH = 9.58
value = Constants.Const()
n = value.n()
F = value.F()
R = value.R()
h = value.h()
kboltzman = value.kboltzman()
Beta = value.Beta()
PhiOX = value.PhiOX()
ChiOX = value.ChiOX()
RhoOX = value.RhoOX()
DFe = value.DFe()
kd_OB = value.kd()
hH2 = value.km() # for now equals to km
Temperature = value.Temperature()
ConcC4H9ONTotal = value.ConcC4H9ONTotal()
FeMolarMass = value.FeMolarMass()
H2MolarMass = value.H2MolarMass()
RhoH2O = value.RhoH2O()
CH2coolant = value.CH2coolant()
TKelvin = self.Temperature + 273.15
if Ce_MP == None:
Ce_MP = self.Ce_MP
if CeH2_MP == None:
CeH2_MP = self.CeH2_MP
if Ce_OB == None:
Ce_OB = self.Ce_OB
if CeH2_OB == None:
CeH2_OB = self.CeH2_OB
if DeltaOX == None:
DeltaOX = self.DeltaOX
# At the oxide-Bulk interface O_B
# Corrosion reaction at the M_P interface; All the reaction are considered in the following direction: O + ne- -> R
# 2H+ +2e- + Fe(OH)2 = Fe + 2H2O
# Fe concentration should be converted to mol/L from g/cm^3 in potential equations g/cm^3 * 1000 / Molar mass Fe = mol/L
EeFe_Fe2p_MP = -1 * DeltaGFe_Fe2_MP * 1000 / (n * F) - math.log(
10
) * 2 * R * TKelvin / (n * F) * pH + R * TKelvin / (n * F) * math.log(
Ce_MP * 1000 / FeMolarMass
) # Equilibrium potential for the oxidation reaction at M_P interface
#in i0 equationsthe unit of concentration should be mol/cm^3 according to Dr. Cookthesis and Olga program ( the power 2/3 is not clear and the units does not match the A/cm^2 ) but according to Lisa Lang program Concentration in mol/lit should be multiplied by diameter of pipe/4 (cm). This would make sense in terms of units. I tried all the variations and they make insignificant changes in the results
i0Fe_Fe2p_MP = F * kboltzman * TKelvin / h * (
math.exp(-1 * self.ActivationEFe_Fe2_MP /
(R * TKelvin)) * math.pow(Ce_MP / FeMolarMass, 0.67) *
math.exp(-1 * Beta * n * F * EeFe_Fe2p_MP / (R * TKelvin)))
#Precipitation of magnetite at M_P interface Fe3O4 + 2H2O + 2H+ + 2e- = 3Fe(OH)2
EeFe2p_Fe3O4_MP = -1 * DeltaGFe2p_Fe3O4_MP * 1000 / (n * F) - math.log(
10) * 2 * R * TKelvin / (n * F) * pH - 3 * R * TKelvin / (
n * F) * math.log(Ce_MP * 1000 / FeMolarMass)
#Hydrogen production 2H+ + 2e- = H2 at M_P;
EeH2_Hp_MP = -1 * DeltaGH2_Hp_MP * 1000 / (n * F) - math.log(
10
) * 2 * R * TKelvin / (n * F) * pH - R * TKelvin / (n * F) * math.log(
(CeH2_MP * 1000 / H2MolarMass) / kHenry
) # At M_P interface, CH2_MP converted to mol/l, g/cm^3 * 1000/Molar mass H2 and then to atm: mol/l/KHenry(mol/l.atm) => atm
i0H2_Hp_MP = F * kboltzman * TKelvin / h * (
math.exp(-1 * self.ActivationEH2_Hp_MP /
(R * TKelvin)) * math.pow(CeHp / 1000, 0.67) *
math.exp(-1 * Beta * n * F * EeH2_Hp_MP / (R * TKelvin)))
# Mixed Potential at the M_P interface
Ecorr = R * TKelvin / (n * F) * math.log(
(i0H2_Hp_MP * math.exp(Beta * n * F * EeH2_Hp_MP / (R * TKelvin)) +
i0Fe_Fe2p_MP * math.exp(Beta * n * F * EeFe_Fe2p_MP /
(R * TKelvin))) /
(i0H2_Hp_MP * math.exp(-1 * (1 - Beta) * n * F * EeH2_Hp_MP /
(R * TKelvin)) +
i0Fe_Fe2p_MP * math.exp(-1 * (1 - Beta) * n * F * EeFe_Fe2p_MP /
(R * TKelvin))))
# Current calculations at M_P interface
iFe_Fe2p_MP = i0Fe_Fe2p_MP * (math.exp(Beta * n * F / (R * TKelvin) *
(Ecorr - EeFe_Fe2p_MP)) -
math.exp(-1 * (1 - Beta) * n * F /
(R * TKelvin) *
(Ecorr - EeFe_Fe2p_MP)))
iH2_Hp_MP = i0H2_Hp_MP * (math.exp(Beta * n * F / (R * TKelvin) *
(EeH2_Hp_MP - Ecorr)) -
math.exp(-1 * (1 - Beta) * n * F /
(R * TKelvin) *
(EeH2_Hp_MP - Ecorr)))
Corrosionrate = iFe_Fe2p_MP * FeMolarMass / (n * F)
#Adjusting Ce_MP for the Ecorr (in mol/l should be calculated back to g/cm^3)
Ce_MPNew = (math.exp(
(Ecorr + DeltaGFe_Fe2_MP * 1000 /
(n * F) + math.log(10) * 2 * R * TKelvin / (n * F) * pH) *
(n * F) / (R * TKelvin))) * FeMolarMass / 1000
#Solubility at M-P: CMPsat assuming that the solubility at the M_P interface is summation of what is diffused at this interface and the equilibrium concentration considering the developed Ecorr
CMP_Diffused = ((0.476 *
(1.101 + PhiOX) * ChiOX * DeltaOX) * Corrosionrate /
(PhiOX * DFe * RhoOX *
(1 - PhiOX))) * FeMolarMass / 1000
CSat_MPNew = CMP_Diffused + Ce_MPNew
#Dissolution of magnetite at O_B interface Fe3O4 + 2H2O + 2H+ + 2e- = 3 Fe(OH)2
EeFe2p_Fe3O4_OB = -1 * DeltaGFe2p_Fe3O4_OB * 1000 / (n * F) - math.log(
10) * 2 * R * TKelvin / (n * F) * pH - 3 * R * TKelvin / (
n * F) * math.log(Ce_OB * 1000 / FeMolarMass)
# Equilibrium potential for dissolution reaction at O_B interface
i0Fe2p_Fe3O4_OB = F * kboltzman * TKelvin / h * (math.exp(
-1 * self.ActivationEFe2p_Fe3O4_OB /
(R * TKelvin)) * math.exp(-1 * Beta * n * F * EeFe2p_Fe3O4_OB /
(R * TKelvin)))
#Hydrogen consumption at O_B interface 2H+ + 2e- = H2
#Standard DeltaGH2_Hp is zero at any temperature
EeH2_Hp_OB = -1 * DeltaGH2_Hp_OB * 1000 / (
n * F) - math.log(10) * 2 * R * TKelvin / (
n * F) * pH - R * TKelvin / (n * F) * math.log(
(CeH2_OB * 1000 / H2MolarMass) / kHenry)
# At O_B interface
i0H2_Hp_OB = F * kboltzman * TKelvin / h * math.exp(
-1 * self.ActivationEH2_Hp_OB / (R * TKelvin)) * math.pow(
CeHp / 1000, .67) * math.exp(-1 * Beta * n * F * EeH2_Hp_OB /
(R * TKelvin))
# Mixed Potential at the O_B interface
Emixed = R * TKelvin / (n * F) * math.log(
(i0H2_Hp_OB * math.exp(Beta * n * F * EeH2_Hp_OB / (R * TKelvin)) +
i0Fe2p_Fe3O4_OB * math.exp(Beta * n * F * EeFe2p_Fe3O4_OB /
(R * TKelvin))) /
(i0H2_Hp_OB * math.exp(-1 * (1 - Beta) * n * F * EeH2_Hp_OB /
(R * TKelvin)) +
i0Fe2p_Fe3O4_OB * math.exp(-1 *
(1 - Beta) * n * F * EeFe2p_Fe3O4_OB /
(R * TKelvin))))
# Current calculations at O_B interface
iFe3O4_Fe2p_OB = i0Fe2p_Fe3O4_OB * (math.exp(
Beta * n * F / (R * TKelvin) *
(EeFe2p_Fe3O4_OB - Emixed)) - math.exp(-1 * (1 - Beta) * n * F /
(R * TKelvin) *
(EeFe2p_Fe3O4_OB - Emixed)))
iH2_Hp_OB = i0H2_Hp_OB * (math.exp(Beta * n * F / (R * TKelvin) *
(Emixed - EeH2_Hp_OB)) -
math.exp(-1 * (1 - Beta) * n * F /
(R * TKelvin) *
(Emixed - EeH2_Hp_OB)))
#Adjusting the concentration of iron species at O_B interface (in mol/l should be calculated back to g/cm^3)
Ce_OBNew = (math.exp(
(Emixed + DeltaGFe2p_Fe3O4_OB * 1000 /
(n * F) + 2 * R * TKelvin * math.log(10) * pH / (n * F)) *
(-1) * n * F / (3 * R * TKelvin))) * FeMolarMass / 1000
CSat_OBNew = Ce_OBNew + CMP_Diffused
#Adjusting the dissolution rate constant of magnetite
ked_OB = kd_OB * math.exp(-1 * (1 - Beta) * n * F *
(Emixed - EeFe2p_Fe3O4_OB) / (R * TKelvin))
#H2 concentration
CH2generated = 0.005659 * Corrosionrate * (7.32 - PhiOX)
#Diffusivity of H2
DH2 = 0.0000222 * TKelvin * math.exp(-12400 / (R * TKelvin))
# H2 concentration at M_P interface assuming high mass transfer to the bulk
CH2coolant = CH2coolant * 1e-6 * H2MolarMass * 101325 * RhoH2O / (
8.314 * 298.15 * 1000
) # cm^3/kg water *1e-6 * H2MolarMass *P/(RT) * rowater/1000 = g H2/cm^3 H2O
CeH2_MPNew = CH2coolant + (0.005659 * Corrosionrate * (7.32 - PhiOX) *
(hH2 * DeltaOX * ChiOX / (RhoOX *
(1 - PhiOX)) +
(DH2 * PhiOX))) / (DH2 * hH2 * PhiOX)
# H2 concentration at O_B interface
CeH2_OBNew = CH2coolant + (0.005659 * Corrosionrate *
(7.32 - PhiOX) / hH2)
return CSat_MPNew, CSat_OBNew, CeH2_MPNew, CeH2_OBNew, Ce_OBNew, Ce_MPNew, ked_OB, Emixed, Ecorr
def ConcentrationsCalculator(self, DeltaOX=None, DMOX=None, Ce_MP=None, Ce_OP=None, Ce_OB=None, pH=None):
gibbsValues = GibbsEnergyClass.gibbs(self.Temperature)
DeltaGFe_Fe2_MP = gibbsValues.getDeltaGFe_Fe2()
DeltaGFe2p_Fe3O4_OP = 80.0 # gibbsValues.getDeltaGFe2p_Fe304() *.33 #( for one mole of Fe(OH)3 instead of 3)
DeltaGFe2p_Fe3O4_OB = 80.0 # gibbsValues.getDeltaGFe2p_Fe304()* .33
DeltaGH2_Hp_OP = gibbsValues.getDeltaGfH2_Hp()
DeltaGH2_Hp_OB = gibbsValues.getDeltaGfH2_Hp()
DeltaGH2_Hp_MP = gibbsValues.getDeltaGfH2_Hp()
kHenry = gibbsValues.getH2HenryConstant()
value = Constants.Const()
n = value.n()
F = value.F()
R = value.R()
h = value.h()
kboltzman = value.kboltzman()
Beta = value.Beta()
PhiOX = value.PhiOX()
ChiOX = value.ChiOX()
RhoOX = value.RhoOX()
DFe = value.DFe()
PhiP = value.PhiP()
ChiP = value.ChiP()
RhoP = value.RhoP()
DeltaP = value.DeltaP()
DP = value.DP()
kd_OB = value.kd()
kd_OP = value.kd()
hH2 = 0.3 # value.km() # for now equals to km
Temperature = value.Temperature()
ConcC4H9ONTotal = value.ConcC4H9ONTotal()
FeMolarMass = value.FeMolarMass()
H2MolarMass = value.H2MolarMass()
RhoH2O = value.RhoH2O()
CH2coolant = value.CH2coolant()
TKelvin = self.Temperature + 273.15
if Ce_MP == None:
Ce_MP = self.Ce_MP
if Ce_OP == None:
Ce_OP = self.Ce_OP
# if CeH2_MP == None:
# CeH2_MP = self.CeH2_MP
if Ce_OB == None:
Ce_OB = self.Ce_OB
# if CeH2_OB == None:
# CeH2_OB = self.CeH2_OB
if DeltaOX == None:
DeltaOX = self.DeltaOX
if DMOX == None:
DMOX = self.DMOX
# if CSat_MP == None:
# CSat_MP = self.CSat_MP
# if CSat_OB == None:
# CSat_OB = self.CSat_OB
if pH == None:
pH = self.pH
CeHp = math.pow(10, -pH)
# Diffusivity of H2
DH2 = 0.0000222 * TKelvin * math.exp(-12400 / (R * TKelvin))
Corrosionrate = DMOX
AP = (RhoP * (1 - PhiP) * DH2 * PhiP) / (DeltaP * ChiP)
AOX = (RhoOX * (1 - PhiOX) * DH2 * PhiOX) / (DeltaOX * ChiOX)
# H2 concentration at M_P interface assuming high mass transfer to the bulk
CH2coolant = (
CH2coolant * 1e-6 * H2MolarMass * 101325 * RhoH2O / (8.314 * 298.15 * 1000)
) # cm^3/kg water *1e-6 * H2MolarMass *P/(RT) * rowater/1000 = g H2/cm^3 H2O
CeH2_MP = (
CH2coolant
+ (0.005659 * Corrosionrate * (7.32 - PhiOX) * (1 / AOX + 1 / hH2))
+ 3.58e-2 * Corrosionrate / AP
)
CeH2_OP = CH2coolant + (0.005659 * Corrosionrate * (7.32 - PhiOX) * (1 / AOX + 1 / hH2))
# H2 concentration at O_B interface
CeH2_OB = CH2coolant + (0.005659 * Corrosionrate * (7.32 - PhiOX) / hH2)
# Dissolution of magnetite at O_B interface Fe3O4 + 2H2O + 2H+ + 2e- = 3 Fe(OH)2
nFe2p_Fe3O4_OB = 2.0
EeFe2p_Fe3O4_OB = (
-1 * DeltaGFe2p_Fe3O4_OB * 1000 / (nFe2p_Fe3O4_OB * F)
- math.log(10) * 2 * R * TKelvin / (nFe2p_Fe3O4_OB * F) * pH
- 3 * R * TKelvin / (nFe2p_Fe3O4_OB * F) * math.log(Ce_OB * 1000 / 90.0)
) # Equilibrium potential for dissolution reaction at O_B interface
i0Fe2p_Fe3O4_OB = (
F
* kboltzman
* TKelvin
/ h
* (
math.exp(-1 * self.ActivationEFe2p_Fe3O4_OB / (R * TKelvin))
* math.pow(Ce_OB * 1000 / 90.0, 1.0)
* math.exp(-1 * Beta * nFe2p_Fe3O4_OB * F * EeFe2p_Fe3O4_OB / (R * TKelvin))
)
)
# Hydrogen consumption at O_B interface 2H+ + 2e- = H2
nH2_Hp_OB = 2.0
EeH2_Hp_OB = (
-1 * DeltaGH2_Hp_OB * 1000 / (nH2_Hp_OB * F)
- math.log(10) * 2 * R * TKelvin * pH / (nH2_Hp_OB * F)
- R * TKelvin / (nH2_Hp_OB * F) * math.log((CeH2_OB * 1000 / 2.0))
) # At O_B interface
i0H2_Hp_OB = (
F
* kboltzman
* TKelvin
/ h
* math.exp(-1 * self.ActivationEH2_Hp_OB / (R * TKelvin))
* math.pow(CeH2_OB * 1000 / H2MolarMass, 1.0)
* math.exp(-1 * Beta * nH2_Hp_OB * F * EeH2_Hp_OB / (R * TKelvin))
)
# Mixed Potential at the O_B interface
i0_H1 = i0H2_Hp_OB * math.exp(Beta * nH2_Hp_OB * F * EeH2_Hp_OB / (R * TKelvin))
i0_H2 = i0H2_Hp_OB * math.exp(-1 * (1 - Beta) * nH2_Hp_OB * F * EeH2_Hp_OB / (R * TKelvin))
i0_Fe1 = i0Fe2p_Fe3O4_OB * math.exp(Beta * nFe2p_Fe3O4_OB * F * EeFe2p_Fe3O4_OB / (R * TKelvin))
i0_Fe2 = i0Fe2p_Fe3O4_OB * math.exp(-1 * (1 - Beta) * nFe2p_Fe3O4_OB * F * EeFe2p_Fe3O4_OB / (R * TKelvin))
# just for Beta = 0.5
Emixed = R * TKelvin / (nFe2p_Fe3O4_OB * F * Beta * 2) * math.log((i0_H1 + i0_Fe1) / (i0_H2 + i0_Fe2))
# print EeFe2p_Fe3O4_OB,i0Fe2p_Fe3O4_OB,EeH2_Hp_OB,i0H2_Hp_OB, Emixed,Corrosionrate
# Adjusting the concentration of iron species at O_B interface (in mol/l should be calculated back to g/cm^3)
CSat_OBNew = (
(
math.exp(
(
Emixed
+ DeltaGFe2p_Fe3O4_OB * 1000 / (nFe2p_Fe3O4_OB * F)
+ 2 * R * TKelvin * math.log(10) * pH / (nFe2p_Fe3O4_OB * F)
)
* (-1)
* nFe2p_Fe3O4_OB
* F
/ (3 * R * TKelvin)
)
)
* 90
/ 1000
)
# print CSat_OBNew
# Adjusting the dissolution rate constant of magnetite
ked_OB = kd_OB * math.exp(-1 * (1 - Beta) * nFe2p_Fe3O4_OB * F * (Emixed - EeFe2p_Fe3O4_OB) / (R * TKelvin))
# # Fe concentration at M_P interface
#
# CMP_Diffused = ((0.476 * (1.101 + PhiOX) * ChiOX * DeltaOX) * Corrosionrate / (PhiOX * DFe * RhoOX * (1 - PhiOX))) * FeMolarMass / 1000
#
# Ce_MP = CSat_MP - CMP_Diffused
# At O_P interface
# Precipitation of magnetite at O_P interface Fe3O4 + 2H2O + 2H+ + 2e- = 3Fe(OH)2
# R * 6 is due to the fact that the n=1/3 in these equation instead of 2
nFe2p_Fe3O4_OP = 0.33 # 1/3
EeFe2p_Fe3O4_OP = (
-1 * DeltaGFe2p_Fe3O4_OP * 1000 / (2 * F)
- math.log(10) * R * TKelvin / (F) * pH
- 3 * R * TKelvin / (2 * F) * math.log(Ce_OP * 1000 / FeMolarMass)
)
i0Fe2p_Fe3O4_OP = (
F
* kboltzman
* TKelvin
/ h
* (
math.exp(-1 * self.ActivationEFe2p_Fe3O4_OP / (R * TKelvin))
* math.pow(Ce_OP * 1000 / 90.0, 1.0)
* math.exp(-1 * Beta * nFe2p_Fe3O4_OP * F * EeFe2p_Fe3O4_OP / (R * TKelvin))
)
)
# Hydrogen production 2H+ + 2e- = H2 at O_P;
nH2_Hp_OP = 0.33 # 1/3
EeH2_Hp_OP = (
-1 * DeltaGH2_Hp_OP * 1000 / (2 * F)
- math.log(10) * R * TKelvin * pH / (F)
- R * TKelvin / (2 * F) * math.log((CeH2_OP * 1000 / 2.0))
)
i0H2_Hp_OP = (
F
* kboltzman
* TKelvin
/ h
* (
math.exp(-1 * self.ActivationEH2_Hp_OP / (R * TKelvin))
* math.pow(CeH2_OP * 1000 / H2MolarMass, 1.0)
* math.exp(-1 * Beta * nH2_Hp_OP * F * EeH2_Hp_OP / (R * TKelvin))
)
)
# Mixed Potential at the O_P interface
i0_H1OP = i0H2_Hp_OP * math.exp(Beta * nH2_Hp_OP * F * EeH2_Hp_OP / (R * TKelvin))
i0_H2OP = i0H2_Hp_OP * math.exp(-1 * (1 - Beta) * nH2_Hp_OP * F * EeH2_Hp_OP / (R * TKelvin))
i0_Fe1OP = i0Fe2p_Fe3O4_OP * math.exp(Beta * nFe2p_Fe3O4_OP * F * EeFe2p_Fe3O4_OP / (R * TKelvin))
i0_Fe2OP = i0Fe2p_Fe3O4_OP * math.exp(-1 * (1 - Beta) * nFe2p_Fe3O4_OP * F * EeFe2p_Fe3O4_OP / (R * TKelvin))
# just for Beta = 0.5
EmixedOP = R * TKelvin / (nFe2p_Fe3O4_OP * F * Beta * 2) * math.log((i0_H1OP + i0_Fe1OP) / (i0_H2OP + i0_Fe2OP))
# Adjusting the dissolution rate constant of magnetite
ked_OP = kd_OP * math.exp(-1 * (1 - Beta) * nFe2p_Fe3O4_OP * F * (EmixedOP - EeFe2p_Fe3O4_OP) / (R * TKelvin))
# Adjusting the concentration of iron species at O_P interface (in mol/l should be calculated back to g/cm^3)
CSat_OPNew = (
(
math.exp(
(EmixedOP + DeltaGFe2p_Fe3O4_OP * 1000 / (2 * F) + R * TKelvin * math.log(10) * pH / (F))
* (-1)
* 2
* F
/ (3 * R * TKelvin)
)
)
* 90
/ 1000
)
Ce_OPNew = CSat_OPNew + (0.476 * (1 - PhiOX) * Corrosionrate / ked_OP) * 90 / 1000
Ce_MPNew = (
(
Corrosionrate
* (value.ChiP() * value.DeltaP())
/ (value.PhiP() * value.DP() * value.RhoP() * (1 - value.PhiP()))
+ Ce_OPNew * 1000 / 90
)
* 56
/ 1000
)
# print EeFe2p_Fe3O4_OP,i0Fe2p_Fe3O4_OP,EeH2_Hp_OP,i0H2_Hp_OP, EmixedOP,Ce_MPNew,Ce_OP
# # At M-P interface
#
# EeFe_Fe2p_MP = -1 * DeltaGFe_Fe2_MP * 1000 / (n * F) - math.log(10) * 2 * R * TKelvin * pH / (n * F) + R * TKelvin / (n * F) * math.log(Ce_MP * 1000 / FeMolarMass) # Equilibrium potential for the oxidation reaction at M_P interface, concentration unit changed from g/cm^3 to mol/L
# #in i0 equationsthe unit of concentration should be mol/cm^3 according to Dr. Cook thesis and Olga program ( the power 2/3 is not clear and the units does not match the A/cm^2 ) but according to Lisa Lang program Concentration in mol/lit should be multiplied by diameter of pipe/4 (cm). This would make sense in terms of units. I tried all the variations and they make insignificant changes in the results
# i0Fe_Fe2p_MP = (F * kboltzman * TKelvin / h) * (math.exp(-1 * self.ActivationEFe_Fe2_MP / (R * TKelvin)) * math.pow(Ce_MP * 1000 / FeMolarMass,1.0) * math.exp(-1 * Beta * n * F * EeFe_Fe2p_MP / (R * TKelvin)));
# Precipitation of magnetite at M_P interface Fe3O4 + 2H2O + 2H+ + 2e- = 3Fe(OH)2
# EeFe2p_Fe3O4_MP = -1 * DeltaGFe2p_Fe3O4_MP * 1000 / (n * F ) - math.log(10) * 2 * R * TKelvin / (n * F) * pH - 3 * R * TKelvin / (n * F) * math.log(Ce_MP * 1000 / FeMolarMass);
nFe_Fe2p_MP = 2
EeFe_Fe2p_MP = (
-1 * DeltaGFe_Fe2_MP * 1000 / (nFe_Fe2p_MP * F)
- math.log(10) * 2 * R * TKelvin / (nFe_Fe2p_MP * F) * pH
+ R * TKelvin / (nFe_Fe2p_MP * F) * math.log(Ce_MPNew * 1000 / FeMolarMass)
) # Equilibrium potential for the oxidation reaction at M_P interface
# in i0 equationsthe unit of concentration should be mol/cm^3 according to Dr. Cookthesis and Olga program ( the power 2/3 is not clear and the units does not match the A/cm^2 ) but according to Lisa Lang program Concentration in mol/lit should be multiplied by diameter of pipe/4 (cm). This would make sense in terms of units. I tried all the variations and they make insignificant changes in the results
i0Fe_Fe2p_MP = (
F
* kboltzman
* TKelvin
/ h
* (
math.exp(-1 * self.ActivationEFe_Fe2_MP / (R * TKelvin))
* math.pow(Ce_MPNew * 1000 / FeMolarMass, 1.0)
* math.exp(-1 * Beta * nFe_Fe2p_MP * F * EeFe_Fe2p_MP / (R * TKelvin))
)
)
# Hydrogen production 2H+ + 2e- = H2 at M_P;
nH2_Hp_MP = 2
EeH2_Hp_MP = (
-1 * DeltaGH2_Hp_MP * 1000 / (nH2_Hp_MP * F)
- math.log(10) * 2 * R * TKelvin * pH / (nH2_Hp_MP * F)
- R * TKelvin / (nH2_Hp_MP * F) * math.log((CeH2_MP * 1000 / 2.0))
) # At M_P interface, CH2_MP converted to mol/l, g/cm^3 * 1000/Molar mass H2 and then to atm: mol/l/KHenry(mol/l.atm) => atm
i0H2_Hp_MP = (
F
* kboltzman
* TKelvin
/ h
* (
math.exp(-1 * self.ActivationEH2_Hp_MP / (R * TKelvin))
* math.pow(CeH2_MP * 1000 / H2MolarMass, 1.0)
* math.exp(-1 * Beta * nH2_Hp_MP * F * EeH2_Hp_MP / (R * TKelvin))
)
)
# having the corrosion rate, the Ecorr can be calculated by solving the following equation:
# Corrosionrate = iFe_Fe2p_MP * FeMolarMass/ (n * F)
# iFe_Fe2p_MP = i0Fe_Fe2p_MP * (math.exp(Beta * n * F / (R * TKelvin) * (Ecorr - EeFe_Fe2p_MP)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (Ecorr - EeFe_Fe2p_MP)))
# i0Fe_Fe2p_MP * (math.exp(Beta * n * F / (R * TKelvin) * (Ecorr - EeFe_Fe2p_MP)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (Ecorr - EeFe_Fe2p_MP)))* FeMolarMass/ (n * F) - Corrosionrate =0.0
A1 = Corrosionrate * (n * F) / (FeMolarMass * i0Fe_Fe2p_MP)
C1 = Beta * n * F / (R * TKelvin)
C2 = math.exp(C1 * (EeFe_Fe2p_MP))
X1 = (A1 + math.pow(math.pow(A1, 2) + 4, 0.5)) / 2
X2 = (A1 - math.pow(math.pow(A1, 2) + 4, 0.5)) / 2
# C1 = Beta * n * F / (R * TKelvin)
# C2 = (1-Beta) * n * F / (R * TKelvin)
# C3 = math.exp(C1 * (EeFe_Fe2p_MP))
# C4 = math.exp(C2 * (EeFe_Fe2p_MP))
# print C1+C2, A1*C3, C2, C3*C4
if X2 > 0:
Ecorr = math.log(X2) / C1 + EeFe_Fe2p_MP
elif X1 > 0:
Ecorr = math.log(X1) / C1 + EeFe_Fe2p_MP
else:
print("error")
if X1 > 0 and X2 > 0:
Ecorr = math.log(X1) / C1 + EeFe_Fe2p_MP
# print A1,C1, C2, X1, X2, Ecorr, Emixed,
# print EeFe_Fe2p_MP,i0Fe_Fe2p_MP,EeH2_Hp_MP,i0H2_Hp_MP,Ecorr
# Current calculations at M_P interface
# iH2_Hp_MP = i0H2_Hp_MP * (math.exp(Beta * n * F / (R * TKelvin) * (EeH2_Hp_MP - Ecorr)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (EeH2_Hp_MP - Ecorr)))
# Adjusting Ce_MP for the Ecorr (in mol/l should be calculated back to g/cm^3)
CSat_MPNew = (
(
math.exp(
(
Ecorr
- EeFe_Fe2p_MP
+ DeltaGFe_Fe2_MP * 1000 / (n * F)
+ math.log(10) * 2 * R * TKelvin * pH / (n * F)
)
* (1)
* n
* F
/ (R * TKelvin)
)
)
* FeMolarMass
/ 1000
)
# CSat_MPNew = Ce_MPNew + CMP_Diffused
# Solubility at M-P: CMPsat assuming that the solubility at the M_P interface is summation of what is diffused at this interface and the equilibrium concentration considering the developed Ecorr
# Current calculations at O_B interface
# iFe3O4_Fe2p_OB = i0Fe2p_Fe3O4_OB * (math.exp(Beta * n * F / (R * TKelvin) * (EeFe2p_Fe3O4_OB - Emixed)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (EeFe2p_Fe3O4_OB - Emixed)));
# iH2_Hp_OB = i0H2_Hp_OB * (math.exp(Beta * n * F / (R * TKelvin) * (Emixed - EeH2_Hp_OB)) - math.exp(-1 * (1 - Beta) * n * F / (R * TKelvin) * (Emixed - EeH2_Hp_OB)));
return (
CSat_MPNew,
CSat_OPNew,
CSat_OBNew,
ked_OB,
ked_OP,
Emixed,
EmixedOP,
Ecorr,
CeH2_MP,
CeH2_OP,
CeH2_OB,
Corrosionrate,
Ce_MPNew,
Ce_OPNew,
)
