w w w . j m r t . c o m . b r
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Original
Article
A
potential
attenuation
equation
for
cathodically
polarized
pipelines
and
risers
Roland
Tolulope
Loto
a,c,∗,
Roy
Olakunle
Loto
b,
Cleophas
Akintoye
Loto
aaDepartmentofMechanicalEngineering,CovenantUniversity,Ota,OgunState,Nigeria
bDepartmentofMetallurgicalandMaterialsEngineering,UniversityofLagos,Akoka,Lagos,Nigeria
cDepartmentofChemical,Metallurgical&MaterialsEngineering,TshwaneUniversityofTechnology,Pretoria,SouthAfrica
a
r
t
i
c
l
e
i
n
f
o
Articlehistory: Received14January2016 Accepted21May2016 Availableonlinexxx Keywords: Alloys Corrosionresistance Cathodicpolarization Electrochemicalproperty Coatingsa
b
s
t
r
a
c
t
Thecathodicprotectionsystemanalysisastoolsforone-dimensionalpipelinesandrisers wasmodeledandderivedincorporatingtherelevantresistanceterms.Thenew expres-sionpertainstopipelineswithsuperimposedanodes.Comparisonsweremadebetween thepotentialattenuationprojectedbythenewexpression,theclassicalequationofUhlig andtheboundaryelementmodelingtechnique.Itwasconfirmedthatthenewlyderived equationismoreconservativethantheboundaryelementmodelingtechniqueduetoits considerationofthemetallicpathresistanceandtheUhligequationbecauseofits consid-erationoftheanoderesistance.
©2016BrazilianMetallurgical,MaterialsandMiningAssociation.PublishedbyElsevier EditoraLtda.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(http:// creativecommons.org/licenses/by-nc-nd/4.0/).
1.
Introduction
Pipelinesare generallyrecognized as the safest,most effi-cientandcosteffectivemeansoftransportationforoiland gasfromfixed productionfacilities [1].Structuraland high strengthsteelsarethemostcommonlyusedmaterialsforthe constructionofmarinepetroleumtransportpipelinesaswell asburiedonshorepipelines[2].However,itsuffersfrom an inherentlackofcorrosionresistanceinanelectrolytesuchas soilorseawater.Thisrequiresthatinordertopreventpipeline failureduetocorrosion,corrosioncontrolsystemshavetobe designedandmaintainedsuchthatahighdegreeofreliability
∗ Correspondingauthor.
E-mail:[email protected](R.T.Loto).
isrealized[3].Theissueofreliabilityisevenmoreimportantin thecaseofdeep-waterinstallations.Severalpublications[4–6] indicatethatthemajorcauseoffailureinpipelineshasbeen corrosion.Mineralsmanagementdate(MMS)datastatesthat over50%offailuresinmarinepipelinestothismode. Approxi-mately63%ofthecaseshaveoccurredonpipelinesasopposed torisersand69%resultedfromexternalasopposedto inter-nalcorrosion.Atthesametime,88%oftheexternalcorrosion occurredonriserswhile12%wereonpipelines[7].
Cathodic protectionscombined withtheuse ofcoatings are the majorsource ofprotectionforoffshoreand buried onshore pipelines [8,9]. It has historically been employed as the corrosion control methodology for the submerged
http://dx.doi.org/10.1016/j.jmrt.2016.05.008
2238-7854/©2016BrazilianMetallurgical,MaterialsandMiningAssociation.PublishedbyElsevierEditoraLtda.Thisisanopenaccess articleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).
portion of petroleum production platforms [10,11]. How-ever, the one-dimensional nature of pipelines entails the use ofcoatings combined withcathodic protection.In the formercase(petroleumproductionplatforms),the fundamen-talparameters importantincathodicprotectiondesignare theanoderesistanceandstructurecurrentdensitydemand. Ontheotherhand,formetallicpipelines,thecoating qual-ityandmetallicpathresistancemustbetakenintoaccount fromageneralpointofview.Cathodicprotectionsystemscan beoftheimpressedcurrenttypeorthegalvanicanodetype [12–15].Impressedcurrentcathodicprotection(iccp)systems aremainlyusedforburiedonshorepipelinesbutnotfeasible foroffshorepipelines.Inbothcasesthelimitingdistanceto whichthecorrosionprotectioncanbeeffectedisafunction ofthevoltagedropalongthemetallicpipeline,whicharises inconjunction with thecurrent return toground. Another factorthataffectsthedistancetowhichcorrosionprotection isaffordedisthequalityoftheprotectivecoating.Thus,the higherthecoatingquality,thelessthecurrentdemandofthe pipeand,asaresult,thelessthevoltagedropforapipelineof agivenlength.However,coatingqualityofmarinepipelines isconsiderablylessthanthatofburiedonshorecounterparts sothatthisdistanceofprotectionisconsiderablylessinthe formercasethaninthelatter.Tomaximizethedistance,to whichprotectionisachieved,theregionofthepipelinenear the rectifier and anode may be overprotected [16,17]. This cancausecoatingdamageintheformofblisteringand dis-bondment,whichinturnincreasesthepipecurrentdemand. Becauseofthese factors, corrosion controlofthe majority ofmarinepipelinesisprovidedbygalvanicanodes,i.e., gal-vanicanodecathodicprotection(gacp).Thisisinvariablyof the bracelet type for structural, economic and installation considerations.
Thedesignnormallyassumesacertainpercentageof coat-ingbarearea andemploysgalvanicbraceletanodesspaced about 250m apart. This relatively short spacing emanates becauseofthelimitationsonthesizeofthebraceletanodes thatcanbedeployedfromalaybargeandthefactthatthe currentdensitydemandisrelativelyhighandservicelivesof 25–30yearsareneeded.Fortheburiedonshorepipelineonthe otherhand,thehighercoatingqualitycombinedwiththeiccp systemissuchthatthemetallicpathgroundreturnresistance isthecontrollingfactorandasaresulttheanodegroundbed spacingof50–100kmcanberealized[18].
Foroffshorepipelineswithcloselyspacedgalvanicanodes, commononnewinstallations,themetallicpathresistanceis negligibleandthedesignprocesshashistoricallyinvolvedthe followingsteps[19,20]:
(1) Calculation of the net pipe current demand from the expression:
Ic=Ac∗fc∗ic (1)
whereIcisthenetpipecurrentdensitydemand[21],Acis thepipesurfacearea,fcisthecoatingbreakdownfactor (theratioofbareareatototalpipesurfacearea),icisthe currentdensitydemand.
(2) Determinationofthetotalanodemass(kg)requiredfrom amodifiedformofFaraday’slaw
M= 8760im·T
u·C (2)
whereTisthedesignlife(years),Cisthecurrentcapacity ofanindividualanode(Ah/kg),uistheutilizationfactor, imisthemeancurrentdensitytopolarizethepipeline. 8760hin365days.Finally,thenumberofanodes,N,is deter-minedasshownbelow;
N=Ia ia
(3)
whereIa is thetotal currentrequired andia isthe current outputfromanindividualanode.
However,formarinepipelinecpretrofits,marinepipelines deployedbyreelingwithsubsequentanodesledplacement, buriedonshorepipelineswithiccpsystems, anodespacing asmentioned above,islikelytobelargeandmetallicpath resistancesignificant.Forthesesituations,numerical meth-odssuchasboundaryelementmodeling(BEM)arecommonly used; first principles basedequations ofMorganand Uhlig arealsoused[22,23].Theseequationsareusedtoprojectthe potential attenuation along the pipelinein order to deter-mine thepolarizationatanypoint alongthepipeline.This thereforeenablesthecalculationoftheprotectiondistance ofthecathodicprotectionsystemalongthepipeline,which in turn enablesthe determinationofaconservative anode spacing,whichwillprovidethedesiredprotection economi-cally.Theyarealsousedtocalculatetheanodecurrentoutput, whichhelpsincalculatingthelifespanoftheanode(incase ofgacpsystem),orthecurrentconsumption(inthecaseof iccpsystem).WiththeMorgan’sequationandalsotheUhlig’s equation, anode resistanceis nottaken into consideration and therearelevelsofuncertainty associatedwiththe cal-culatedmagnitudeofpolarizationatthedrainagepoint,the mid-anodespacingandatavariablepointzfromadrainage point.Therearealsolevelsofuncertaintyassociatedwiththe currentdensitydemandprojectedbytheequation.
The BEM on the other hand incorporates anode resis-tance andexpressesthe closedcircuitcathodepotentialas a functionofdistance from thedrainage point but it does notaccommodatethemetallicpathresistance[21].Henceit predictsaconstantpolarizationbeyondthefieldoftheanode whereas,theoreticallygiventhefactthatthepipelinehasa finiteresistivity,thepolarizationwilldecreasewithdistance.It isthereforepossibleforthepolarizationatsomepointtohave risenabove−0.80VAgClwiththeBEMprojectingasignificantly higher value [24,25]. This means that there are situations wherethepotentialatapointalongthepipelineisaffectedby allfourresistancetermsearlierdescribed.Usinganyofthese methodswouldgiveresultsthatincorporatesomeamountof error.Inlightofthelimitationsofthemethodsdescribed,this research aims toderive a first principle based attenuation equation that incorporates all the four resistance terms (anoderesistance,coatingresistance,polarizationresistance andmetallicpathresistance).Thisistoaccuratelycalculate
the pipeline polarization behavior and the pipe current demand.
2.
Methodology
The modeling research was carried out in two parts: (i) developmentofaninclusivepotentialattenuationmodelfor pipelinecathodicprotectionand(ii)verificationofthe atten-uationmodel.
2.1. Developmentofaninclusivepotentialattenuation modelforpipelinecathodic
2.1.1. Protection
AccordingtoPiersonetal.[26],theelectrode(pipelineorriser) potentialcanberepresentedasthechargegradientassociated withanelectricdoublelayeror
Ec(z)=Em(z)−Ee(z)+Kref (4) whereEm(z)andEe(z)arethepotentialsofthemetallicpipe andelectrolyterespectively,atadistancezfromthedrainage pointofanoffsetanodeorthecentrelineofananode super-imposedon thepipelineinwhich theanodes areidentical andequallyspacedandKref(constant)isthepotentialofthe referenceelectrode.Inaddition,
c(z)=Ec(z)−Ecorr (5)
whereEcorr isthe freecorrosion potential and c(z)is the degreeofpolarizationfrom thefreecorrosion potential.By takingthesecondderivativeofEqs.(4)and(5)andsubstituting
forEc(z)theexpressionbelowisobtained.
c(z)=Em(z)−Ee(z) (6) Themethodadoptedwastoderiveamathematical expres-sionforthethreecomponenttermsasshowninEq.(6). 2.2. Expressionforcathodicpolarization
Thec(z)termwasassumedforsimplicitytohavealinear relationshipwiththecurrentdensityic(z)as
c(z)= ˛
f ·ic(z) (7)
where˛isthepolarizationresistance,fisthecoating break-downfactor,ic(z)isthecurrentdensityatpointzalongthe pipeline.
2.3. Expressionforelectrolytepotentialvariation
The expression for Ee(z) was developed by considering a pipeline,whichisprotectedbyasphericalanodeofradiusra locatedatanoffsetdistanceyoffromthepipeline.Aspherical anodewasassumedformathematicalsimplicityalthough,in reality,braceletanodesareusedformarinepipelinesoranode sledinthecaseofretrofits.However,theeffectofthegeometry onthecathodicprotectionsystemisnegligibleiftheresistance
toremoteearthofthebraceletanodeandthesphereanode are thesame.Therefore,theequivalent sphereanodefora givenbraceletanodecanbecalculatedbyreferringtoMcCoy’s formula[27].Forabraceletanodeofagivensurfacearea,the approximateresistancetoremoteearthisgivenbyMcCoy’s formulaas
R=0.315e
√
A (8)
whereeistheresistivityoftheelectrolyte,Aistheexposed surfaceareaofthebraceletanode.Byassumingasphereof thesamesurfaceareatothebraceletanode,theequivalent sphereradiusra(eq)iscalculatedas
ra(eq)=
Sa 4 (9) or ra(eq)=0.282Sa (10)whereSaisthespheresurfacearea.
Ee(z)isconsideredtobetheproductoftheresistanceRe(z)
betweentheanodeandaradialoutwarddistancedinthe elec-trolyteandthenetcurrentintheelectrolyte(Ia(z))atthatpoint inaccordancewithOhm’slawwhered≡
z2+y2of
i.e. Ee(z)=Re(z)∗Ia(z) (11)
Thepotentialdifferencebetweentwopointsr1and r2is givenbytheclassicalequationforpotentialdropassociated withasphericalelectrodeinanelectrolyteofresistivityeas
Er1→r2=−
r2 r1 ˇedr=− r2 r1 e·I 4r2dr= e·I 4 1 r1− 1 r2 (12)whereˇeistheelectricfieldintensityandIisthetotalcurrent dischargedbytheanode.Thus,uponsubstitutingr1withra andr2with(y2of+z2)
1/2
anddividingtheresultingexpression byI, theresistancebetweentheanodesurfaceand aradial outwarddistanced,Re(z)wasobtainedas
Re(z)= e 4
1 ra − 1 (y2 of+z2) 1/2 (13)An expressionforIa(z) wasobtainedbyconsidering the regioninFig.1,whichencompassestheentirecurrentfield oftheanode. Whereanodesareequallyspaced, theregion willintersectthepipelineatthemidpointbetweentheanodes (i.e.z=LwhereListhe semi-anodespacing).Sincethereis conservationofcharge,thenetcurrentpassingthrougha pla-narsurfaceperpendiculartothepipelineatz=z1,Ia(z1)was assumedtoadheretotheexpression
Ie(z1)=Ip(z1)=2rp
Lz1
ic(z)dz (14)
whereIp(z1)isthetotalcurrententeringthepipebetweenthe pointsz1andLonthepipeorthenetcurrentflowingtothe
Ψ ra yof d z=0 z Drainage point la(z) le(z) le(z1) z1
Fig.1–Currentfieldandindividualcurrentelementsofan offsetgalvanicanodeassociatedwithapipe.
anodeatz1,ic(z)isthecathode(pipe)currentdensityandrpis metallicpipeouterradius.Thenetcurrentintheelectrolyte
atz1,Ia(z1)isdoublethatofIe(z1)duetothecontributionfrom
bothsidesoftheanodei.e.
Ia(z1)=2Ip(z1)=4rp·
Lz1
ic(z)dz (15)
Duetothecoatingonthepipeline,thebareportionis signif-icantlysmallerthaninasituationwherethereisnocoating atall.Thiswasexposedwiththecoatingbreakdownfactor f,which isthefractionofthepipelinethatisbare(i.e.not coated).Atthesametime,somepartsofthebareportionare anodicsiteswhileothersarecathodicsites.Whenthepipeline ispolarized,theanodicreaction(ontheanodicsites)would havebeensuppressedandthecathodicreaction(onthelocal cathodicsites)willbeaccelerated.Itwasassumedthatonthe basisofthis,current fromtheanodeentersthepipelineat thelocalsiteshencethecurrentdensityterminEqs.(14)and (15)wasthecurrentdensityofthelocalcathodicsitesofthe bareportion.Thismadenecessarytoevaluatetheareaofthe localcathodicsitesforagivenbareportion(becausethelocal anodicareausuallydiffersfromthelocalcathodicareainsize). Thiswasdonebyderivingthefollowingexpressionsfromthe combinedactivation polarizationcurve forthecathodically polarizedmetal. id=ieexp 2.303 (Ecorr−Ee) bc
(16) icorr=ioexp 2.303 (Ecorr−Eo) ba (17)whereidthecurrentdensityisdemandofthelocalcathodic sitesandicorristhedissolutioncurrentdensityforthelocal anodicsites, bcisthe Tafelconstant forthecathodic reac-tionandbaistheTafelconstantfortheanodicreaction.Eqs. (16)and(17)werere-writtentoexpresstheareaandabsolute currentofthelocalanodicandcathodicareasofthepipe.
id=Acieexp
2.303(Ecorr−Ee) bc (18) icorr=Aaioexp 2.303(Ecorr−Eo) ba (19)Acistheelectrodeactivesurfaceareaforthecathodicsite, whileAaistheelectrodeactivesurfaceareafortheanodicsite. Fromliterature,totalanodicreactionisequaltototalcathodic reaction;henceitwasconsideredthatforpipelinewithouta cathodicprotectionsystem(anunpolarizedpipeline)
Id=icorr (20)
Fromthisexpression[Eq.(20)],Eq.(21)wasderived Abc= Ab
(21)
whereAbc=areaofexposedcathodicsitesonthepipemetal, =ratiooftotalpipesurfaceareatoareaofexposedcathodic sitesonthepipemetal,Ab=totalpipesurfacearea.
= ieexp((2.303(Ecorr−Ee))/bc) ioexp((2.303(Ecorr−Eo))/ba)+
1 (22)
Eq.(15)wasthenrewrittenintermsoftheareaofthelocal cathodicsitesofthebareportionandthecoatingbreakdown factor. Ia(z)=2Ip(z)=4rp·f ·
L z ic(z)dz (23)Ia(z)wassubstitutedwithIc(z)usingEq.(23)andRe(z)wasalso substitutedusingEq.(11).Upondifferentiatingtheproductof theresultingequationstwice(toobtaintheseconddifferential of[Ia(z)·Re(z)]),theexpressionofEe(z)wasobtainedtobe
Ee(z)= e·rp·f
d d 2−3z2 d5 · L z ic(z)dz+ 1 ra− 1 d ic(z)+ z d3ic(z) (24)2.4. Expressionformetallicpipepotentialvariation Ohm’slawwasagaintakenintoconsiderationinderivingan expressionforEm(z).AccordingtoOhm’s law;thepotential changealongapipeatapointzisgivenby
∂Em
∂z =−Rm·Ip(z) (25)
GiventhefactthatRmisconstant,Em(z)wasobtainedby combining theaboveexpressionwithEqs.(23)and(6)after whichitwasdifferentiatedtwicetoobtain
Em(z)=
2f·rp·Rm·ic(z)
(26)
2.5. Thegoverningequation
The expressions forthe electrolytic and metallic potential gradient alongthe pipeline[Eqs. (24)and (26)]were substi-tuted into Eq.(6)thenic(z)substituted withc(z)according
toEq.(7).Thetermswerebroughttogether andrearranged thus providingan expressionfor the potential attenuation alongapipelineas:
c(z)=
c(z)+ e·rp k1 ra− 1 d c(z) +d d2−3z2 d5 · L z c(z)dz ·rp k 2Rm− 2z·e d3 (27) where k= ˛· f (28)
kistheeffectualcoatingresistivity.
Eq.(27)holdsforthepipewithlengthbetweenz=raand z=Lforpipelinesprotectedbysuperimposedanodesandfor the entirepipe length forpipelines protectedby displaced anodes.Forsuperimposedanodeshowever,onlytheportion betweenz=laandz=Lisrequiredtobecalculated;wherela isthereal anode’slength(length ofthebraceletanode for instance).Thisisbecausetheregionbetweenz=0andz=la inthecaseofpipelinesprotectedbysuperimposednodesis coveredbytheanodeandthereforeisnotexposedtothe elec-trolyte,henceitdoesnotcorrode.
TherewasnosolutionknowntoEq.(27),thusitwassolved numericallyusinganexplicitfinitedifferenceschemethatis baseduponfirstandsecondderivativesinspace[28].Thefirst derivativewasrepresentedbybackwardfinitedifference
∂ ∂z=
mi +1−mi−+11
∂z (29)
andthesecondderivativeby ∂2 ∂2z= m i+1−2m+ 1 i +m+ 1 i−1 ∂2z (30)
Eqs.(28)and(29)weresubstitutedintoEq.(27)toyield
mi +1=(−QHdz/2)(
N−1 j=12mi+j+mN)+(im−+11/dz)H((1/ra)−(1/di))−(m+ 1 i−1 /∂z 2)−(m i+1/∂z 2) (−2/∂z2)+(H/dz)((1/ra)−(1/d i))+((2ziH/di3)+B)+(QHdz/2) (31) where Q=d d2−3z2 d5 (32) H=e·rp k (33) B= 2rp·Rm k (34)wherenisthenumberofnodesonthecathodebetweenthe anodesurfaceandmidpoint(thenumberofelementsoflength dzplus1),mistheiterationstepandireferstotheinternal nodeoverthelengthofthecathodebetweentheanode sur-faceandthemidpoint.Eq.(30)providedanexplicitmeansfor calculatingthecathodepolarizationateachinternalnodefor
thenextiterationstep(iterationstepm+1)basedonthevalues ofthepresentiterationstep(iterationstepm)atthenodes.
Therepresentationoftheboundaryconditionsattheend nodeswasdoneusing
c(z=0)=Ea−Ecorr=a (35)
Atthebeginning(i.e.theanode) ∂c(z) ∂z
z=L =0 (36)Atthemidpointbetweentheanodes,thederivative bound-aryconditionisrepresentedas
mi=+11= 4· m+1 i=n−1−m+ 1 i=n−2 3 (37)
Theelementclosesttothe anodei=0 wasconsideredto beequaltoa.Basedontheanodepolarizationaccordingto Eq.(34), a wasalsoassigned toeveryelement discretizing the cathode forthe initialiteration step m=1as aninitial estimate. The iteration succession was ended when rms becamelessthan10−9.Thetruesolutionofpotential atten-uationalongacathodicallypolarizedpipelineisdifficultand expensivetomeasureinsitu,hencethevalidityofthe pro-posedmodelwasdeterminedbycomparisonwithalternative modelingtechniques–boundaryelementmodeling(BEM)and Uhlig’sequationunderconditionswheretheywereconsidered accurate.
2.6. Verificationofthepotentialattenuationmodel 2.6.1. Theeffectofparameterkandanodespacingonthe potentialattenuation
Theattenuation model[Eq. (27)]was comparedwith alter-native modeling techniques – BEM and Uhlig’s equation. Comparisonswereforkvaluesof4,20,100and1000 m2fora hypotheticalpipe.Thiscoverstherangeforapipeline,which
isverydifficulttopolarize,tothatwhichislikelytobemet inpractice.Otherparametersforthehypotheticalsystemare listedinTable1.Thepotentialattenuationalongthepipeline obtainedbythesethreemethodsforeachvalueofkwas plot-tedasafunctionofdistancealongthepipelinefromtheanode (i.e.asafunctionofz)toverifytheaccuracyofEq.(27).
UsingthesameparametersinTable1fork=100 m2but withL=3000m,theattenuationprofilesfromBEMandEq.(27), neglectingmetallicpathresistance(i.e.assumingm=0),were plotted.Thiswasdonetoshowtheeffectofthemetallicpath resistance,whichisnotconsideredbytheBEM,onthe polar-izationofapipelineinsituationswhereitisnotnegligible.By relatingc(z)tothecurrentdensitydemandofthepipe,the anodecurrent outputwasdetermined byEq.(27)usingthe pipe andelectrolyteparametersthatare inTable1and the
Table1–Pipelineandelectrolyteparametersfora hypotheticalsystemusedinanalysis[27].
Pipe/CPparameter Example
Pipelineouterradius,m 0.136
Pipelineinnerradius,m 0.128
Anodespacing(2L),m 244
Coatingbreakdownfactor 0.04
Equivalentsphereradiusofanode,m 0.201
Electrolyteresistivity, m 0.30
Piperesistivity, m 1.7×10−9
Freecorrosionpipepotential,VAg/AgCl −0.65
Anodepotential,VAg/AgCl −1.05
resultobtainedwasplottedagainstk.Theresultsobtainedfor theUhligequationandtheBEMwerealsoplottedonthesame graphforcomparison.
2.6.2. Theeffectofpipecurrentdemandandanode separationdistanceupon
2.6.2.1. Potential attenuation and anode current output. The attenuationprofilesforone-halfanodespacingof1000,2000 and3000mwithk=100 m2andalsoforone-halfanode spac-ingof2000,6000,and10,000mwithk=1000 m2usingthe pipeandenvironmentparametersinTable1,asobtainedfrom Eq.(27)underthe aboveconditions, werealsoplottedas a functionofone-halfanodespacinginordertoinvestigatethe accuracyoftheanodecurrentoutputprojectedbyEq.(27).
TofurthercompareEq.(27)totheBEM,aplotofthe poten-tialdifference betweenBEM and Eq.(27) atthe mid-anode pointasafunctionofkwasmadeforvariousanodespacing rangingfrom250to10,000musingthepipeandelectrolyte parametersinTable1.Thepercentagedifferencebetweenthe anodecurrentoutputscalculatedusingtheBEMandEq.(27) wasalsoplottedagainstkforvariousanodespacingranging from250to10,000m.
2.6.3. Effectofoffsetdistanceuponpotentialattenuation alongapipeline
Eq.(27)was comparedwiththe BEM forasituation where apipelineisprotectedbyoffsetanodes. Thisisthecasein practicewhenapipelineisprotectedbyananodearraysuchas inthecaseofretrofits.Plotsofpipepotentialversusdistance fromdrainagewereobtainedfromEq.(27)andBEMusingthe parametersinTable1,exceptwithra=0.170m,e=0.15and 1 m,kvaluesof100and1000 m2,anodespacingsof200, 500,1000,2000m,andyofvaluesof1,5and10m.
When usingoffset anodes, the anode potentialand the drainagepointpotentialaredifferentsothedrainagepoint potentialwascalculatedusingclassicalequationforpotential dropandassociatedwithasphericalanodeinanelectrolyte Era→yof= e·I 4
1 ra − 1 yof (38)whereIaistheanodecurrentoutputandra(eq)istheradius oftheequivalentanode.
Theanodecurrentoutputwasobtainedbycalculatingitfor asituationwherethesameanodewassuperimposedonthe pipeline(sinceitwouldbethesameasinthecaseofanoffset anode).However,inthecaseofdisplacedanodes,nopartof
–0.60 –0.65 –0.70 –0.75 –0.80 –0.85 –0.90 P otential,V(Ag/AgCl) Distance, m Equation 27 BEM Uhlig Eqn. –0.95 –1.00 –1.05 0 20 40 60 80 100 120 140
Fig.2–ComparisonofpotentialprofilesprojectedbyEq.
(27),BEMandUhlig’sequation.
thepipelinewascoveredbytheanode.Thusthepipelineis completelyexposedtotheelectrolyte.Hence,wheninitially calculating theanode currentbyfirstassuminga superim-posedanode, thevalueofra wasaddedtothe semi-anode spacing. Thiswas done because Eq. (27) holds from z=ra, hence,theresultsobtainedwouldreflectthecurrentdemand bythe entirepipe notexcluding theregion coveredbythe anode(sincethedistancebetweenz=raandz=L+raisequal tothatbetweenz=0andz=L).Thevalueobtainedwasthen substitutedintoEq.(38).
3.
Results
and
analysis
Eq. (38) was comparedto the boundary element modeling (BEM) and Uhlig’s equation in order to verify its accuracy. The comparison was made by looking into the potential projected bythe threeequations undervarious conditions. Theconditionsincludedvariouscombinationsofpolarization resistance,anode spacing, offsetdistanceand pipe current demand.Theanodecurrentoutput(Ia),calculatedfromthese threeequationsundersomeoftheaboveconditions,werealso compared.
3.1. Effectofpolarizationresistanceandanode spacingpotentialattenuation
Fig.2presentsaplotofpipepotentialasafunctionofdistance fromananodesurfacedeterminedfromtheUhligequation, boundaryelementmodelingandEq.(27)forkvaluesof4,20, 100and1000 m2inascendingorderforahypotheticalpipe. OtherpipeparametersarelistedinTable1.
Undertheaboveconditions,itwasobservedthattheUhlig equation isrelatively insensitive topolarization resistance and the current density demandofthe pipe (the polariza-tionresistanceisequaltothecurrentdensitydemand).The Uhlig equation was also non-conservativewhen compared withBEMandEq.(27)duetothefactthatitpredictsagreater cathodicpolarization.ThereasonforthisbeingthattheUhlig modeldoesnottaketheanoderesistanceintoconsideration. ThepotentialprofileobtainedfromBEMandEq.(27)wasvery closeandhenceingoodmutualagreementwitheachother.
–0.55 –0.65 –0.75 –0.85 –0.95 P otencial, V(Ag/AgCl) –1.05 0 500 1000 1500 Distance, m 2000 2500 3000 3500 Equation 27 Equation 27 wfo resist BEM
Fig.3–ComparisonofpotentialprojectedbyEq.(27)and BEMwheretheformerexhibitsthepresenceandabsence ofthemetallicpathresistanceterm.
Theyfeaturedasignificantpotentialdropwithinroughlythe first 10–15m ofthe anode, the magnitude ofwhich varied directlywiththepipecurrentdemand(i.e.polarization resis-tanceandcurrentdemand).Beyondthefieldoftheanode(the regionwheretheanoderesistanceisinfluential),thepotential fortheremainingportionofthepipelinewasrelatively con-stant,althoughthepotentialdropinthisregionforEq.(27)was slightlygreaterthanthatofBEM,themagnitudeofthis differ-encewasinverselyrelatedtothecurrentdemandofthepipe. Thereasonforthisisbecauseoftheinclusionofthemetallic pathresistanceinEq.(27)anditsexclusioninBEM.Inthis sit-uation,however,thedifferencebetweenthepotentialprofiles obtainedfromthesetwomethodsisnotofanyoperative sig-nificance.ItfollowsthatsinceBEMisaproventechniquefor characterizingthepotentialfieldofacathodicallyprotected pipelineorriser,Eq.(27)isaviablemeansforprojectingthe potentialattenuationalongpipelinesaswellasanodecurrent output.
Fig.3presentsthepotentialattenuationprofilesfrom(1) BEM,(2)Eq.(27),and(3)Eq.(27)neglectingmetallicpath resis-tance (i.e.assuming m=0in this case) for the same pipe and electrolyteparameters used inFig. 2with k=100 m2 andL=3000m.Ecversuszforthethreecasesshowa signif-icantpotentialdropintheimmediatevicinityoftheanode. Beyond this region, the BEM and Eq. (27) without metallic pathresistanceexhibitaconstantpotentialwhereasEq.(27) withmetallicpathresistanceincludedischaracterizedby fur-ther potentialattenuation along thelength ofthe pipeline althoughofamuchlessermagnitude.Hence,itfollowsthat ofthethreeequations,Eq.(27)withthemetallicpath resis-tancegavethe mostaccuratepresentationofthepotential profile.SinceUhlig’sequationdoesnottaketheanode resis-tanceintoconsideration,itcanbesaidtobenon-conservative whentheanoderesistanceisnon-negligibleandtheBEMcan alsobesaidtobenon-conservativewherethemetallicpath resistanceisnon-negligible.Eq.(27)canbesaidtobethemost accuratemethodforsituationswhereboththemetallicpath resistanceandanoderesistancearesignificant.
Sincetheanodecurrentoutputisafunctionofthe poten-tial, thecurrent demandwas determined from Eq.(27) for thesamepipe andelectrolyteparametersutilizedinFig.2.
100 Equation 27 BEM Uhlig Eqn. 10 1 Current output, A 0.1 0.01 0 200 400 600 k 800 1000 1200
Fig.4–Comparisonofanodecurrentoutputasprojectedby Eq.(27),BEMandUhlig’sequation.
Fig.4presentsaplotofIaagainstkasdeterminedbyUhlig’s expression,BEMandEq.(27)baseduponthesamepipeand electrolyteparametersasFig.2.Itshowsthattheanode cur-rentoutputprojectedbyEq.(27)andtheBEMareverycloseto eachother,i.e.,theyareinagreement.Itisexpectedthatthe BEM willover-estimatetheanodecurrentoutputforlonger pipe lengthswherethemetallicpathresistanceisnot neg-ligible (since the BEM over-estimatesthe potential insuch situations).TheUhligequationontheotherhandclearly over-estimates the anode current output at very low values of polarizationresistance(i.e.intheregionofk=100 m2)but moreaccuratelyathigher valuesofpolarizationresistance (aroundk≥100˝m2).ThiswasprobablyduetoUhlig equa-tionneglectingthesignificanteffectofthenear-fieldatlower valuesofpolarizationresistance.
3.2. Theeffectofpipecurrentdemandandanode separationdistanceuponpotential
3.2.1. Attenuationandanodecurrentoutput
Fig. 5presents the attenuation profiles for different semi-anodespacingrangingfrom1000to3000mwithk=100 m2 projectedbytheBEM andEq.(27)usingthepipe and envi-ronment parameters in Table 1. Fig. 6 on the other hand
–0.60 –0.65 –0.70 –0.75 –0.80 –0.85 –0.90 P otential, V(Ag/AgCl) –0.95 –1.00 –1.05 0 500 1000 1500 2000 Equation 27 BEM Distance, m 2500 3000 3500
Fig.5–ComparisonofpotentialprofilesfromEq.(27)and BEMforpipelineswithanodespacingfrom1000to3000m andk=100m2.
–0.60 –0.65 –0.70 –0.75 –0.80 –0.85 –0.90 P otential, V(Ag/AgCl) –0.95 –1.00 –1.05 0 2000 4000 6000 8000 Equation 27 BEM Distance, m 10 000 12 000
Fig.6–ComparisonofpotentialprofilesfromEq.(27)and BEMforpipelineswithanodespacingfrom2000to 10,000mandwithk=1000m2.
presentsattenuationprofilesforsemi-anodespacingranging from1000to10,000mwithk=1000 m2projectedbythesame twomethods.
Figs.5and6indicatethatpotentialprofilesprojectedby Eq.(27)isdifferentfromthatprojectedbytheBEMandthe magnitudeofthisdifferenceisdirectlyproportionalto(1)the distancefrom theanode(2) thesemi-anodespacingand is inverselyproportionaltothepolarizationresistance.Eq.(27) isconsideredtobethemoreaccurateofthetwomethods espe-ciallywhenthemetallicpathresistanceissignificantbecause itincorporatestheterm.Itfollowsthatonpipeforinstance, theBEMcanindicateprotectionalongtheentirelengthofthe pipeline,whereasthepipeisunder-protectedbeyondacertain point.Figs.7and8showtheplotsofanodecurrentoutputasa functionofsemi-anodespacingforthesameconditionsused forFigs.5and6,respectively.TheplotsshowthattheBEMand Eq.(27)areingoodmutualagreementforrelativelyshort spac-ing.However,forlongersemi-anodespacing,theBEMprojects thatthecurrentincreaseswhereas,Eq.(27)projectsthatthe currentincreasestoamaximumandthendecrease.Thelatter phenomenonismoreprominentthanthelowerpolarization resistancecase.Theloweranodecurrentoutputprojectedby Eq.(27)inFigs.7and8comparedwiththeBEMcorresponds withthepotentialprojectedbytheformer inFigs. 5and6,
3500 3000 2500 2000 1500 Current output, mA 1000 500 0 0 500 1000 1500
Half anode spacing, m
Equation 27 BEM
2000 2500 3000 3500
Fig.7–AnodecurrentoutputasprojectedbyEq.(27)and BEMasafunctionofhalfanodespacingandfor
k=100m2. 300 250 200 150 Current output, mA 100 50 0 0 200 400 600
Half anode spacing, m
Equation 27 BBM
800 1000 1200
Fig.8–AnodecurrentoutputasprojectedbyEq.(27)and BEMasafunctionofhalfanodespacingandfor
k=1000m2.
whichhadalessermagnitude.Thedifferenceisalsodueto theBEMnotconsideringthemetallicpathresistance.
Fig. 9 presents a plot of difference in potential at the midpointbetweenthevalueprojectedbyEq.(27)andthat pro-jectedbytheBEMasafunctionofkforvariousanodespacing between250and10,000m.Thecurvesshowthat,exceptfor thelargestanodespacing(10,000m),thedifferencein poten-tialincreaseswithdecreasingk.Thistrendaroseduetothe anodecurrentoutputincreasingwithdecreasingkwiththe BEMnotaccountingfortheincreasingvoltagedropalongthe pipelineasaresultoftheincreasinganodecurrentoutput. However,areversetrendwasobservedfor2L=10,000mand k<600 m2.Thiswasduetothefactthatasaresultofthe relativelylowkandthelargepotentialdropfromtheanode tothemid-anodepoint,thepolarizationatthemidpointwas verysmallforboththeBEMandEq.(27)andthedifferencein potentialprojectedbythetwowassmall.
Fig.10presentsthepercentagedifferenceinanodecurrent outputbetweentheBEMandEq.(27)asafunctionofkforthe sameanodespacingusedinFig.9.Itwillbeobservedthat,for allthevaluesofanodespacing,thedifferencedecreasedwith increasingvalueofk.Thistrendisparticularlysignificantin
0.07 0.06 0.05 0.04 0.03 0.02
Mid-anode spacing potencial diff
erence , V 10 000 m 5000 m 1000 m 500 m 250 m 0.01 0 200 400 600 k 800 1000 1200 0
Fig.9–Plotofthedifferencebetweenpotentialprojectedby Eq.(27)andBEMatthemid-anodepointfordifferentanode spacingasafunctionofk.Apositivedifferenceindicatesa relativelypositivepotentialprojectedbyEq.(27).
60
50
P
ercentage diff
erence in anode current output
40 30 20 10 0 0 200 400 10 000 m 5000 m 1000 m 500 m 250 m 600 800 1000 1200 k
Fig.10–Plotofthepercentdifferenceinanodecurrent outputasprojectedbyEq.(27)andBEMasafunctionofk.A positivedifferenceindicatesrelativelygreatercurrent projectionbyBEM.
thetwolargestanodespacing.Thistrendcorrespondswith Figs.7and8wheretheanodecurrentoutputprojectedbythe twomethodsdifferwithincreaseinanodespace;theeffect beinggreaterthelowerthevalueofk.
3.3. Effectofoffsetdistanceuponpotential attenuationalongapipeline
Fig.11showsthepotentialattenuationprofilesforyofvalues at1,5and10mwithe=0.15andk=100 m2,ra=0.170mand anodespacingsof200,500,1000and2000masobtainedfrom Eq.(27)andtheBEManalysis.Theyallshowexact similari-ties.TheplotsshowthattheBEMprojectsalargerpolarization valuealongthepipelinethanEq.(27).Themagnitudeofthe differencebeingdirectlyproportionaltotheanodespacing. Thisdifference isbecause oftheBEM analysisneglectsthe metallicpathresistance.Theresultsalsoshowthatthegreater theoffsetdistance,themorepositivethepotentialatz=0and thefarfieldpotentialisessentiallyindependentoftheoffset distance. –0.75 –0.80 –0.85 –0.90 –0.95 –1.00 P otential, V(Ag/AgCl) –1.05 0 200 400 600 800 Equation 27 BEM Distance, m 1000 1200
Fig.11–Potentialattenuationprofilesfora0.272m diameterpipelinewithe=0.15m,k=100m2,and
0.340mdiameterequivalentsphericalanodeoffsetat1m, 5mand10m. –1.000 –1.005 –1.010 –1.015 –1.020 –1.025 –1.030 –1.035 –1.040 –1.045 P otential, V (Ag/AgCl) –1.050 0 200 400 600 800 Equation 27 BEM Distance, m 1000 1200
Fig.12–Potentialattenuationprofilesfora0.272m diameterpipelinewithe=0.15m,k=100m2,and
0.340mdiameterequivalentsphericalanodeoffsetat1m, 5mand10m.
Fig.12 showsplotswiththesameparameters asFig. 11 butwith,k=1000 m2.Incomparisonwiththetrendobtained inFig.11,itwillbeobservedthat(1)theattenuationinthe immediate vicinityofthe anode decreases withincreasing polarizationresistance,(2)theattenuationprofileismore neg-ativeand(3)theprotectiondistanceisgreater,thusthehigher thepolarizationresistance.
The newly proposed equation can beused forpipeline cathodic protectiondesignfor(1) marine pipelinecathodic protection retrofits (2) buried onshore pipelines with impressed current cathodic protection systems and (3) marinepipelinesdeployedbyreelingwithsubsequentanode sledplacementwherethemetallicpathresistanceis signifi-cant.Thisisduetothefactthatthenewlyproposedequation is more conservative than any of the other two methods mentionedinthissituationandthereforemoreaccurate.
4.
Conclusion
Anewlyproposedpotentialattenuationequationfor cathodi-callyprotectedpipelinesandriserswasderived.Theequation wassolvedusingafinitedifferencemethodnumerical pro-cedure. The improved accuracy of this equation over the boundary element modeling (BEM)and the Uhlig equation isconfirmedwithexampleanalysisprovided.Itwasproven thattheequationisingoodagreementwiththeBEMwhen themetallicpathresistanceisnegligiblebutmoreaccurate insituationswherethereisfinitevalueforthemetallicpath resistance.Itwasalsoshowntobemoreaccuratethan the Uhligequation.Themagnitudeisproportionaltothedifficulty withwhichapipelineispolarized.
Conflicts
of
interest
Acknowledgements
Theauthors acknowledgethe Department of Metallurgical and Materials Engineering, Faculty ofEngineering, Univer-sity of Lagos, Akoka, Lagos, Nigeria and the Department ofMechanicalEngineering, Covenant University,Ota,Ogun State,Nigeriafortheprovisionofresearchfacilitiesforthis work.
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