Physico-Chemical Modelling (PCM) Industrial Electrical

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Physico-Chemical Modelling (PCM) – a literature review

Kimberly Solon

Division of Industrial Electrical Engineering and Automation Faculty of Engineering, Lund University

Physico-ChemicalModelling(PCM)

–ALITERATUREREVIEW

KIMBERLYSOLON2016May12

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CONTENTS

PAGENO.

1. Introduction ……………………………………… 21.1 Physico-chemicalprocesses ……………………………………… 21.2 Applicationsofphysico-chemicalmodelling ……………………………………… 21.3 Motivationforthedevelopmentofphysico-

chemicalmodelsforwastewatertreatmentprocesses

……………………………………… 2

1.4 Availablemodelsonphysico-chemistryforwastewatertreatmentmodelling

……………………………………… 3

2. Contentofthephysico-chemicalmodelling

framework……………………………………… 4

2.1 Acid-basereactions ……………………………………… 42.2 Ionspeciation/pairing ……………………………………… 42.3 Ionactivity ……………………………………… 62.4 Precipitationandre-dissolution ……………………………………… 82.5 Stripping ……………………………………… 122.6 Temperaturecorrection ……………………………………… 12

3. Implementationdetailsandnumericalissues ……………………………………… 15

4. Availablesoftwaretoolsanddatabases ……………………………………… 16

5. Conclusionsandfutureperspective ……………………………………… 18

6. References ……………………………………… 19

Physico-ChemicalModelling–ALiteratureReview

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1. INTRODUCTIONThis literaturereviewisapartoftheself-studycourserequiredformyPhDstudiesat IEA,LTH.Thespecificobjectivesofthistaskaretoresearchonandcriticallysummarizecurrentand significant papers about physico-chemistry, especially focusing on its application tomodellingwastewatertreatmentprocesses.1.1. Physico-chemicalprocesses

Physico-chemical refers to both physical and chemical properties/aspects, as opposed tobiochemical.Physico-chemicalprocessesarenon-biologicallymediatedprocesses.Theycanbe liquid-liquid processes (i.e. ion association/dissociation), gas-liquid processes (i.e. gas-liquidtransfer),andliquid-solidprocesses(i.e.precipitationandsolubilisation)(Batstoneetal.,2012).1.2. Applicationsofphysico-chemicalmodellingThe advances in themodelling aswell as data collection related to physico-chemistry aregenerally established in the field of geochemistry. Most of the software products aboutwaterchemistryaredirectedforgeochemicalapplications.Inthefieldofwastewatertreatmentmodelling,thephysico-chemicalmodelsthatareusedare the sameones used as for geochemistry. And for this reason, publicationswrittenonphysico-chemicalmodellingasappliedtowastewatertreatmentmodellingare,mostofthetime,similar.Becauseoftheenormityofavailabledatabasesandprocessesinvolvedinthephysico-chemicalmodels, themain difference on these for-wastewater-treatmentmodelswouldbethereductionofthedatabase/processesdependingonthespecificobjective/sofwhythesephysico-chemicalmodelswereincludedinthefirstplace.1.3. Motivation for the development of physico-chemical models for

wastewatertreatmentprocessesDuetothecurrentneedstoexpandtheIWAmodels(e.g.ActivatedSludgeModels(Henzeetal.,2000),AnaerobicDigestionModel(Batstoneetal.,2002)),modelformulationsalsoneedto be modified and expanded. For example, anaerobic digestion systems dealing withsulfate-richwastestreamsneedtoaccountforan increase inthenumberofstatevariablebeingconsidered(Fedorovichetal.,2003;Barreraetal.,2015).Inaddition,oneshouldalsotake into account theother relevant and significant components that are interactingwith


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sulfur, in this case iron (Fe) (Flores-Alsinaetal.,2016). Incertainconditions, sulfur (in theform of sulfate) can be removed by precipitating it with S or aluminium (Al). Suchprecipitation reactions are physico-chemical processes. Modelling precipitation reactionsshould consider ion activities instead of the actual concentrations (KazadiMbamba et al.,2015a; 2015b) (more detail on this in the next sections) and this involves non-idealitycorrections that are inherent to geochemical software. Thus, in this highly interlinkingmechanism, addition of certain components on standard models would increasedramaticallythenumberofcomponentsandsubmodelstobe included(Solonetal.,2015;Flores-Alsinaetal.,2015).Mostimportantly,systemswithnon-idealconditionsaremodelledcorrectlywhenphysico-chemicalmodelsareincluded(Batstoneetal.,2012).1.4. Available models on physico-chemistry for wastewater treatment

modellingCurrently, a significant number of IWA models have been extended to include physico-chemicalmodelling. Flores-Alsinaet al. (2015)haveextended theactivated sludgemodelsASM1, ASM2d, ASM3 with a physico-chemical framework that allows for the dynamiccalculation of pH.However, it does not includeprecipitation yet. Solon et al. (2015) haveextendedtheanaerobicdigestionmodelno.1 (ADM1)aswell inorder toaccount fornon-idealities.Theseresultshaveshownasignificantdifferencewiththeresultsfromtheoriginalmodelimplementation,especiallyforthecaseswheremulti-valentionsareconsidered.Theresults have proved that such physico-chemical modelling is needed in order to produceaccurate results. Kazadi Mbamba et al. (2015a; 2015b) have also shown using the PCMframeworkandtogetherwiththedevelopedprecipitationframeworktheeffectofmultiplemineral precipitation on simulation results and have further validated this extendedmodellingwithexperimentaldata.Lizarralde et al. (2015) have also developed a model to describe physico-chemicaltransformations,which includes liquid-gas transfer and liquid-solid transfer. The acid-baseequilibriumaswellastheion-pairingequilibriumarefurtherdescribedinaphysico-chemicalmodel. In their study, the components added in order to describe the physico-chemicalreactionsaredependentonthebiochemicalprocesseschosen.Musvoto et al. (1997; 2000a; 2000b; 2000c) have also used theweak acid/basemodel inorder to extend the applications of ASM1 in cases where pH estimation is consideredimportant(Sötemannetal.,2005a).


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2. CONTENTSOFTHEPHYSICO-CHEMICALMODELLINGFRAMEWORK

A physico-chemical model involves the phase changes in the reactions products. Thisincludes precipitation/re-dissolution reactions and gas stripping. Modelling these phasechangesreactionsrequiretheuseofwaterchemistry(acid/baseandionpairingreactions).Thus, physico-chemical modelling includes: liquid-liquid processes, liquid-solid processes,andgas-liquidprocesses.LIQUID-LIQUIDPROCESSES2.1. Acid-basereactions(chemicalequilibriumdissociation)

StummandMorgan(1996)givethegeneralprincipleofdissociationreactions:

HA ↔ H! + A! Eq.2.1.1The chemical equilibrium is solved either with ordinary differential equations (ODEs) oralgebraicequations(AEs).Thedissociationprocessesofacid/basereactionsandionpairingreactionscanbedescribedusingODEs.Theyaregivenhighkinetic rateconstants toshowthatthesereactionsoccurinstantaneously(Musvotoetal.,1997;2000a;2000b).2.2. Ionspeciation/pairing

TheTableaumethod(MorelandHering,1993)showsanorganizedwayofrepresentingallequationsrequiredforacid/baseandionpairingreactions.Itallowstoeasilydeterminetheexpressionforasolution(representedbythetotal),giventhecomponents.Componentsarethebasicbuildingblocksfromwhichallspeciesareformed.ThiscanbeorganisedusingtheTableaumethod,whichisbasicallyatableofstoichiometriccoefficients,whereeachcolumnrepresents an individual component, and each row represents a separate species. Thecompositionofthesolutioniswrittenintermsofthecomponents.Inapropercomponentset:

1. Allspeciescanbeexpressedstoichiometricallyasafunctionofthecomponents,thestoichiometrybeingdefinedbythechemicalreactions.

2. Eachspecieshasauniquestoichiometricexpressionasafunctionofthecomponents.3. H2O should always be chosen as a component. Although it is omitted from the

Tableau,itisimplicitlyincludedinthecomponentset.4. H+shouldalwaysbechosenasacomponent.


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In some cases, there are also non-electrostatic interactions between ions, which form ioniccomplexesasnewchemicalspecies.Theseionpairsorioncomplexesaredifferentfromthefreeions (suchas abiphosphate ion,HPO4

2-,which is different from freeorthophosphate, PO43-) in

solution.Forthisreason,ioncomplexescanincreaseordecreasethechemicaldrivingforceforaspecificreactiontooccur,dependingonwhetherthefree-form(suchasphosphateprecipitatingasstruvite)ortheioncomplex(suchasammoniumprecipitatingasstruvite)isthechemicalspeciesthatparticipates in theparticular reaction. IonpairingeffectsonpHpredictionsareconsideredsignificant in systems with high total dissolved solids concentrations (Musvoto et al., 2000a),indicatinghighionicstrengthssuchasinhigh-strengthanaerobicdigestionliquors,seawater,andconcentratedindustrialwastewater(Table2.2.1).Table2.2.1.Non-idealitycorrections(Batstoneetal.,2012)toachieveapHerroroflessthan5%.LEVEL IONICSTRENGTH(M) WASTEWATERTYPE APPROACH

1 < 0.001 drinkingwatercleannaturalfreshwater

nocorrectionrequired–assumeideal

2 < 0.1 weakindustrialwastewateralldomesticwastewater non-iterativesimplecorrection

3 < 1(onlyionactivity)

seawateranaerobicdigesters

fulliterativecalculationofionactivity

4< 1

(activitywithnon-valents)

asabove,withgastransfersimilartolevel3,usingnon-valentformofExtendedDebye-Hückelequation

5 < 5strongindustrialwastewaterlandfillleachateRObrine

similartolevel4,butincludingionpairs

Asub-modelcontainingthemostcommonionpairspresentinwastewaterisset-uptodescribeion-pairingbehaviour(seesomeexamplesinTable2.2.2).Thisisimplementedinasimilarfashionasweakacid/basereactionswhereanalgebraicprocedure(Ikumietal.,2011)isusedbasedontheassumption that ion pairs are in a state of equilibrium at all times. Total concentrations aredeterminedbymassbalancesandsubsequently,theionicconcentrationsarecalculatediterativelyusing a speciation sub-routine from a set of algebraicmass balances and equilibrium constantrelationships(Taitetal.,2012).Table2.2.2.Commonion-pairreactionstakenfromMusvotoetal.(2000a).

IonpairreactionsCa!! + OH! ↔ CaOH! Ca!! + PO!!!

↔ CaPO!!(aq)

-

Mg!! + OH! ↔ MgOH!Ca!! + CO!!!

↔ CaCO!(aq)Ca!! + H!PO!!↔ CaH!PO!!(aq)

Mg!! + CO!!!↔ MgCO!(aq)Ca!! + HCO!! ↔ CaHCO!! Mg!! + HPO!!!

↔ MgHPO!(aq)

Mg!! + HCO!!↔ MgHCO!!

Ca!! + HPO!!!↔ CaHPO!(aq)

Mg!! + PO!!!↔ MgPO!!(aq)

-


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2.3. Ionactivity

The effect of ionic strength, also known as ion activity (S ! ), is defined as the effectiveconcentrationofanyparticularkindofioninsolutionandiscausedbyelectrostaticinteractionsbetweenions.Itiscalculatedbymultiplyingtheconcentrationofioni(S[!])byacorrectionfactorwhichiscalledtheactivitycoefficient(γ!):

𝑆 ! = γ! ∙ 𝑆[!] Eq.2.3.1Consideringachemicalequilibriumreaction:

bB+ cC ↔ dD Eq.2.3.2Ininfinitelydilutesolutions,theionactivitiescanbeapproximatedbytheconcentrations,astheactivitycoefficientapproachesunity.Theequilibriumconstant(K!")isexpressedas:

K!" = 𝑆[!]!

𝑆[!]! ∙ 𝑆[!]! Eq.2.3.3

However,fornon-idealsolutionsK!"iscalculatedas:

K!" = 𝑆 !

!

𝑆 !! ∙ 𝑆 !

! = γ! ∙ 𝑆[!]

!

γ! ∙ 𝑆[!]! ∙ γ! ∙ 𝑆[!]

! Eq.2.3.4

Therehavebeennumerousstudiesonthistopic,whichhavedevelopedempiricalcorrelationsofexperimentaldatathatallowpredictionofactivitycoefficientsatvarioussolutionconditions(seeTable 2.3.1). All of these expressions have shown that ion activity is dependent on the ionicstrength(I)ofthesolution,whichcanbedeterminedasfollows:

I = 12 𝑆[!] ∙ z!! Eq.2.3.5

wherez!isthechargeofioni.Thecalculatedactivityshouldbeusedinequilibriumequationsaswellasweakacid/basepairingandioncomplexationreactions.Thecurrentphysico-chemicalmodelsapplied forwastewater treatmentprocessmodellinghasusedtheDaviesequation(Davies,1939;MerkelandPlaner-Friedrich,2008)todescribetheactivityofthecomponentsinsteadoftheconcentrations.Thisisbecausethisequationissimpleanddoesnotneedother constantsunlike theextendedandWATEQDebye-Hückel


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equation (Truesdell and Jones, 1973). In addition, theDaviesequation is valid for a largerrange of ionic strength. The differences between the different ion activity correctionequationsandtheirrelationshipwithionicstrengthandiontypes(i.e.monovalent,divalent,etc.)areshowninFigure2.3.1.ItwouldhavebeenbettertousetheWATEQDebye-Hückelequation since it is valid for thewidest range of ionic strength, however, it requires twoadditional parameters for each ion type considered which are not always available inliterature.Thus,theDaviesequationseemstobethemostfittingchoice.Table2.3.1.Expressionsforcalculatingactivitycoefficientsandtheircorrespondingrangeofapplicability.

EQUATIONNAME EQUATION APPLICABILITY

Debye-Hückel(DebyeandHückel,1923) log γ! = −A ∙ z!! ∙ I I < 0.005

molL

ExtendedDebye-Hückel(DebyeandHückel,1923) log γ! = −A ∙ z!! ∙

I1+ B ∙ α! ∙ I

I < 0.1 molL

Güntelberg(Guntelberg,1926) log γ! = −A ∙ z!! ∙

I1+ I

I < 0.1 molL

Davies(Davies,1939) log γ! = −A ∙ z!! ∙

I1+ I

− 0.3 ∙ I I < 0.5 molL

WATEQDebye-Hückel(TruesdellandJones,1973) log γ! = −A ∙ z!! ∙

I1+ B ∙ α! ∙ I

+ β! ∙ I I < 1 molL

where: γ! Activitycorrectioncoefficientofioni;A,B Temperature-dependentconstants;z! Chargeofioni;α!, β! Ion-specificparameters;I Ionicstrength.AnothertechniquefordeterminingactivitycoefficientsisthePitzerequation(Pitzer,1973).Itisapplicableforpredictingactivitycoefficientvaluesofionsinsolutionwithveryhighionicstrength.Itimplementsaspecificioninteractionmodel,whereinbothionicstrengthandthecharacteristics of the solution are considered. This means that this is a more precisemodellingofmeanactivitycoefficientdataandequilibriumconstants.However, there isamuchgreaternumberofparametersincludedinPitzerequationswhichmeansthattheyaremoredifficulttocalibratebasedonavailablemeasurements.Asanexampleoftypicalionspresentinwastewater,Na+andMg2+areshowninFigure2.3.1.Insolutions with low ionic strength (I < 0.1 mol/L), the activity corrections applied to the


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concentrationsareprecise for thedifferentexpressionswhenusedwithin their ionicstrengthvalidityrange.However,correctionfactorsbecomeverydifferentathigherionicstrengthsevenwhenusedwithintheirvalidranges,andcanbeobservedbythecurvescorrespondingtoDaviesandWATEQDebye-Hückelequations.

Na+ (a) Mg+2 (b)

Figure2.3.1.Activitycoefficientsof(a)sodiumand(b)magnesiumionsascalculatedfromfiveactivitycoefficientcorrelations.

LIQUID-SOLIDPROCESSES2.4. Precipitationandre-dissolutionOpposite to liquid-liquid processes, liquid-solid processes are assumed to occur slowly toreachequilibrium.Inordertomodelprecipitationreactions,thepossibilityofprecipitationiscalculatedfirstbytesting if thesolution issupersaturatedornot.TheSaturation Index(SI)indicatesifasolutionis inequilibrium,undersaturatedorsupersaturatedwithrespecttoamineral (i.e. whether a mineral precipitation might occur or not) (Merkel and Planer-Friedrich, 2008; Stumm and Morgan, 1996; Alley, 1993). If SI < 0, the liquid phase isundersaturated,thusamineralmightdissolveintotheliquidphase.IfSI=0,theliquidphaseissaturatedoratequilibrium.WhileifSI>0,theliquidphaseissupersaturatedandmineralprecipitationmightoccur.Itiscalculatedby:


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SI = logIAPK!"

Eq.2.4.1

where: IAP Ion-activityproduct,

theproductoftheactivitiesoftheelementscomprisingthemineralprecipitate;K!" Solubilityproductconstantofthemineral.Consideringanequilibriumreaction:

xM!! + yA!! ↔ M!A! Eq.2.4.2TheIAPis:IAP = M!! ! ∙ A!! !whiletheKSPis:K!" = M!!

!! ∙ A!! !

!.Note that SI only indicates what could happen thermodynamically. However, it does notindicate the rate by which the process will proceed. This means that a solution may besuper-saturated for a very long time (i.e. it will take a long time for the mineral toprecipitate).Thereareseveralequationsavailableforkineticratesdescribingcrystallization.The general form of the crystallization rate given by Koutsoukos et al. (1980) and firstproposedbyDaviesandJones(1955)asappliedtosilverchlorideprecipitationis:

r! = k!"#$%s M!! ! ∙ A!! !!! − M!!

!! ∙ A!! !

!!!!

Eq.2.4.3

where: r! Precipitationkineticrate;k!"#$% Precipitationrateconstant;s Proportionaltothetotalnumberofavailablegrowthsitesontheadded

seedmaterial;M!! !, A!! ! activitiesoflatticeionsattimet;M!!

!!, A!! !

! activitiesoflatticeionsatequilibrium;v v++v-;n Aconstant,typically2.Thisrateequationhasbeenusedforkineticstudies(Koutsopoulos,2002;Koutsoukos,1980;NancollasandReddy,1971)aswellas inmodellingofprecipitationreactionsasappliedtowastewater treatment (Barat et al., 2011; Musvoto et al., 2000a; 2000b, 2000c). Theequationdescribeshowthedifferencebetweentheconcentrationofcontributingionsina


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solution and their equilibrium concentrations is the thermodynamic driving force thatdictatestheoccurrenceofprecipitation.KazadiMbamba et al. (2015a; 2015b) have adapted the crystallization rate presented byNielsen(1984):

r! = k!"#$% X!"#$%M!! ! ∙ A!! !

KSP

!!− 1

!

Eq.2.4.4

Lizarraldeetal.(2015)havepresentedthekineticrateforprecipitationreactionsbasedontheworkbyKoutsoukosetal. (1980)and improveditwithaconsiderationoftheeffectofcrystalseeding,TSSconcentration,anddelayofnucleation(spontaneousnucleation).

r! = k!"#$%s M!! ! ∙ A!! !!! − M!!

!! ∙ A!! !

!!!!

∙ A!" + A!"## + A!" Eq.2.4.5

where: A!"

s!M!A!

M!A! !+ K!

A!"## X!""X!"" + K!

A!" Mv+ x ∙ Av− y

Mv+ x ∙ Av− y + K!

K1,K2,K3 Constantswithverysmallvaluestoguaranteenumericalstability.s! isanadimensionalparameter representing thegrowthof interfacialareaconcentrationassumingaconstantsizedistribution.The first two added terms (A!",A!"##) represent crystal growthwhen supportmaterial isaddedandthelastterm(A!")isaddedtodescribecrystalgrowthwhennoseedmaterialisadded.Building the model also requires identification of the possible precipitates or else theremightbemissedcomponentsorthemodelwillbeoverlycomplex.Thisistheadvantageofusinganexternalsoftwaretoolbecauseallpossibleprecipitatescouldbereflected.That isalsothereasonwhyitisadvantageoustohavepriorknowledgeandprocessunderstanding.Musvotoetal.(2000a;2000b;2000c)haveidentifiedprecipitates(seeTable)thatarelikelytooccurinwastewatertreatmentplants.Thetypesofprecipitatesarealsohighlydependentontheinfluentandtheuseofmetalsforchemicalprecipitation.


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SomeexamplesofmineralswhichareidentifiedtoprecipitateduringwastewatertreatmentarelistedinTable2.4.1.Table2.4.1.Commonmineralprecipitatesformedduringwastewatertreatment.

Mineral ChemicalformulaAmorphouscalciumphosphate(ACP) Ca3(PO4)2

Calcite(CCM) CaCO3Struvite MgNH4PO4

Newberryite MgHPO4:3H2OHydroxyapatite(HAP) Ca5(PO4)3OH

Dicalciumphosphatedihydrate(DCPD) CaHPO4:2H2OOctacalciumphosphate(OCP) Ca4H(PO4)3:3H2O

Aragonite(ACC) CaCO3Magnesite MgCO3K-struvite KMgPO4Ironsulfide FeS

Ironphosphate FePO4Aluminiumphosphate AlPO4

Vivianite Fe3(PO4)2:8H2OLettingaetal.(1997);Musvotoetal.(2000a,2000b,2000c);MaurerandBoller(1999);

Carlssonetal.(1997);Clarketal.(1997);Wangetal.(2016)Unlike the equilibrium constants used in aqueous phase chemistry, the constants used tomodel precipitation reactions usually vary between databases and literature references.Because of this, one should take caution when using the solubility product constants forcertainminerals.Inaddition,notallinformationcanbefoundinonedatabase.Forexample,theKSPvalueforK-struviteandIronphosphatecannotbefoundinthedatabaseofMINTEQ(Gustafsson,2010).Onereasonisthatthisdatabaseismostlyusedforgeochemicalstudiesandsuchprecipitatesarenotcommoningroundwater.Onecouldaddmanuallythevaluesfoundelsewhere(e.g.literature)forthesemineralsinthedatabase.Dissolutioncanbeconsideredas the inverseof theprecipitationkinetics. In linewith this,thedissolutionrateequationcouldbeexpressedinthegeneralformas:

r! = −k!"##s M!! ! ∙ A!! !!! − M!!

!! ∙ A!! !

!!!

!

Eq.2.4.6

Lizarralde et al. (2015)mathematically expressed the dissolution rate similar to Eq. 2.4.5withtheexceptionofactivationtermsA!"##andA!", whichdonotaffectthere-dissolution

r! = −k!"##s M!! ! ∙ A!! !!! − M!!

!! ∙ A!! !

!!!

!

∙ A!" Eq.2.4.7


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LIQUID-GASPROCESSES2.5. Stripping/volatilizationandabsorptionMass transfer between the liquid phase and gas phase are modelled to describe thedissolutionofthegaseouscomponentsformedduringbiologicalreactionsintotheaqueousphase(i.e.absorption)aswellasthemasstransferofthedissolvedformofthesegaseouscomponentsintothegasphase(i.e.volatilization-duetonaturalphenomenon;stripping-duetoamechanicaldevice).DerivedfromFick’sfirstlaw(Fick,1855),theequationbelowisaverycommonformofthekineticrateequationfortheliquid-gastransfer:

r!,!/! = k! ∙ a ∙ K!,! ∙ P! − C! Eq.2.5.1where: r!,!/! Masstransferratebetweenthegasandliquidphase;k!a Gastransfercoefficient;k! Masstransferrate;a Contactareabetweentheliquidandgasphase;K!,! Henry’sconstant;P! Partialpressure;C! Dissolvedconcentrationofthegaseouscomponent.TheproductoftheHenry’sconstantandthepartialpressureofthegas K!,! ∙ P! givesthesaturation concentration (CS). The gas transfer coefficient k!a is dependent on thetemperature. ASCE (1993) gives the widely-used relationship between kLa (in d-1) andtemperature(°C):

k!a T = 1.024(!!!") ∙ k!a (15 °C) Eq.2.5.2The most common gaseous components which can be considered during modelling ofstripping processes in wastewater treatment are: O2, CO2, NH3, N2, H2, CH4, and H2S(Lizarraldeetal.,2015).OTHER2.6. TemperaturecorrectionFor most of the kinetic rate used, temperature is an important variable that affects thekineticparameterssuchaspresentedinEq.2.5.2.TheArrheniusequationiswidelyusedto


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quantify the effect of temperature on the kinetic rates of chemical and biochemicalreactions(Fogler,2005;Pelegetal.,2012).Therelationshipisgivenby:

k T = A ∙ exp−ER ∙ T Eq.2.6.1

where: k T Rateconstantasafunctionoftemperature;A Pre-exponentialfactor;E Activationenergy;R Gasconstant;T Temperature(K).In several processes occurring in biological systems, a variant of this equation has beenwidelyused(Sheridanetal.,2012;Henzeetal.,2000).TheequationhasbeenreformedintowhatiscalledtheModifiedArrheniusFunction(MAF),givenby:

k T = k!" ∙ θ(!!!") Eq.2.6.2where: k!" Rateconstantat20°C;θ Temperaturecorrectionfactor.ThisequationhasbeeneasiertousesincethetemperatureisgivenindegreesCelsiusandthe correction factor is fairly measureable (Kadlec and Knight, 1996; Kadlec et al., 2009;Nguyen et al., 2014). However, the main assumption is that the operating temperatureshouldbeclosetoorlowerthan20°C.Sheridanetal.(2012)hasshownthatanerrorashighas25%isfoundifusedattemperaturesupto50°C.Sheridan et al. (2012) thus suggested a further modified Arrhenius equation for use inbiologicalsystemsandapplicableforalltemperatureranges:

k T = k!"# ∙ exp βT− 293293 ∙ T Eq.2.6.3

where: k!"# Rateconstantat293K(20°C);

β ER

On the other hand, the expanded square root equation of Ratkowsky (Ratkowsky et al.,1983)canalsobeusedtodescribethetemperatureeffectsduringmicrobialgrowth:


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µ T = b T− T!"# 1− e!(!!!!"#) ! Eq.2.6.4

where: µ T Growthconstantasafunctionoftemperature;b, c Empiricalparameter;T!"# Minimumtemperatureatwhichgrowthisobserved;T!"# Maximumtemperatureatwhichgrowthisobserved.Hiatt (2006)andHiattandGrady (2008)haveused thisRatkowskyequation fordescribingthe growth of heterotrophs and autotrophs. According to Ratkowsky et al. (1983), theequation eliminates the need for setting upper limits on the valid temperature range.Betweenthetemperaturesof5°Cand25°C,thebehaviouroftheArrheniusandRatkowskyequationsaresimilar.At temperatureshigher than25 °C thevaluesarestill increasing forArrhenius.However,using theRatkowskyequationat temperatureshigher than25 °C thevalues are slowly increasinguntil it reaches amaximumpre-defined temperature andwillthen start to decrease at further higher temperatures. Some models, such as SHARON(SinglereactorsystemforHighactivityAmmoniaRemovalOverNitrite)process,typicallyisoperatedathightemperatures(i.e.30-40°C)(Hellingaetal.,1999;Volckeetal.,2007)andtheuseoftheRatkowskyequationispreferable.


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3. IMPLEMENTATIONDETAILSANDNUMERICALISSUESThe implementationof thephysico-chemicalmodelwithinplant-wideframeworks leadstosomenumericalissuethatneedtobeaddressed.Theaqueousphasechemistryreactsveryquickly,whilethebiologicalprocesses(aswellasprecipitation)reactsveryslowlygivingrisetoastiffsystem(Solonetal.,2015;Flores-Alsinaetal.,2015;Lizarraldeetal.,2015).Implicitsolverscouldbeusedtosolvethesetypesofsystems,butnotveryefficiently fordynamicinputs, noise and controller characteristics (if used for a control perspective). Rosenet al.(2006)havesolvedthistypeofissueinthepastforADM1bysolvingpHandhydrogenstatesas independent algebraic equations and used an explicit Runge-Kutta solver for the otherordinary differential equations (ODE). However, this approach cannot be used due to thelargeinterdependencieswithinthealgebraicsystem.Solonetal.(2015)andFlores-Alsinaetal. (2015)haveusedamulti-dimensionalversionoftheNewton-Raphsonmethod(Pressetal., 2007) to solve the highly interdependent nonlinear algebraic system while thedifferential equations are solved separately with an ODE solver. Musvoto et al. (2000a;2000b), Sötemann et al. (2005a; 2005b), and Poinapen and Ekama (2010) have usedchemical species in their state vectors and thus have expressed all processes as ODEs.Lizarralde et al. (2015) have calculated fast reactions (e.g. aqueous phase chemistry) asdifferentialalgebraicequations(DAEs)ateachtimestepsimilartoBatstoneetal.(2002)andRosen et al. (2006) for ADM1. Components are included in the state vector and arecalculatedusingODEs,butateachtimestep,theconcentrationofeachspeciesiscalculatedusingalgebraicequations(DAEs).Solonetal.(2015)andFlores-Alsinaetal.(2015)haveusedexternalsoftware(i.e.MINTEQ)in order to check the precision of their results in relation to the values obtained fromMINTEQ,bothresultsagreeinginvalues.Lizarraldeetal.(2014)havefurthershownboththeusesofanexternalsoftwaretooloratailoredcodetosolvetheaqueousphasechemistry.Both have shown similar simulation results, however, it was with the tailored code thatshowed the fastest simulation times (Lizzaralde et al., 2014; 2015). It should be noted,however,thatintheirstudyspecieswhichareconsideredinsignificantwereremovedfromthetailoredcodeinordertoimprovethesimulationspeed.Nevertheless, it is stillanon-goingdiscussionwhetherusinga tailoredcodeor includingageochemical software as a sub-routine to the biochemical model. It could be that thegeochemical software could be faster as long as the number of components is alsodecreasedbasedontheneedsofthebiochemicalmodel.


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4. AVAILABLESOFTWARETOOLSPHREEQC (Parkhurst and Appelo, 1999) is a computer program for speciation, batch-reaction,one-dimensionaltransport,andinversegeochemicalcalculationspublishedbytheUS Geological Survey (USGS). It is designed to perform a wide variety of aqueousgeochemical calculations. Italso includes thePitzeraqueousmodelwhichcanbeused forhigh-salinitywatersthatarebeyondtherangeofapplicationfortheDebye-Hückeltheory.Itismentionedintheirlimitationsthelackofinternalconsistencyinthedatainthedatabases.Nosystematicattemptshavebeenmadetodeterminethecurrentconsistencyofthedatawiththeoriginalexperimentaldata.MINTEQorVisualMINTEQ(Gustafsson,2010)isalsoafreewarechemicalequilibriummodelforthecalculationofmetalspeciation,solubilityequilibria,sorption,etc.fornaturalwaters.Furthermore,itincludesstate-of-the-artcomplexationmodelstoestimatebindingofionstohydroxidesurfacesandorganicmatter.Thecode that isbeingusedwasoriginallybuiltonUSEPA’sMINTEQA2softwareand ismaintainedattheRoyal InstituteofTechnology(KTH),Stockholm,Sweden. It runson theWindowsplatform. It also interactswithExcel fordataimportandexport.MINTEQA2 (Allisson et al., 1991) is an equilibrium speciation model that can be used tocalculate the equilibrium composition of dilute aqueous solutions in the laboratory or innatural aqueous systems. The model is useful for calculating the equilibrium massdistributionamongdissolved species, adsorbed species, andmultiple solidphasesunderavarietyofconditionsincludingagasphasewithconstantpartialpressures.Acomprehensivedatabase is included that is adequate for solving a broad range of problemswithout theneedforadditionaluser-suppliedequilibriumconstants.CHEAQSNext(Verweij,2008)isthenextgenerationofthespeciationprogramCHEQSPro.CHEAQSNextisacomputerprogramforcalculatingCHemicalEquilibriainAQuaticSystems.You supply input data, the program calculates the chemical speciation for you. CHEAQSNext is freeware. You can calculate the concentration of complexes, but you can alsocalculate:redoxequilibria,complexationbynaturalorganicmatter,solidsthatareformedduetooversaturation,adsorption(surfacecomplexationmodel).

The Geochemist’s Workbench® (Lee and Goldhaber, 2011) is an integrated geochemicalmodelling package for balancing chemical reactions, calculating stability diagrams andequilibriumstatesofnaturalwaters,modellingreactivetransport,plottingcapabilitiesanddata storage. It makes use of the thermodynamic datasets from other tools, such asPHREEQC,WATEQ4F,andVisualMINTEQ.Itcanalsocalculateflowfieldsdynamically,whichmakesituniqueamongtheothertools.


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Asidefromthese,therearestillnumerousgeochemicalmodellingprogramsinusesuchas:MINEQL+,WHAM,WATEQ4F, SOLMINEQ, CrunchFlow, CHEPROO, ECOSAT, Aqion, CHESS,HSCChemistry®,HYDROGEOCHEM,ChemPlugin,ChemEQL,TOUGHREACT.


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5. CONCLUSIONSTheimportantaspectsofaphysico-chemicalmodelare:

§ Weakacid-baseandwastewatersolutionchemistry(i.e.speciationandionpairing);§ Solid-liquidmasstransferprocesses(i.e.precipitationanddissolution);§ Gas-liquidmasstransferprocesses(i.e.stripping/volatilizationandabsorption);§ Non-idealitycorrections(i.e.ionicstrengtheffects/activitycorrection).

Thisliteraturereviewcouldbepartofageneralframeworkthatcouldbeusedforphysico-chemicalmodellingasapplied towastewater treatmentsystems. Itcouldserveasa listofrecommendationsforincludingtheaspectslistedaboveinaPCMthatonewantstodevelopfor his/her own system. It does not have to include everything that is written here, butrathertheusershould lookathis/herownsystemonacase-to-casebasisandusewhat isneededandsuitable.


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