C H G 3 3 3 1 D R A N D R E W S O W I N S K I

FractionalConversionofPackedBedReactorsolvedwithVBAShailJoshi(7282674)HusseinHaider(7534475)

Tuesday,December8th,2015

08Fall

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ExecutiveSummary Thisreportexploresthedesignofapackedbedreactorthatisgovernedbya

systemofdifferentialequations,allofwhichrelatepressure,temperature,and

fractionalconversion.Fractionalconversionreferstothereactionthattakesplaceinside

thereactor;itisthemassormolarpercentageoftheinletreactantsthatconverttothe

products.Allthreeofthesegoverningparameters(referredtoasX,PandT)areagain

relatedtotheweightofthecatalystpresentinthereactor.

Giventhatthethreegoverningequationsaregiven,itisstandardengineering

practicetofirstmodeltheprocess-inthiscaseareactiontakingplaceinsideapacked-

bedreactor–toexplorethedynamicsofthesystem.Alotofthevariablesofthereactor

arefixed(suchastheinletconcentration,flowrate)asonlyonetypeofreactorisbeing

explored,howeverthecodeconstructedinVBAconsidersthesetobevariableswhose

valuesmaybechangedintheirrespectivecellstoproveforarobustfunction.And

buildingarobustcodeisoneoftheaimsofthisproject,aswillbeexploredinthe

relevantsectionsofthisreport.

Runge-Kutta4methodisusedtosolveforX,PandT.Preliminarilytowritingthe

codeinVBA,pseudo-codewaswrittenandavalidationsheetmade,asareferencefor

latercalculations.ThecodeinVBAwasthenslowlywrittenwithonemainfunction

callingonthethreefunctionsrelatingX,PandT,iteratinguntilthefinalvaluesofthese

threeparameterswerefoundwhenthetotalweightofthecatalystwasinsidethe

reactor.

Thecodewasbuilttoberobustanduser-friendly,alongwithgraphsintheexcel

sheetsothattheusercoulddeduceatwhichpointdidaddingmorecatalystweight

becomeinefficientinregardstoincreasingfractionalconversionofthereactants.

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TableofContentsExecutiveSummary.....................................................................................................1

TableofContents.........................................................................................................2

Background..................................................................................................................3Runge-KuttaMethodsforODEs...........................................................................................................................5

Design..........................................................................................................................7FinalDesign...............................................................................................................................................................10

Validation..................................................................................................................11

ImprovementsandExtensions...................................................................................13

Conclusion.................................................................................................................14

Appendices................................................................................................................17Code..............................................................................................................................................................................17ListofTablesandFigures...................................................................................................................................19TaskAllocationSheet............................................................................................................................................20Shail..............................................................................................................................................................................20Hussein........................................................................................................................................................................20PersonalEthicsStatement..................................................................................................................................21SampleCalculations(stepsize0.1).................................................................................................................22References..................................................................................................................................................................24

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Background

Theprojectinvolvesapackedbedreactorandahypotheticalfirstorder

reaction.Aschemicalengineersitisourtasktodeterminehowtooptimizethe

processsothatityieldsthehighestfractionalconversionandisstilleconomically

viable.Thefractionalconversion(X)ofthereactingspecies,basicallyrelatesthe

molesfedintothereactorwiththemolesreacted.Itisdefinedasfollows

𝑋 =𝑚𝑜𝑙𝑒𝑠 𝑟𝑒𝑎𝑐𝑡𝑒𝑑

𝑚𝑜𝑙𝑒𝑠 𝑓𝑒𝑑 𝑡𝑜 𝑟𝑒𝑎𝑐𝑡𝑜𝑟

Thefractionalconversionisdependentonthecatalystmass(W)asisthe

pressureandtemperatureofthereaction.Thusasystemof3-coupledordinary

differentialequationscanbeformedasfollows:

Where X is the conversion, W is the catalyst mass (kg), 𝐶!!is the initial

concentration of species A (mol/m^3), 𝐹!! is the initial molar flow rate of the species A,

k’ is the modified rate constant (m3/kg·s), 𝐸! is the activation energy (kJ/mol), R is the

gas constant (kJ/mol·K), ε is a parameter based on the difference of moles reacted and

produced, T is the temperature at a given point in the reactor (K), P is the pressure at a

given point of the reactor (Pa), α is a constant that is characteristic of the particle bed

properties (kg‐1), ΔHr is the heat of reaction (kJ/mol), and 𝐶! is the average head

capacity of the reactant and products (kJ/mol·K).

Alltheparametersoftheprocessaresummarizedbelow

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Table1:TestParameters

To simplify the system of ODEs a dimensionless system will be used and

substituted into equations 1,2,3. This will make the general analysis of the system clearer

thus resulting in an easier application of Runge-Kutta. The dimensionless system is as

follows:

Where 𝑊! is the total mass of the catalyst. These diminsionaless parameters will

now be plugged back into equations 1,2,3. This results in the follwing equations:

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Sinceitisunknownwhichcatalystweightwillyieldthehighestconversion,

themainaimforthisprojectwillbetodeterminewhatcatalystweightcorresponds

tohighestfractionalconversion,thusoptimizingtheprocess.UsingRunge-Kutta

methods,thefractionalconversion,pressureandtemperaturewillbedeterminedat

differentdimensionlesscatalystweightsfrom0to1.P(hat) and T(hat) will finally be

solved using runge kutta technique and graphed with respect to W(hat). Thenbasedon

theresultanoptimumcatalystweightwillbechosenthatissatisfactoryboth

economicallyandfromachemicalstandpoint.

Runge-KuttaMethodsforODEs

WidelyusedinengineeringmathematicsandotherdisciplinesRunge-Kutta

methodsaresomeofthemostcommon,simpleandrobustmethodstoapproaching

systemsofODEs.Runge-Kuttamethodsfallintoawidebranchofiterative

techniquesthatareimplicitandexplicit.Euler,Heun’sandMidpointmethodscanall

beclassifiedasRunge-Kuttatechniques.Howeverforthisprojectthemoreclassical

andexplicitRK4techniquewillbeused.

ThemainprocedureinRK4withnODEsisgivenbelow:

[10]

TheRK4methodusesfourkvaluestoreachafinalestimationofthevalueofa

certainvariable.Thesearereferredtointhefunctionask1,k2,k3andk4.Onceall

thekvaluesaredeterminedtheyaresimplysubstitutedintoequation10summed

andthefinalvaluefortheODEisdetermined.Thusthefirststepinusingthis

techniqueistocalculatethekconstants,theyaredefinedinliteraturebelow:

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[11]

[12]

[13]

[14]

RK4isconsiderablymorepowerfulthanothertechniquessuchasEuler’sor

midpoint,andforthisreasonitconvergesfasteranddoesnotrequirealowstepsize

togetanaccurateanswer.HoweverthemaindrawbackwithRK4isthatitistime

consumingespeciallywhentherearelotsofsteps.Consequentlyitisalsoveryeasy

tomakeanerrorwhencalculatingbyhand,ifamistakeismadeearlyoninthe

initialcalculationthenitisoftencarriedthroughtherestoftheprocedureresulting

inanexponentialincreaseinerror.Thususeofaniterativeprogramsuchasexcel

minimizesthechanceorerrorandsubsequentlyreducesthecalculationtime.

HowevercreatinganRK4programusingVBAorsettingupanexcelsheetis

neverthelessconsiderablymorechallengingincomparisontoothermethods.This

projectusedRK4primarilyduetoitsefficiencyandeffectivenessinsolvingmultiple

ODEequations.

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DesignThecodeconstructedwasbasedontheRunge-Kutta4thorder(RK4)method

tosolveasystemofdifferentialequations.Therearethreedimensionlessvariables;

X,P(hat)andT(hat),allofwhicharedifferentiatedwithrespectedtothesame

parameter,W(hat).Thefunctionsare:

Thesethreedifferentialequationsgovernthepropertiesofthepackedbed

reactor.SinceitissoughttosolveforthethreevaluesofX,P(hat),andT(hat)fora

certaintotalweightWt,theRK4methodwillbeusedforeachvariablewiththethree

functionsabove,andanoverallgoverningprocedurethatthecodewillbebasedon,

thiswillthecode’smainfunction.

Oncethesevaluesarecalculated,theyarethenusedtofindtheRK4estimate

forthevariable’snextvalue:

RK4estimate

Theseestimatesarecalculateduntilavalueisreached,orconvergeduntilthe

proportionalweightvalueW(hat)isequalto1,meaningthefullvalueofthetotal

weight.ThisistheentiretyoftheRK4method.Althoughitmayseemsimple,itis

quitetime-consumingifdonemanually,sincetheprocessmustberepeatedforeach

ofthethreevariablesX,P(hat)andT(hat).

Sincetheprocedureforcalculatingthenextestimationofeachofthethree

variablesfollowsthesameorder;thecodemustreflectthat.Itmustbeoneskeleton

forallthreevariables(andpossiblyanyothervariablethatwoulddefinethepacked

bedreactorsystem;ifadifferentgoverningfunctionshouldbespecifiedsimilarto

equations[7],[8],and[9]).Thisistheapproachthatthecodewasconstructedwith.

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Firstoffthethreefunctionsareconstructedthatcorrespondtoequations[7],

[8]and[9]:Public Function dXdW(parameters As Range, X As Double, Ph As Double, Th As Double, wt As Double) As Variant

Dim Cao, Fao, k, e, a, Ea, Hr, Cp, X0, T0, P0 As Double

Cao = parameters(1, 1) Fao = parameters(2, 1) k = parameters(3, 1) e = parameters(4, 1) a = parameters(5, 1) Ea = parameters(6, 1) Hr = parameters(7, 1) Cp = parameters(8, 1) X0 = parameters(9, 1) T0 = parameters(10, 1) P0 = parameters(11, 1)

If Cao <= 0 Or Fao <= 0 Or Ea <= 0 Or T0 = 0 Or Cp = 0 Then dXdW = "PARAMETERS INVALID" Else dXdW = (Cao / Fao) * k * Exp(Ea * ((1 / T0) - (1 / (T0 * Th)))) * (1 - X) * Ph * wt / ((1 + (e * X)) * Th) End If End Function

Andtheremainingtwofunctionsfollowasimilarstructure,withtheir

correspondingequations:

dTdW = (Hr / Cp) * (Cao / Fao) * k * Exp(Ea * ((1 / T0) - (1 / (T0 * Th)))) * (1 - X) * (Ph * wt / ((1 + (e * X)) * T0 * Th)) dPdW = -(a / 2) * (1 + e * X) * (Th / Ph) * wt

Thesethreefunctionsaretobecalledupon,takinginthearrayofcellsthat

theusermaysimplyscrolloverintheExcelspreadsheet,checkingfirstthatnoneof

theinputcellsareinvalidvalues,andthencalculatingthefunctionvalueforthe

purposesoffindingaKoranestimationvalue.

Themainfunctionthenconstructsaforloopthatperformsaseriesof

calculationsforanumberoftimesdeemedno_of_steps,whichistheinverseofthe

stepsizeh,sincethisiswhatdetermineshowextensivelythecodewilliterate

estimationsfor(forexample,astepsizeofh=0.01willtake100steps).Itshouldalso

takeintheinitialestimatesofXi,Pi,Ti.Themainfunctionthenholdstheframe:Public Function RungeKutta(ByVal parameters As Range, ByVal h As Double, ByVal wt As Double, ByVal Xi As Double, ByVal Pi As Double, ByVal Ti As Double) As Variant

If wt < 0 Or h <= 0 Or dXdW(parameters, Xnew, Thnew, Phnew, w) = "PARAMETERS INVALID" Then

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RungeKutta = "CHECK PARAMETERS" Else

For i = 1 To no_of_steps …………… …………… ……………… ……………… ……………… ……………… Next i

End If RungeKutta = Runge End Function

Theusercansimplyselectthearrayof11parameters,alongwithstepsize,

totalweightandinitialestimates.Insidetheloop,theRunge-Kutta4methodis

simply“translated”intocodeform:X = Xnew

Ph = Phnew Th = Thnew k1X = dXdW(parameters, X, Ph, Th, wt) k1P = dPdW(parameters, X, Ph, Th, wt) k1T = dTdW(parameters, X, Ph, Th, wt) X = Xnew + (k1X * 0.5 * h) Ph = Phnew + (k1P * 0.5 * h) Th = Thnew + (k1T * 0.5 * h)

AspertheRK4method,Kvaluesarecalculatedbycallinguponthethree

previouslydefinedfunctions.ThefirstK1valuesarecalculatedonlywiththeinitial

estimatesandthesubsequentKvaluesmusttakeinmodifiedestimates.Thesesteps

arefolloweduntilall4Kvaluesarecalculatedtothencalculatetheestimationsand

outputthemasathree-cellarrayinexcel:Xnew = Xnew + (h / 6) * (k1X + (2 * k2X) + (2 * k3X) + k4X) Thnew = Thnew + (h / 6) * (k1T + (2 * k2T) + (2 * k3T) + k4T) Phnew = Phnew + (h / 6) * (k1P + (2 * k2P) + (2 * k3P) + k4P) w = w + h Runge(i, 1) = Xnew Runge(i, 2) = Phnew Runge(i, 3) = Thnew

TheVBAcodeisthereforeasimpleone,followingthestructureand

procedureoftheRK4method.Itperformsaloopwithacontrollednumberof

iterations.ThethreefunctionsdXdW,dPdWanddTdWarecalleduponforeachK

value.Theusersimplyneedstoselecttheparameters,stepsize,weightandinitial

valuesofX,P(hat),T(hat)bydraggingandclickingontheappropriatecellsonly

onceandthecodefinishestherest.

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WhengraphingthevaluesofX,P(hat),T(hat)obtainedbythethe

RungeKuttafunction,withrespecttoW(hat),itisapparentthatXconvergestoit’s

highestapproximatevaluenear1at0.65.Meaningthatreadingthetableinthe

“Design”taboftheexcelsheet,conversionXreaches0.99whentheproportion

W(hat)is0.65,or65%ofthetotalweight(inthiscase35kg).Anidealcatalyst

weightforthispackedbedreactoristherefor65%(35kg)=22.75kg.

FinalDesign

Theidealcatalystweightwasdeterminedbasedonallofthefractional

conversions,pressuresandtemperaturescalculatedintheexcelsheet.Froma

purelythermodynamicstandpointitwasseenthatincreasingcatalystweight

correspondedtoahigherfractionalconversionandtemperature.Pressure,onthe

otherhand,graduallydecreasedasweightwasincreased.

Thusthetotalcatalystweightof35kgyieldsthehighestamountofproduct.

Howeveritwasseenthattheincreaseinfractionalconversionwastrivialasthe

proportionalweightreachedthetotalweight.Thiscanbeseenonthegraphwhere

theconversionandtemperatureslinesleveloutafteraW(hat)ofabout50.Thisis

becauseitimpossibletohaveaconversionthatis100percentbasedon

thermodynamiclaws.Thereforeitwouldnotmakesensetobuy35kgofcatalystifit

onlyincreasesthefractionalconversionbyanegligibleamount.Inapackedbed

reactorthatisoperatingonamacro-scalethisdifferencewouldbeinfinitesimal.

Thusthefinalcatalystweightwaschosentobearound22.75kg.Thisweightyieldsa

conversionof99%ensuringahighfractionalyieldisobtainedwhilebeingas

economicalaspossible.Thetablebelowsummarizesthepressure,temperatureand

pressurethatcorrespondtoaweightof22.75kgTable2ChosenOperatingConditions

W(hat) X P(hat) T(hat)0.65 0.990098208 0.840611223 1.96259548

ERROR 0.1088% 0.4874% 0.0534%Weight X P(Pa) T(K)22.75kg 99% 85174.93 883.168

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ValidationAseparateRK4wasdoneonthesystemwithoutusinganyprograming

techniquestocompareandvalidatethevaluesobtainedusingVBA.Averysmallstep

sizeof0.01wasusedtoensurethemethodwasasaccurateaspossible.Sinceno

programingwasusedthespreadsheetofthevalidationisconsiderablylargerthanthat

oftheVBA.Thisisbecausenumerouscolumnsandrowshadtousedinorderto

accommodatethesheernumberofvaluesandconstantsthatneededtobecalculated.

TheValidationvalueswerethencomparedwiththevaluesobtainedthroughVBA.This

wasdonetoensurethecodewasworkingandoutputtingthecorrectvalues.Asummary

ofthecomparisonissummarizedinthetablebelow:

Table3ComparisonsBetweenValidationandVBA

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Initiallytheerrorwashighduetotheinitialguessfortheconversion,P(hat),and

T(hat).Howeverforthemajorityofcasesthedifferencebetweenthevalues

calculatedusingVBAandexcelisminute.Theerrorisalsoverysmalltherefore

confirmingthattheprogramisoutputtingcorrectvalues.

Ascanbeseenbelow,bothtechniquesyieldalmostanidenticalgraph:Figure1ComparisonsBetweenValidationandVBA

Additionallythecodeitselfwasdesignedtovalidateandeliminateanyusererrorduring

theentryofparameters.

If wt < 0 Or h <= 0 Or dXdW(parameters, Xnew,

Thnew, Phnew, w) = "PARAMETERS INVALID" Then

RungeKutta = "CHECK PARAMETERS"

Thisensuresthatusersdon’tentervaluesthatcantbephysicallyormathematically

possiblesuchasnegativevalues.

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ImprovementsandExtensions Afterhavingcompletedtheproject,itwasclearthattherewasaminute

differencebetweentheresultsofthevalidationandthatofthecode.Asseeninthe

“Comparison”taboftheexcelfile:Table4ComparisonBetweenValidationandVBAatW(hat)1

Validation VBA ErrorW(hat) X P(hat) T(hat) X P(hat) T(hat) X P(hat) T(hat)

1 0.999778 0.6773 1.972 0.99976 0.68237 1.97199 0.0022% 0.7494% 0.0011%

Thediscrepancyofthevaluesobtainedbybothmethodsisnegligible,asseen

bytheerrorvaluescomparingthetwomethodsonthethreeright-mostcolumns.

ThegreatesterrorisintheP(hat)pressurevaluesatapproximately0.75%.

Nevertheless,extrastepscouldhavebeentakentoavoidthesediscrepancies.One

exampleisintegratingadaptivestepsizingintothecode.Thiswasinitiallyavoided

asitwasdeemedunnecessaryandaninefficientuseoftime,asitwouldrequirea

verycomplexmodificationtothecurrentcode.Howeverhavingdonethiscould’ve

improvedthecodebytakingstepsizesbasedonthedesirederrorinvaluesand

precisionofconvergence.Consequentlyitwouldhaveincreasedthecode’s

complexityconsiderablyandmadeitpossiblylessuserfriendly.Thustaking

uniformstepswasusedasitwassimpleranddidn’toutputabigerror.

Anideatoconsiderwhenitcomestoextendingtheprojectisthatconcerning

cost:asdesignengineersitisimportanttoconsiderthecostofmaterialsandthe

costofthereactor.AsthefractionalconversionXbeginstoconvergetoitsupper

limitataround65%ofthetotalweight(asseeninExcel).Assuchthecostperunit

weightofthematerialcouldbeintegratedintotheVBAcodeandthegraph’s

relatingXtoweightW(hat)toseethatatacertainpoint,spendingmoreonthe

catalystweightwillnotyieldasmuchreactionproduct

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Conclusion

Inconclusiontheconversionanddimensionlesspressureandtemperature

weresolvedforthegivencatalystweightof35kg.Thecodewasrobustand

versatilesothatausercouldinputanydesiredweight,aswellmakechangestoany

ofthedifferentdependantparameters.Thevalidationwasdoneinsuchamanner

aswellsotheusercouldchangeanyofthegivenparameters.Thisensuresthatour

programandexcelsheetisadaptableandcanbeappliedonamultitudeof

situations.Makingitidealforuseinchemicalindustrywhereconstantsand

parametersneedtobeconstantlyvariedtoensurethesystemisasoperatingatits

peakefficiency.

Theexcelsheetwasdesignedtooutputgraphsoftheresultsthuschoosing

theidealoperatingconditionseasierandmakingtheoverallprogrammoreuser

friendly.Thefractionalconversion,dimensionlesspressureandtemperaturewere

plottedagainsttheproportionalweight(W(hat)).AscanbeseeninFigures1

through3thefractionalconversionincreaseswithW(hat)howeverafterabouta

W(hat)of0.5thefunctionstartstoleveloffasitapproachesitshorizontal

asymptoteof1.Thisisbecauseitischemicallyandphysicallyimpossibletohavea

conversiongreaterorequalto1.Asimilartrendwasseenwiththedimensionless

temperature,howeverthetemperatureapproachedahorizontalasymptoteof2.

Thissuggeststhereactionwasexothermicsinceanincreaseinproductsledtoa

highertemperature.Ascanbeseeninthegraphsbothconversionand

dimensionlesstemperatureoutputnearlyidenticallines,theonlydifferenceliesin

thelocationoftheasymptoteandthestartingpositionofthelines.The

dimensionlesspressureontheotherhandwasseentodecreasewithincreasing

catalystweightandincreasingconversion.Thissuggestsperhapsahighpressure

wouldholdthereactionback.Thustoshiftthereactionequilibriumtofavourthe

products,thepressurecouldbedecreasedinordertooptimizetheprocess.

Thegraphswereusedtodeterminetheidealcatalystweighttooptimizethe

process.Aweightof22.75kgwaschosenwhichcorrespondsW(hat)onthegraphs.

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Thiswaschosenbecauseitwasseenthatafterthispointtheincreaseinconversion

wastrivial.Thusitdidnotmakesenseexpendingadditionalmoneyfor35kgof

catalystifyieldincreaseisinfinitesimal.Theoperatingconditionsofthisweightare

summarizedintable2shownbelow:

Table2:ChosenOperatingConditions

W(hat) X P(hat) T(hat)0.65 0.990098208 0.840611223 1.96259548

ERROR 0.1088% 0.4874% 0.0534%Weight X P(Pa) T(K)22.75kg 99% 85174.93 883.168

Chemicalengineersoperateinindustrytooptimizeprocessesboth

economicallyandchemically.Forthisreasontheseoperatingconditionswere

chosen.Thoughtheydonotyieldthehighestamountofproducttheyarethemost

idealwhencostofcatalystistakenintoaccount.Ascanbeseeninthetable,the

conversionisat99%thusyieldingahighproduct.Theerrorvaluesofallthree

parametersarealsosufficientlylowthereforemakingitsafetooperateunderthese

conditions.

TheresultscouldhavebeenimprovedbyusingahigherorderRunge-Kutta

technique;orbyusingadaptivestepsizinginthecode,buttheseweredeemed

unnecessarysincetheRK4methodthatwasimplementedoutputtedsatisfactory

valueswithlowerrors.

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Figure2ValidationGraph

Figure3VBAGraph

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Appendices

Code

18

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ListofTablesandFiguresTable1:TestParameters.......................................................................................................................4Table2ChosenOperatingConditions............................................................................................10Table3ComparisonsBetweenValidationandVBA.................................................................11Table4ComparisonBetweenValidationandVBAatW(hat)1..........................................13Figure1ComparisonsBetweenValidationandVBA...............................................................12Figure2ValidationGraph....................................................................................................................16Figure3VBAGraph................................................................................................................................16

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TaskAllocationSheetHusseinHaider(7534475)&ShailJoshi(7282674)

ShailSetuptheinitialexcelsheet,withequations,namingvariables,and

doingtheinitialvalidations.

WritescodefordX/dW,dP/dWaswellasstartedwritingthemain

Runge-Kuttacode.

Startedwritingreport.

HusseinHelpedmakethemainexcelsheetclearer.

WritescodefordT/dW,aswellashelpingwiththemainRunge-Kutta

code.

Implementingarrayandrangeoutputsinthecodewhereneeded,

makingthecodemoreefficient.

Willwriteanequalhalfofthefinalreport.

Whenwearedonewritingeachofourrespectivepartsofthereport,we

willgatherourVBAcodes,functionsandwrittenreportparts.Wewill

discussourfindings,combineourreports,peerrevieweachother’s

work,makingourreportsandcodesclearerandmoreefficient.Weplan

toholdregularbi-latstoupdateeachotherastoourprogress.

Wewillalsousethird-partyresources,suchasonlinetutorialsforVBA,

allofwhichwillbeproperlycitedaccordingtoUniversityofOttawa

standards.

21

PersonalEthicsStatement

7282674SHAIL JOSHI

HUSSEIN HAIDER

08/12/2015

7534475

08/12/2015

22

SampleCalculations(stepsize0.1)

23

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ReferencesChapra,StevenC.,andRaymondP.Canale.NumericalMethodsforEngineers.6thed.Boston:McGraw-HillHigherEducation,2010.