1. (25 pts) Thecompound tanksystemshown in Figure1 consistsof asphericaltank ofradius R1and acylindrical tankofdiameterD2. A liquid ofconstantdensityis fed atavolumetricrateF1in intothetopof aspherical tankandvolumetric rateF2in intothetop ofthecylindricaltank. The sphericalandcylindrical tanks interactthrough thepipeconnectingthem.Theflowrates intotheconnecting pipedepend ontheheightsoftheliquid in thetanks. Thevolumetric flowrate outof thesphericaltank
intothepipeis given byF1out =k1
h1 , whilethevolumetric flow rateoutofthecylindricaltankinto
thepipeis given byF2out =k1
h2 whereh1 and h2 arethe heightsoftheliquid in thesphericaland
cylindrical tanks respectivelyandk1 is thecommonvalvecoefficient.Thecylindricaltank alsohas a
drain on theright-hand sidewhich hasvolumetricflowrateF3out =k2
h2 wherek2 is thevalve
a) Obtain a dynamicmodelthatdescribes theheightsoftheliquid in thetanks. Is this a linearornonlinearmodel?
b) For constantinputflow rates, F1in and F2in, analytically determine thesteady-statevaluesofh1 and
h2. Dotheshapesand dimensionsof thetanksaffectthesteady-statevalues?
Notethatyoudonotneedtosolvethedifferential equations for thesteady stateanalysis.
c) Simulatethesystem and plottheheightsoftheliquidin thetanks versustimeforconstantinputflowratesusing thevaluesgiven in thetablebelow.(Run thesimulation for2000sec)
2. (25pts) Chaoticsystemsareones forwhich small changes eventuallylead toresults thatcan bedramaticallydifferent. The Rösslersystemisoneofthesimplestsets ofdifferential equationsthatexhibits chaoticdynamics. In addition totheir theoretical valuein studyingchaotic systems,theRösslerequations areusefulin several areasofphysicalmodeling including analyzing chemicalkineticsforreaction networks. Consider thereaction network:
whereX,Y, and Z represent thechemical specieswhoseconcentrationsvaryandA1, A2, A3, A4, and A5 arechemicalspecies whoseconcentrationsareheldfixed bylargechemical reservoirs, serving tokeepthe system outofthermodynamicequilibrium.ki andk–i denote theforward and inversereaction rates.Thesystem of differentialequations thatdescribetheconcentrations x, y, and z(for chemicalspeciesX,Y, and Z) are:
a) Simulatethis systemfor a=0.380, b=0.300,and c=4.280with initial conditionsx(0)=0.1,
y(0)=0.2, z(0)=0.3. Run thesimulation for200seconds using a fixed-step size algorithm witha stepsizeof0.001seconds.Plottheconcentrationsx, y, andzversustimeononefigurewiththreesubplots. Additionally, in separategraphs, plotthephase-spaceplots:xversus y,xversusz, and yversus z. Finally,makea3-Dplotof xvs yvszusing theMatlab graphics command “plot3”
b) Illustratethesensitivityofthesolution to variations intheinitial conditions byrepeating thesimulation ofpart(a) withx(0)=0.0999and thenwith x(0)=0.1001. (A0.1%changein thevalueoftheinitial condition in either direction.)Keeptheinitial conditions for y(0) and z(0) thesameas inpart(a). Showthesensitivitybysuperimposing theplots forthenew valuesyouobtain forx(t),y(t),and z(t)with theoriginal plots forxvs t,yvs t,and zvs t. In addition,makeplotsof thedifferences:x(t)– xorginal(t) vst,y(t) –yorginal(t) vst, and z(t) –zorginal(t) vs t.
c) Illustratethesensitivityofthesolution to variations inparametervalues byrepeating thesimulationof part(a)withc=4.280001. [Use theoriginal initial conditions from part(a).]Show thesensitivitywith thesamesetof plotsas in part(b).
Qualitatively describethesensitivity toinitial conditions and parametervalues.
3) (25 pts) Acontinuous stir tank reactor(CSTR)is used toproducea product P fromchemicals Aand B.Thereactionis A+BàP. Ais inexcess andtherateof decompositionofBisgiven by:
(1 k2 x2 )
wherek1and k2areconstants and x2is theproductconcentration.Theequations describing thesystemaregiven by:
1 1 2 2
The parameters are: Cb1 =24.9,Cb2=0.1 ,k1=k2= 1,andu1 =u2 = 1.The initial conditions are: x1(0) =10and x2(0)=0.
a) First, simulatetheequation for x1onlysinceitdoesnot depend onx2.Notethesteady-statevaluethatyou find forx1.
b) Inthemodelforx2, initializetheintegrator for x1to thesteady-statevalue you found in part(a) andinitialize x2(0)=0. Run thesimulation for 1000seconds to determinex2(t)and thesteady-statevalueof x2.
c) Repeatthesimulation for x2 using an initial value x2(0)=10. Notethenewsteady-statevalueyoufindfor x2.
d) Bothof thepreviouscasesarestable.Thereis anothersteadystatecorresponding to everything elsebeingthesameand thesteadystates forx2and x1being x2= 2.793and
x1=100.This is an unstablesteadystate.Demonstratethis bysetting theinitialvalueoftheintegratorforx2tox2(0) =2.80andshowthatthesimulation goesto theuppersteady-statevalue.Repeatthesimulationwithx2(0)=2.79 and showthatitgoes tothelowersteady-statevalue.Thismeansthatanysmall fluctuation willcausethesystemtofalltosteadystateofpart(b)or risetothesteady stateof part(c).
e) Demonstratethatthesteadystatewith x2=2.793is unstablebylinearizing theequation forx2aboutthesteady-statevaluesandshowingthatthelinearsystem thatresultsis unstable. Youcan do this by either solving thedifferential equationor by simulating thesystem witha small initial Dx2.
4) (25 pts) Considerthefollowing linear system
G(s)=–8s+6 2s3 +9s2 +13s+6
For this problem, usea unit step input.
a) Createamodel for this systemusingonly integrators and run itfor10seconds.
b) Replacetheintegratorswith discreteintegratorsand investigatetheeffectof forward, backward,and trapezoidal integratorswith sampling timesof0.01, 0.1, and0.2seconds.
Notethatintheconfiguration parametersyouwillneedtospecify thestepsizeand changetheintegratortodiscretestates only.Also,setallof thesampling times inthevarious blocksequal tothediscretesampling time.
c) For each case, comparethetruesolution tothediscretesolution intwo ways:
(i) Plotthetruesolution (in blue) and discretesolution (in red)ona singleplot.
(ii) Plotthedifferencebetween thetruesolution and thediscretesolution.
If youwantto bereallyfancy, usethesubplotcommand toshowboth plotsin asinglefigure.
d) Computeanestimateof theintegral squareerror foreach caseand createa tableofthesedifferences.
e) Whatconclusionscanyoudrawregarding theaccuracyofthedifferentmethodsand step sizes?
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