SYSEN533

DeterministicModelingandSimulationExam1

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

coefficientfortheright-hand sidedrain.

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)

Figure1.Compound TankSystem

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:

k1

A1+Xƒ2X

k–1

k2

X+Yƒ2Y

k–2

k3

A5+YƒA2

k–3

k4

X+ZƒA3

k–4

k5

A4 +Zƒ2Z

k–5

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:

dx =–y–zdt

dy =x+aydt

dz =b–cz+xzdt

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).

Whyisthis systemnon-linear?

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:

rb  =

k1 x2

+         2

(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.