Opportunity cost of your resource is defined as “the value associated with the next best use of that resource (which you must give up).” Decisions you make should reflect your opportunity cost, and not just your out-of-pocket costs. For example, if you decide to spend two hours of your time watching TV, the opportunity cost of your time is the value associated with its next […]
Imagine you are a health care manager for the Krahe Health Care facility. You have emailed your assistant manager, Michelle, the survey data for patient satisfaction, quality of care, and likelihood to recommend. You have asked Michelle to review the data and create a presentation for the staff addressing the impact of the data on the facility. Michelle reviews the data but questions the importance […]
Module 2 – Case ARGUMENTATION Case Assignment Note: Module 2 SLP should be completed before the Module 2 Case. No secondary sources are to be used in the process of writing this first paper; it is to be entirely written based on one’s point of view and supported with body points and details. Reflecting on the four gender-related videos for Module 2 (see below), write a […]
Read the following articles attached 1 & 2. Write a 8 page paper in which you: 1. Consider that Data Security and Policy Assurance methods are important to the overall success of IT and Corporate data security. a. Determine how defined roles of technology, people, and processes are necessary to ensure resource allocation for businesscontinuity. b. Explain how computer security policies and data […]
ART330 Essay Module 8 Directions: Please choose one (1) of the following essays to answer. Each essay must consist of at least 700 words (this equals two double-spaced pages in Times New Roman font 12), must be in complete sentences and where necessary, appropriate information must be properly documented in MLA style. This essay, worth forty (40) points, must be submitted to the […]
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Assignment verified through turn it In Please use lesson plan template attached DIRECTIONS: After reviewing the module resources and the readings, complete this assignment. Consult additional research. For this assignment, you will design or redesign an Interdisciplinary Lesson Plan you can use in your setting and with your students. Interdisciplinary lessons are especially interesting to gifted students since they present a unified approach to learning, […]
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)
|
F1in |
2.0 ft3/s |
F2in |
1.0 ft3/s |
|
R1 |
10.0 ft |
D2 |
20.0 ft |
|
k1 |
2.0ft5/2/s |
k2 |
3.0ft5/2/s |
|
h1(0) |
4.0 ft |
h2(0) |
4.0 ft |
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+X2X
k-1
k2
X+Y2Y
k-2
k3
A5+YA2
k-3
k4
X+ZA3
k-4
k5
A4 +Z2Z
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.
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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