Colorado School of Mies CHEN43 Secial Modelig Techiques Secial Modelig Techiques Summary of Toics Deviatio Variables No-Liear Differetial Equatios 3 Liearizatio of ODEs for Aroximate Solutios 4 Coversio to Deviatio Variables & Liearizatio of ODEs i a Combied Oeratio 5 Liear Examle: Heated Stirred Tak with Costat Ilet Flow 6 No-Liear Examle: Heated Stirred Tak with Variable Ilet Flow 7 No-Liear Examle: Flow Through Valve with No-Liear Characteristics 7 Summary of Toics There are two secial toics that are useful ad ossibly required to use Lalace trasform techiques to solve our rocess dyamics roblems: Deviatio variables Liearizatio of ODEs Deviatio Variables It may be coveiet to use deviatio variables whe modelig a roblem Deviatio variables are simly the differece of the actual variable to the origial value Mathematically: y t y t y y t y whe our iitial coditio corresods to a iitial steady state ( y y ) This is coveiet because it greatly simlifies the alicatio of the Lalace trasform Trasformig a -th order derivative to Lalace sace requires the first (-) derivatives at t If we use deviatio variables, these will all be zero So, a -th order ODE i the form of: 2 d y d y d y dy a a a 2 a 2 a y f t simly becomes a -th order olyomial i s : 2 2 a s a s a s a s a y f t Joh Jechura (jjechura@miesedu) - - Coyright 27 Aril 23, 27
Colorado School of Mies CHEN43 Secial Modelig Techiques How do we covert our origial ODE to be i terms of deviatio variables? Substitute i & searate out the terms ivolvig the iitial coditios If we start from a steady state coditio the the combiatio of terms ivolvig the iitial coditio will be equal to zero & disaear For the -th order ODE let s isert the sum of the deviatio variable & the steady state value i for the variables: d y d y dy a a a a y f t d y y d y y d y y a a a a y y f f Sice the time derivative of steady state variables are zero: d y d y dy a a a a y y f f Let s grou the steady state variables together: d y d y dy a a a a y f f a y But sice at the steady state variables are related by f a y : d y d y dy a a a a y f It aears that all we ve doe is chage the origial variables directly to the deviatio variables, but we ve really doe more tha just that As a examle, let s look at the equatio that comes from the mass balace o a uright cylidrical tak with a liquid of costat desity This equatio will be: A F F where h, F, ad F ca all be fuctios of time Sice it is a ODE, we also eed a boudary coditio If the coditio is set at t, the it is a iitial coditio If we cosider that we are modelig a erturbatio to the steady state coditio, the the iitial coditio is for steady state, or: Joh Jechura (jjechura@miesedu) - 2 - Coyright 27 Aril 23, 27
Colorado School of Mies CHEN43 Secial Modelig Techiques IC: F F If we relate the outlet flow to the tak level, the we ca get our steady state level, assume the outlet flow is roortioal to the level, or F C h The: A FC h F IC: F Ch h C Let s defie the deviatio variables: h t h t h F t F t F Now the ODE & IC are: d A h h F F C h h A A F Ch F Ch h Let s ad: with: A F Ch h With deviatio variables, the iitial coditios will always be zeros, which may be easier to work with to solve the ODEs (esecially whe usig Lalace trasforms) No-Liear Differetial Equatios No-liear ODEs are oes with at least oe o-liear term (other tha those ivolvig t ) So, the followig ODE is o-liear: dy y t 2 2 whereas the followig is still liear: Joh Jechura (jjechura@miesedu) - 3 - Coyright 27 Aril 23, 27
Colorado School of Mies CHEN43 Secial Modelig Techiques dy yt 2 Deviatio variables are ot ecessarily easier to work with whe dealig with o-liear ODEs For examle, for the tak roblem with a o-liear valve: A FC h whe we aly deviatio variables we get: A F F C h h I this case we caot factor out & elimiate the steady state values Liearizatio of ODEs for Aroximate Solutios There are more methods for solvig liear ODEs tha for solvig o-liear ODEs Sometimes the solutio to the liearized ODE may be close eough to the solutio of the origial o-liear ODE that it may be of use This is esecially true whe the drivig fuctio is a disturbace aroud the steady-state value Presumably, the chages to the state variable are fairly small Make use of Taylor series exasio to exad the fuctio f x aroud x x : 2 df d f d f f x f x x x x x x x 2 2 dx xx 2! dx! dx xx xx We ca liearize by droig all of the 2 d order terms ad higher So: df f x f x x x dx xx As a examle, let s look at flow through a tak with o-liear valve characteristics: A F F F Cv h A Cv h F The h term makes the ODE o-liear However, let s do a Taylor series exasio of terms aroud the steady state value h : Joh Jechura (jjechura@miesedu) - 4 - Coyright 27 Aril 23, 27
Colorado School of Mies CHEN43 Secial Modelig Techiques 2 3 df d F 2 d F 3 F h F h h h h h h h 2 3 2 6 h h h h h h We will deote F h F F (sice, at steady state, the flows i & out are equal) We also kow that: df Cv df Cv F Cv h 2 h 2 h h h So, keeig oly the first or liear term from the exasio: C v h h A F F F F 2 h C v h h A F F 2 h Usig deviatio variables h h h ad F F F, the: C A v h F 2 h Eve though there is still a o-liear term i the exressio, ot affect the liear/o-liear ature of the ODE h, it is a costat so does There are other terms that may be eed to be liearized: Effect of geometry: horizotal cylider volume vs liquid level / Effect of temerature o reactio rate: E A RT ka ka,e 2 Effect of o-liear reactio rate kietics: r k C A A A Coversio to Deviatio Variables & Liearizatio of ODEs i a Combied Oeratio We ca liearize the ODE & isert deviatio variables directly i a sigle combied oeratio that works just as well for liear ad o-liear ODES The rocedure is: Cosider the ODE to have the form of: Joh Jechura (jjechura@miesedu) - 5 - Coyright 27 Aril 23, 27
Colorado School of Mies CHEN43 Secial Modelig Techiques dy C F x, x2,, xn, y where C is a costat Fid the total differetial of the right-had-side fuctio: F F F F df dx dx2 dxn dy x x2 xn y where the artial derivatives are foud by holdig all other variables costat Aroximate the total differetial as the fiite differece from the iitial steady state, ie, as the fiite deviatio variable: df F F F F dx x x x x ad so o Ay artial derivative that is ot a costat is to be set equal to its value at the iitial steady state: F F F F df F x x2 x2 y x x2 x y Chage the ODE from terms of the origial variable to that of the deviatio variable Isert the aroximate form of the total differetial: dy F F F F C x x2 x2 y x x2 x y This rocedure will work whether the ODE is liear or o-liear Liear Examle: Heated Stirred Tak with Costat Ilet Flow The heat balace equatio for a stirred tak with a steam heatig coil is: ˆ dt VC ˆ F C T T UA Ts T Joh Jechura (jjechura@miesedu) - 6 - Coyright 27 Aril 23, 27
Colorado School of Mies CHEN43 Secial Modelig Techiques If the ilet flow rate F is a costat the the oly variables besides T are T ad T s The total differetial to the right-had-side fuctio is: d F C ˆ ˆ ˆ T T UA Ts T F C dt UA dts F C UA dt Sice all of the terms i the brackets are costats they ca be iserted ito the origial ODE without ay real chage to the accuracy of the origial ODE: ˆ dt VC ˆ ˆ F C T UA Ts F C UA T No-Liear Examle: Heated Stirred Tak with Variable Ilet Flow This heat balace equatio is the same: ˆ dt VC ˆ F C T T UA Ts T but ow the ilet flow rate F is a also a variable The total differetial to the right-hadside fuctio is: ˆ d F C ˆ T T UA Ts T C T T df ˆ ˆ F C dt UA dts F C UA dt Now the oly bracketed term that is costat is the oe i frot of the dt s differetial all other bracketed terms have some variable term i them The terms ivolvig T, T, or F evaluated at the origial steady state coditio ad the iserted ito the origial ODE: ˆ dt VC ˆ ˆ ˆ C T T F F C T UA Ts F C UA T Now the origial ODE has bee modified the effect of the flow rate has bee liearized No-Liear Examle: Flow Through Valve with No-Liear Characteristics Let s look at our flow through a tak with o-liear valve characteristics agai: A F F F Cv h Joh Jechura (jjechura@miesedu) - 7 - Coyright 27 Aril 23, 27
Colorado School of Mies CHEN43 Secial Modelig Techiques We kow that h is a variable the ilet flow F is the other ossible variable The total differetial to the right-had-side variable is: Cv d F C v h df 2 h The bracketed term i frot of is a variable so we will use its costat value evaluated at h Isertig ito the origial ODE & covertig differetials to deviatio variables: C A F v h 2 h Sice we are holdig the coefficiet o h costat we have aroximated the origial oliear ODE with a liearized versio Joh Jechura (jjechura@miesedu) - 8 - Coyright 27 Aril 23, 27