CHBE320 LECTURE IV MATHEMATICAL MODELING OF CHEMICAL PROCESS. Professor Dae Ryook Yang

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1 CHBE320 LECTURE IV MATHEMATICAL MODELING OF CHEMICAL PROCESS Professor Dae Ryook Yang Spring 208 Dep. of Chemical and Biological Engineering CHBE320 Process Dynamics and Conrol 4-

2 Road Map of he Lecure IV Basics of Process Modeling + - Conroller Acuaor PROCESS Lecures IV o VII Sensor Mahemaical Modeling Seady-sae model vs. Dynamic model Degree of freedom analysis Models of represenaive processes, ec. CHBE320 Process Dynamics and Conrol 4-2

3 THE RATIONALE FOR MATHEMATICAL MODELING Where o use To improve undersanding of he process To rain plan operaing personnel To design he conrol sraegy for a new process To selec he conroller seing To design he conrol law To opimize process operaing condiions A Classificaion of Models Theoreical models (based on physicochemical law) Empirical models (based on process daa analysis) Semi-empirical models (combined approach) CHBE320 Process Dynamics and Conrol 4-3

4 DYNAMIC VERSUS STEADY-STATE MODEL Dynamic model Describes ime behavior of a process Changes in inpu, disurbance, parameers, iniial condiion, ec. Described by a se of differenial equaions : ordinary (ODE), parial (PDE), differenial-algebraic(dae) Iniial Condiion, x(0) Inpu, u() Parameer, p() Dynamic Model (ODE, PDE) Oupu, y() Seady-sae model Seady sae: No furher changes in all variables No dependency in ime: No ransien behavior Can be obained by seing he ime derivaive erm zero CHBE320 Process Dynamics and Conrol 4-4

5 Conservaion law MODELING PRINCIPLES Wihin a defined sysem boundary (conrol volume) rae of rae of rae of accumulaion inpu oupu rae of rae of generaion disappreance Mass balance (overall, componens) Energy balance Momenum or force balance Algebraic equaions: relaionships beween variables and parameers CHBE320 Process Dynamics and Conrol 4-5

6 MODELING APPROACHES Theoreical Model Follow conservaion laws Based on physicochemical laws Variables and parameers have physical meaning Difficul o develop Can become quie complex Exrapolaion is valid unless he physicochemical laws are invalid Used for opimizaion and rigorous predicion of he process behavior Empirical model Based on he operaion daa Parameers may no have physical meaning Easy o develop Usually quie simple Requires well designed experimenal daa The behavior is correc only around he experimenal condiion Exrapolaion is usually invalid Used for conrol design and simplified predicion model CHBE320 Process Dynamics and Conrol 4-6

7 DEGREE OF FREEDOM (DOF) ANALYSIS DOF Number of variables ha can be specified independenly N F = N V -N E N F : Degree of freedom (no. of independen variables) N V : Number of variables N E : Number of equaions (no. of dependen variables) Assume no equaion can be obained by a combinaion of oher equaions Soluion depending on DOF If N F = 0, he sysem is exacly deermined. Unique soluion exiss. If N F > 0, he sysem is underdeermined. Infiniely many soluions exis. If N F < 0, he sysem is overdeermined. No soluions exis. CHBE320 Process Dynamics and Conrol 4-7

8 LINEAR VERSUS NONLINEAR MODELS Superposiion principle,, and for a linear operaor, L Then L( x ( ) x ( )) L( x ( )) L( x ( )) 2 2 Linear dynamic model: superposiion principle holds,, u () y () and u () y () Easy o solve and analyical soluion exiss. Usually, locally valid around he operaing condiion Nonlinear: No linear 2 2 u ( ) u ( ) y ( ) y ( ) 2 2,, x (0) y ( ) and x (0) y ( ) 2 2 x (0) x (0) y ( ) y ( ) 2 2 Usually, hard o solve and analyical soluion does no exis. CHBE320 Process Dynamics and Conrol 4-8

9 ILLUSTRATION OF SUPERPOSITION PRINCIPLE u () u 2 () u() + y () y 2 () y() Valid only for linear process For example, if y()=u() 2, (u ()+.5u 2 ()) 2 is no same as u () 2 +.5u 2 () 2. CHBE320 Process Dynamics and Conrol 4-9

10 TYPICAL LINEAR DYNAMIC MODEL Linear ODE dy() d Nonlinear ODE y ( ) Ku ( ) ( and K are conan, s order) n n d y() d y() a 0 () n n a y n d d m m d u() d u() bm bm bu 0 ( ) (nh order) m m d d dy() y d 2 () Ku () dy() dy() y () K u () d d dy() y () y ()sin( y) Ku () d e y () Ku() CHBE320 Process Dynamics and Conrol 4-0

11 MODELS OF REPRESENTATIVE PROCESSES Liquid sorage sysems Sysem boundary: sorage ank Mass in: q i (vol. flow, indep. var) Mass ou: q (vol, flow, dep. var) V h No generaion or disappearance (no reacion or leakage) Area = A No energy balance dh A qi q qi f( h) d DOF=2 (h, q i ) - = If f ( h) h/ RV, he ODE is linear. (R V is he resisance o flow) If f ( h) CV gh / gc, he ODE is nonlinear. (C V is he valve consan) q i Conrol volume Mass ou rae Oule flow is a Mass in rae Accumulaion rae in ank funcion of head q CHBE320 Process Dynamics and Conrol 4-

12 Coninuous Sirred Tank Reacor (CSTR) Liquid level is consan (No acc. in ank) Consan densiy, perfec mixing Reacion: A B (r = k 0 exp(-e/rt)c A ) Sysem boundary: CSTR ank Componen mass balance dca V q( c c ) Vkc d Energy balance DOF analysis Ai A A c Ai, q i, T i V, T No. of variables: 6 (q, c A, c Ai, T i, T, T c ) No. of equaion:2 (wo dependen vars.: c A, T) DOF=6 2 = 4 Independen variables: 4 (q, c Ai, T i, T c ) Parameers: kineic parameers, V, U, A, oher physical properies Disurbances: any of q, c Ai, T i, T c, which are no manipulaable h Cooling medium, T c dt VCp qcp( Ti T) ( H) VkcA UA( Tc T) d c A, q, T CHBE320 Process Dynamics and Conrol 4-2

13 STANDARD FORM OF MODELS From he previous example dca q ( c Ai c A) kc A f ( c A, T, q, c Ai ) d V dt q q UA ( T T) ( H) kc ( T T) f ( c, T, q, T, T) d V C C Sae-space model i A c 2 A c i p p x dx/ d f( x, u, d) where x[ x,, x ], u[ u,, u ], d[ d,, d ] T T T n m l x: saes, [c A T] T u: inpus, [qt c ] T d: disurbances, [c Ai T i ] T y: oupus can be a funcion of above, y=g(x,d,u), [c A T] T If higher order derivaives exis, conver hem o s order. CHBE320 Process Dynamics and Conrol 4-3

14 CONVERT TO ST -ORDER ODE Higher order ODE n n d x() d x() an a 0x() bu 0 () n n d d Define new saes n x x, x x, x x,, x x 2 3 A se of s -order ODE s x x x 2 x 2 3 n ( ) x a x a x a x b u n n n n2 n 0 0 CHBE320 Process Dynamics and Conrol 4-4

15 CHBE320 Process Dynamics and Conrol SOLUTION OF MODELS ODE (sae-space model) Linear case: find he analyical soluion via Laplace ransform, or oher mehods. Nonlinear case: analyical soluion usually does no exis. PDE Use a numerical inegraion, such as RK mehod, by defining iniial condiion, ime behavior of inpu/disurbance Linearize around he operaing condiion and find he analyical soluion Conver o ODE by discreizaion of spaial variables using finie difference approximaion and ec. TL TL dt v ( Tw TL ) L j v v ( ) ( ) z d z z HL HL ( ) TL j TL j Tw HL TL TL( j) TL( j) ( j, N) z z 4-5

16 LINEARIZATION Equilibrium (Seady sae) Se he derivaives as zero: 0 f( x, u, d) Overbar denoes he seady-sae value and ( xud,, ) is he equilibrium poin. (could be muliple) Solve hem analyically or numerically using Newon mehod, ec. Linearizaion around equilibrium poin Taylor series expansion o s order 0 f f fxu (, ) fxu (, ) ( xx) ( uu) x ( xu, ) u ( xu, ) Ignore higher order erms Define deviaion variables: x xx, u uu f f x x u AxBu x u ( xu, ) ( xu, ) Jacobian f f x x n f x fn fn x xn CHBE320 Process Dynamics and Conrol 4-6