Process Modeling. Improving or understanding chemical process operation is a major objective for developing a dynamic process model
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1 Process Modelng Improvng or understandng chemcal process operaton s a major objectve for developng a dynamc process model
2 Balance equatons Steady-state balance equatons mass or energy mass or energy enterng leavng = 0 a system a system Specfy the system Mcroscopc Macroscopc Dynamc balances rate of mass or energy mass or energy mass or energy accumulaton n enterng leavng = a system a system a system dm de dn or or 2
3 Integral balances and Instantaneous balances mɺ n ( t ) Integral balances t+ t t+ t = t+ t t ɺ n t t M M m mɺ (useful for dstrbuted parameter system) out Instantaneous balances dm = m ɺ n m ɺ or out dv ρ = F ρ F ρ n n out out (F: volumetrc flowrate) 3
4 Materal balances Ex1. Lqud Surge Tank Develop a model that descrbeshow the volume of tank vares as a functon of tme. rate of change of mass flowrate of mass flowrate of mass of water n tank = water nto tank water out of tank 4
5 dv ρ = F ρ F ρ Assume ρ's are constant. dv = F F In order to solve the problem, we must specfy the nputs: F( t) & F ( t) and the ntal condton V (0). It may be desrable to have tank heght, h, as the state varable Express the tank volume as V Ah, we obtan: dh F F = A A If we also know the flowrate out of the tank s proportonal to the heght of lqud n the tank ( F dh F β h = A A where h = the state varable F = the nput varable β & A = the parameters = β h), we have: V = state varable F, F = nput varables Modelng equatons and varables depend on assumptons and objectves 5
6 Ex 2. An sothermal chemcal reactor A + 2B P Develop a model that descrbeshow the reactor concentraton of each speces vares as a functon of tme. Overall materal balance dv ρ = F ρ Fρ Assume ρ = ρ. dv = F F (2) (1) 6
7 Component materal balances It s convenent to work n molarunts when wrtng components balances, partcularly f chemcal reactons are nvolved. Recall the stochometrc equaton: A + 2 B P. dvc dvc dvc A B P = FC FC + Vr, wth r = - kc C (3) A A A A A B = FC FC + Vr, wth r = -2 kc C (4) B B B B A B = FC FC + Vr, wth r = + kc C (5) P P P P A B 7
8 Expandng the LHS of Eq. 3. dvca dca dv = V + CA Combne Eqs 2, 3 and 6: (6) dca F = ( CA CA) kcacb (7) V Smlarly, we have: dc dc B P F = ( C C ) 2kC C V F = ( CP CP ) + kcacb V B B A B V, C A, C B, C P = state varables F, F, C A, C B, C P = nput varables k = parameter 8
9 If the speces B s mantaned n a large excess,.e., C B Smplfyng Assumptons Assume a constant volume Q: dv = 0 reduce one equaton constant, what are the resultant equatons? r = -kc C -k C where k = kc A A B 1 A 1 B dc B dc dc = 0 The resultng equatons are A P F = ( C C ) k C V A A 1 A F = ( C C ) + k C V P P 1 A ( k = k C ) 1 B C A can be solved ndependently 9
10 Ex 3. Gas Tank Develop a model that descrbes how the pressure n the tank vared wth tme Assumpton: deal gas law (IG) PV = nrt Pv = RT v = 3 or ( molar volume, e.g., cm /mol) dn d( PV / RT ) = q q or = q q Assume T=constant, V dp dp RT = q q or = ( q q ) RT V (q, q : molar rate) P = state varable q, q = nput varables V, T, R = parameters 10
11 Consttutve Relatonshps (used n Ex.1-3) -The requred relatonshps, more than smple materal balances,to defne the modelng equatons. Gas Law IG law: Pv = RT v = 3 ( molar volume, e.g., cm /mol) VDW (van der Waal's) equaton of state: a ( P + )( v b) = RT 2 v 11
12 Chemcal reacton knetcs reacton knetcs: A+2B C+3D reacton rate (rate per unt volume, e.g., mol/(volume*tme)) r = k( T ) C C A A B where r A =rate of reacton of A (mol A/(volume*tme) k = reacton rate constant (e.g., (volume/mol)/tme) C =concentraton of (mol /volume) Arrhenus rate expresson: k( T ) = k e 0 E / RT where k = reacton rate constant ((volume/mol)/tme) k0 =frequency factor or preexponental factor (same unt as k) E =actvaton energy (cal/gmol) R =deal gas constant (1.987 cal/(gmol K)) T =absolute temperature (K or R) r B = 2r = -2kC A A C B r C = -r A = kc A C B r D = -3r A = 3kC A C B 12
13 Phase Equlbrum Vapor Lqud Equlbrum (VLE) y where y x K = K x = vapor phase mole fracton of component = lqud phase mole fracton of component = equlbrum constant for component K = f (C, T) A constant relatve volatlty assumpton s often made K1 Ideal bnary VLE usng relatve volatlty ( α = > 1) K (based on lght component) α x y = 1 + ( α 1) x 2 13
14 Heat transfer Rate of heat transfer Q where Q U A T = UA T = rate of heat transfer from hot flud to cold flud (kj/s) 2 = overall heat transfer coeffcent (kj/(s m K)) (functon of flud propertes and veloctes) 2 = heat transfer area (m ) = temperature dfference (K) through a vessel wall separatng two flud (a jacketed reactor) 14
15 Flow through a valve Lqud flow through a valve F = C f ( x) where F C v v Pv s. g. = volumetrc flowrate (gallon per mnute, GPM) = valve coeffcent x = fracton of valve openng (0 x 1; stem poston) P = pressure drop across the valve (ps) v s. g. = specfc gravty f ( x) = flow characterstc (0 f ( x) 1) lnear f ( x) = x quck-openng f ( x) = x equal-percentage f ( x) = x 1 α α = 50 15
16 Materal and energy balances Necessary when thermal effect s mportant Bascs TE = U + KE + PE or TE = U + KE + PE where 2 KE = mv 2 (knetc energy) PE = mgh (potental energy) (per mass) (usually neglected when there s thermal effect; two orders of magntude less than nternal energy) For flowng systems (work wth enthalpy) P H = U + PV or H = U + PV = U + snce ρ = ρ where H = enthalpy per mass U = nternal energy per mass V = volume per mass 1 V 16
17 Example Materal balance accumulaton = n - out dv ρ = F ρ Fρ 17
18 Energy balance accumulaton = n by flow out by flow + n by heat transfer + work down on system dte = TE TE + Q + W = F ρ TE F ρ TE + Q + W Neglect the knetc and potental energy: T T (1) du = F ρ U F ρ U + Q + W T (2) The total work done on the system conssts of shaft work and flow work: W = W + F P FP (3) T s Substtute Eq 3 nto Eq 2: du = P P F ρ ( U + ) F ρ( U ) Q Ws (4) ρ + ρ
19 Snce H = U + PV and neglects W, Eq 4 can be rewrtten as: dh dpv P P = F ρ ( U + ) F ρ( U + ) + Q (5) ρ ρ T T ref ( ) p ref Snce V s constant and P does not change much (good assumpton for lqud system), Eq 5 becomes: dh = F ρh F ρh + Q (6) The defntons for H and H are: H = V ρh (7) Select an arbtrary reference temperature and assume the heat capacty s constant ( ) H(T) = c dt = c T -T H = c T -T p p ref (8) s 19
20 Eq. 6 becomes: dv ρc p ( T Tref ) = F ρc p ( T Tref ) FρC p ( T Tref ) + Q (9) Assume constant densty and volume (so F = F). dt V ρc p = FρC p ( T T ) + Q (10) dt F Q = ( T T ) + (11) V V ρc p Assumptons: 1. Neglect knetc and potental energy. 2. Ignore the change n PV. 3. C p s not a functon of temperature. 4. Vs constant. 5. ρs constant. 20
21 Dstrbuted parameter system Tubular reactor Mole balance of speces A (assumng a frst-order reacton) t+ t ( V ) C ( V ) C = [( FC FC ) kc V ] A t+ t A t A V A V + V A t Usng the mean value theorem of ntegral and dvdng by t, ( V )[ CA t+ t CA t ] = FCA V FCA V + V kca V t Dvdng by V and lettng t and V go to zero, CA FCA = kca t V wth dv = Adz and F = Av, we have: C t v C = kc z A z A A z V V V + V mean value theorem of ntegral b ( ) f ( t ) = f ( x ) b a a 21
22 Smlarly, the overall materal balance can be found as: ρ vzρ = t z If the densty s constant: v = constant CA CA = vz kca t z To solve the problem, we must know ntal condton and boundary condton. C ( z, t = 0) = C ( z) A A0 C (0, t) = C ( t) A An z 22
23 Dmensonless Form Models typcally contan a large number of parameters and varables that may dffer by several orders of magntude. It s often desrable to develop models composed of Dmensonless parameters and varables. Consder a constant volume, sothermal CSTR modeled by a smple 1st order reacton: dca F = ( C Af C A) kc A V Defnng x C / C, we fnd: Af,0 dx F F = x f ( + k) x V V Let τ = t t = t V F. res A ( ) dx dx F dx F F = = = x f ( + k) x V d( τ ) V dτ V V F C Af,0 = steady-state feed concentraton of A xf = CAf CAf tres,0 = V F = resdence tme 23
24 One obtans: dx Vk k = x f (1 + ) x = x f (1 + ) x dτ F F / V ( Vk F ) s a dmensonless term and whch s also Damkholer number (Da). dx = x f (1 + Da) x dτ known as Remarks: Ths mples a sngle parameter, Da, can be used to characterze the behavor of all 1 st order, sothermal chemcal reactons. Explct soluton Explct solutons to nonlnear dfferental equatons can rarely be obtaned (except for few examples). F dh β h = If there s no nlet flow, A A 24
25 General form of dynamc models General models consst of a set of 1st order, nonlnear ODEs. (often called as state space equaton) xɺ = f ( x,, x, u,, u, p,, p ) 1 1 n 1 m 1 r xɺ = f ( x,, x, u,, u, p,, p ) 2 1 n 1 m 1 xɺ = f ( x,, x, u,, u, p,, p n where x u p 1 n 1 m 1 r = state varables = nput varables = parameters r ) 25
26 State varables A state varable arses naturally n the accumulaton termof a dynamc materal or energy balance. (e.g. temperature, concentraton ) Input varables A nput varable normally must be specfed before a problem solved or a process can be operated. Input varables are often manpulateo acheve desred performance. (e.g. flowrates, compostons, temperatures of streams ) Parameters A parameter s typcally a physcal or chemcal propertyvalue that must be specfed or known to solve a problem. (e.g. densty, reacton rate constant, heat-transfer coeffcent) 26
27 Vector notaton General models consst of a set of 1st order ODEs. xɺ = f ( x,u,p) where x = state varables u = nput varables p = parameters The above equaton can also be used to solve steady-state problems. xɺ = 0 f ( x,u,p) = 0 The steady-state solutons are often used for ODEs. as the ntal condtons 27
28 State varable form for Ex.2 dv dc dc dc A B P = F F F = ( C C ) kc C V A A A B F = ( C C ) 2kC C V B B A B F = ( C C ) + kc C V 4 states 5 nputs 1 parameter P P A B Vɺ F F F Cɺ ( C A C A) kc A C B A V = F Cɺ B ( CB CB ) 2kCACB V Cɺ F P ( CP CP ) + kcac B V xɺ 1 u1 u2 u 1 xɺ ( ( ) 2 u x ) p x x f x, u, p x1 f2 (,, ) u x u p = = 1 xɺ ( 3 u 4 x 3) 2 p 1 x 2 x 3 f3 ( x, u, p) x1 f4 (,, ) u x u p 1 xɺ 4 ( u5 x4) + p1x2 x 3 x 1 28
29 Homework #1 1. Irreversble consecutve reactons A B C occur n a jacked, strred-tank reactor as shown n Fgure. Derve a dynamc model based on the followng assumptons, and ndcate the state varables, nput varables, parameters. () The contents of the tank and coolng jacket are well mxed. The volumes of materal n the jacket and n the tank do not vary wth tme. () The reacton rates are gven by E1 RT r = k e C, heat of reacton = H 1 1 A 1 E2 RT r = k e C, heat of reacton = H 2 2 B 2 () constant physcal propertes and heat transfer coeffcent can be assumed. 29
30 Homework #1 2. Consder a lqud flow system consstng of a sealed tank wth noncondensblegas above the lqud as shown n Fgure. Derve a dynamc model relatng the lqud level hto the nput flow rate q.is operaton of ths system ndependent of the ambent pressure P a? What about for a system open to the atmosphere? You may make the followng assumptons: () The gas obeys the deal gas law. A constant amount of (m g /M)moles of gas are present n the tank. () The operaton s sothermal. () A square root relaton holds for flow through the valve ( q = C P ). v 30
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