Chaptr 2: Exampls Mathmatical Mdls r Chmical Prcsss In this chaptr w dvlp mathmatical mdls r a numbr lmntary chmical prcsss that ar cmmnly ncuntrd in practic. W will apply th mthdlgy discussd in th prvius chaptr t guid th radr thrugh varius xampls. Th gal is t giv th radr a mthdlgy t tackl mr cmplicatd prcsss that ar nt cvrd in this chaptr and that can b und in bks listd in th rrnc. Th rganizatin this chaptr includs xampls systms that can b dscribd by rdinary dirntial quatins (ODE), i.. lumpd paramtr systms llwd by xampls distributd paramtrs systms, i.. ths dscribd by partial dirntial quatins (PDE). Th xampls cvr bth hmgnus and htrgnus systms. Ordinary dirntial quatins (ODE) ar asir t slv and ar rducd t simpl algbraic quatins at stady stat. Th slutin partial dirntial quatins (PDE) n th thr hand is a mr diicult task. But w will b intrstd in th cass wr PDE's ar rducd t ODE's. This is naturally th cas whr undr apprpriat assumptins, th PDE's is a n-dimnsinal quatin at stady stat cnditins. It is wrth t rcall, as ntd in th prvius chaptrs, that th distinctin btwn lumpd and distributd paramtr mdls dpnds smtims n th assumptins put rward by th mdlr. Systms that ar nrmally distributd paramtr can b mdld undr apprpriat assumptins as lumpd paramtr systms. This chaptr includs sm xampls this situatin. 28
2.1 Exampls Lumpd Paramtr Systms 2.1.1 Liquid Strag Tank Cnsidr th prctly mixd strag tank shwn in igur 2.1. Liquid stram with vlumtric rat F (m 3 /s) and dnsity ρ lw int th tank. Th utlt stram has vlumtric rat F and dnsity ρ ο. Our bjctiv is t dvlp a mdl r th variatins th tank hldup, i.. vlum th tank. Th systm is thrr th liquid in th tank. W will assum that it is prctly mixd and that th dnsity th lunt is th sam as that tank cntnt. W will als assum that th tank is isthrmal, i.. n variatins in th tmpratur. T mdl th tank w nd nly t writ a mass balanc quatin. ρ F ρ F Figur 2-1 Liquid Strag Tank Sinc th systm is prctly mixd, th systm prprtis d nt vary with psitin insid th tank. Th nly variatins ar with tim. Th mass balanc quatin can b writtn thn n th whl systm and nt nly n a dirntial lmnt it. This lads t thrr t a macrscpic mdl. W apply th gnral balanc quatin (Eq. 1.2), t th ttal mass m = ρv. This yilds: Mass lw in: ρ F (2.1) 29
Mass lw ut: ρ F (2.2) Accumulatin: dm d(ρv = ) (2.3) Th gnratin trm is zr sinc th mass is cnsrvd. Th balanc quatin yilds: ρ F = ρ F d(ρv ) + (2.4) Fr cnsistncy w can chck that all th trms in th quatin hav th SI unit kg/s. Th rsulting mdl (Eq. 2.4) is an rdinary dirntial quatin (ODE) irst rdr whr tim (t) is th nly indpndnt variabl. This is thrr a lumpd paramtr mdl. T slv it w nd n initial cnditin that givs th valu th vlum at initial tim t i, i.. V(t i ) = V i (2.5) Undr isthrmal cnditins w can urthr assum that th dnsity th liquid is cnstant i.. ρ = ρ =ρ. In this cas Eq. 2.4 is rducd t: dv = F F (2.6) Th vlum V is rlatd t th hight th tank L and t th crss sctinal ara A by: V = AL (2.7) 30
Sinc (A) is cnstant thn w btain th quatin in trms th stat variabl L: dl A = F F (2.8) with initial cnditin: L(t i ) = L i (2.9) Dgr rdm analysis Fr th systm dscribd by Eq. 2.8 w hav th llwing inrmatin: Paramtr cnstant valus: A Variabls which valus can b xtrnally ixd (Frcd variabl): F Rmaining variabls: L and F Numbr quatins: 1 (Eq. 2.8) Thrr th dgr rdm is: Numbr rmaining variabls Numbr quatins = 2 1 = 1 Fr th systm t b xactly spciid w nd thrr n mr quatins. This xtra rlatin is btaind rm practical nginring cnsidratins. I th systm is pratd withut cntrl (at pn lp) thn th utlt lw rat F is a unctin th liquid lvl L. Gnrally a rlatin th rm: F = α L (2.10) culd b usd, whr α is th discharg cicint. I n th thr hand th liquid lvl is undr cntrl, thn its valu is kpt cnstant at crtain dsird valu L s. I F is usd t cntrl th hight thn a cntrl law rlats F t L and L s : 31
F = F (L,L s ) (2.11) Fr instant, i a prprtinal cntrllr K c is usd thn th cntrl law is givn by: F = K c (L L s ) + F b (2.12) Whr F b th bias, i.. th cnstant valu F whn th lvl is at th dsird valu i.., L = L s. Nt that at stady stat, th accumulatin trm is zr (hight ds nt chang with tim), i.., dl/ = 0. Th mdl th tank is rducd t th simpl algbraic quatin: F 0 = F (2.13) 2.1.2 Stirrd Tank Hatr W cnsidr th liquid tank th last xampl but at nn-isthrmal cnditins. Th liquid ntrs th tank with a lw rat F (m 3 /s), dnsity ρ (kg/m 3 ) and tmpratur T (K). It is hatd with an xtrnal hat supply tmpratur T st (K), assumd cnstant. Th lunt stram is lw rat F (m 3 /s), dnsity ρ (kg/m 3 ) and tmpratur T(K) (Fig. 2.2). Our bjctiv is t mdl bth th variatin liquid lvl and its tmpratur. As in th prvius xampl w carry ut a macrscpic mdl vr th whl systm. Assuming that th variatins tmpratur ar nt as larg as t act th dnsity thn th mass balanc Eq. 2.8 rmains valid. T dscrib th variatins th tmpratur w nd t writ an nrgy balanc quatin. In th llwing w dvlp th nrgy balanc r any macrscpic systm (Fig. 2.3) and thn w apply it t ur xampl stirrd tank hatr. Th nrgy E(J) any systm (Fig. 2.3) is th sum its intrnal U(J), kintic K(J) and ptntial nrgy φ(j): E = U + K + φ (2.14) Cnsquntly, th lw nrgy int th systm is: 32
ρ F ( U + K + φ ) (2.15) whr th ( ) dnts th spciic nrgy (J/kg). F,T, ρ L Q T st Hat Supply Figur 2-2 Stirrd Tank Hatr Inlts F 2 F 1 Q Systm F 1 utlts F 2 F n Ws Figur 2-3 Gnral Macrscpic Systm F 3 Th lw nrgy ut th systm is: ρ F ( U + K + φ ) (2.16) Th rat accumulatin nrgy is: 33
( V ( U + K + φ ) d ρ ) (2.17) As r th rat gnratin nrgy, it was mntind in Sctin 1.8.4, that th nrgy xchangd btwn th systm and th surrundings may includ hat ractin Q r (J/s), hat xchangd with surrundings Q (J/s) and th rat wrk dn against prssur rcs (lw wrk) W pv (J/s), in additin t any thr wrk W. Th lw wrk W pv dn by th systm is givn by: W pv = F P F P (2.18) whr P and P ar th inlt and utlt prssur, rspctivly. In this cas, th rat nrgy gnratin is: Q + Q W + F P F P ) (2.19) r ( Substituting all ths trms in th gnral balanc quatin (Eq. 1.7) yilds: d ( ρv ( U + K + φ ) = ρ F ( U + K + φ ) ρ F ( ) U + K + φ ) + Q + Q r ( W + F P F P ) (2.20) W can chck that all trms this quatin hav th SI unit (J/s). Equatin (2.20) can b als writtn as: d ( ρ V ( U + K + φ) ) = ρ F ( U + K + φ ) ρ F ( U + K + φ ) + Q + Q r W F ρ P ρ + F ρ P ρ (2.21) 34
Th trm V = 1/ ρ is th spciic vlum (m 3 /kg). Thus Eq. 2.21 can b writtn as: ( V ( U + K + φ )) = ρ F ( U + P V + K + φ ) ρ F ( d ρ U + PV + K + φ ) (2.22) + Q + Q W r Th trm U + PV that appars in th quatin is th spciic nthalpy h. Thrr, th gnral nrgy balanc quatin r a macrscpic systm can b writtn as: d ( ρ V U + K + φ) ) = ρ F ( h + K + φ ) ρf ( h + K + φ ) + Q + Qr W ( (2.23) W rturn nw t th liquid stirrd tank hatr. A numbr simpliying assumptins can b intrducd: W can nglct kintic nrgy unlss th lw vlcitis ar high. W can nglct th ptntial nrgy unlss th lw dirnc btwn th inlt and utlt lvatin is larg. All th wrk thr than lw wrk is nglctd, i.. W = 0. Thr is n ractin invlvd, i.. Q r = 0. Th nrgy balanc (Eq. 2.23) is rducd t: ( VU ) d ρ = ρ F h ρf h + Q (2.24) Hr Q is th hat (J/s) supplid by th xtrnal surc. Furthrmr, as mntind in Sctin 1.12, th intrnal nrgy U r liquids can b apprximatd by nthalpy, h. Th nthalpy is gnrally a unctin tmpratur, prssur and cmpsitin. Hwvr, it can b saly stimatd rm hat capacity rlatins as llws: 35
h = Cp( T ) (2.25) T r whr C p is th avrag hat capacity. Furthrmr sinc th tank is wll mixd th lunt tmpratur T is qual t prcss tmpratur T. Th nrgy balanc quatin can b writtn, assuming cnstant dnsity ρ = ρ = ρ, as llws: ( V ( T T )) d r (2.26) ρ C p = ρf C p( T Tr ) ρfc p( T Tr ) + Q Taking T r = 0 r simplicity and sinc V = AL rsult in: ( LT ) d (2.27) ρc p A = ρf C pt ρfc pt + Q r quivalntly: ( LT ) d Q A = F T FT + ρc p (2.28) Sinc ( LT ) d( L) d( T ) d A = AT + AL (2.29) and using th mass balanc (Eq. 2.8) w gt: dt AL + T ( F F ) = F T FT + ρc p Q (2.30) r quivalntly: 36
dt Q AL = F ( T T ) + ρc p (2.31) Th stirrd tank hatr is mdld, thn by th llwing cupld ODE's: dl A = F F (2.32) dt Q AL = F ( T T ) + ρc p (2.33) This systm ODE's can b slvd i it is xactly spciid and i cnditins at initial tim ar knwn, L(t i ) = L i and T(t i ) = T i (2.34) Dgr rdms analysis Fr this systm w can mak th llwing simpl analysis: Paramtr cnstant valus: A, ρ and C p (Frcd variabl): F and T Rmaining variabls: L, F, T, Q Numbr quatins: 2 (Eq. 2.32 and Eq. 2.33) Th dgr rdm is thrr, 4 2 = 2. W still nd tw rlatins r ur prblm t b xactly spciid. Similarly t th prvius xampl, i th systm is pratd withut cntrl thn F is rlatd t L thrugh (Eq. 2.10). On additinal rlatin is btaind rm th hat transr rlatin that spciis th amunt hat supplid: Q = UA H (T st T ) (2.35) 37
U and A H ar hat transr cicint and hat transr ara. Th surc tmpratur T st was assumd t b knwn. I n th thr hand bth th hight and tmpratur ar undr cntrl, i.. kpt cnstant at dsird valus L s and T s thn thr ar tw cntrl laws that rlat rspctivly F t L and L s and Q t T and T s : F = F (L, L s ), and Q = Q (T, T s ) (2.36) 2.1.3 Isthrmal CSTR W rvisit th prctly mixd tank th irst xampl but whr a liquid phas chmical ractins taking plac: A k B (2.37) Th ractin is assumd t b irrvrsibl and irst rdr. As shwn in igur 2.4, th d ntrs th ractr with vlumtric rat F (m 3 /s), dnsity ρ (kg/m 3 ) and cncntratin C A (ml/m 3 ). Th utput cms ut th ractr at vlumtric rat F, dnsity ρ 0 and cncntratin C A (ml/m 3 ) and C B (ml/m 3 ). W assum isthrmal cnditins. Our bjctiv is t dvlp a mdl r th variatin th vlum th ractr and th cncntratin spcis A and B. Th assumptins xampl 2.1.1 still hld and th ttal mass balanc quatin (Eq. 2.6) is thrr unchangd F ρ ο C A V F ρ C A C B A k B Figur 2.4 Isthrmal CSTR 38
Th cmpnnt balanc n spcis A is btaind by th applicatin (Eq. 1.3) t th numbr mls (n A = C A V ). Sinc th systm is wll mixd th lunt cncntratin C A and C B ar qual t th prcss cncntratin C A and C B. Flw mls A in: F C A (2.38) Flw mls A ut: F C A (2.39) Rat accumulatin: dn d( VC = A) (2.40) Rat gnratin: -rv whr r (mls/m 3 s) is th rat ractin. Substituting ths trms in th gnral quatin (Eq. 1.3) yilds: d( VCA) (2.41) = F CA F CA rv W can chck that all trms in th quatin hav th unit (ml/s). W culd writ a similar cmpnnt balanc n spcis B but it is nt ndd sinc it will nt rprsnt an indpndnt quatin. In act, as a gnral rul, a systm n spcis is xactly spciid by n indpndnt quatins. W can writ ithr th ttal mass balanc alng with (n 1) cmpnnt balanc quatins, r w can writ n cmpnnt balanc quatins. 39
Using th dirntial principls, quatin (2.41) can b writtn as llws: d( VCA) d( C d V V A) ( ) (2.42) = + CA = F CA F CA rv Substituting Equatin (2.6) int (2.42) and with sm algbraic manipulatins w btain: V d( CA) (2.43) = F ( C A C A) rv In rdr t ully din th mdl, w nd t din th ractin rat which is r a irst-rdr irrvrsibl ractin: r = k C A (2.44) Equatins 2.6 and 2.43 din th dynamic bhavir th ractr. Thy can b slvd i th systm is xactly spciid and i th initial cnditins ar givn: V(t i ) = V i and C A (t i ) = C Ai (2.45) Dgrs rdm analysis Paramtr cnstant valus: A (Frcd variabl): F and C A Rmaining variabls: V, F, and C A Numbr quatins: 2 (Eq. 2.6 and Eq. 2.43) Th dgr rdm is thrr 3 2 =1. Th xtra rlatin is btaind by th rlatin btwn th lunt lw F and th lvl in pn lp pratin (Eq. 2.10) r in clsd lp pratin (Eq. 2.11). Th stady stat bhavir can b simply btaind by stting th accumulatin trms t zr. Equatin 2.6 and 2.43 bcm: 40
F 0 = F (2.46) F ( C C ) rv (2.47) A A = Mr cmplx situatins can als b mdld in th sam ashin. catalytic hydrgnatin thyln: Cnsidr th A + B P (2.48) whr A rprsnts hydrgn, B rprsnts thyln and P is th prduct (than). Th ractin taks plac in th CSTR shwn in igur 2.5. Tw strams ar ding th ractr. On cncntratd d with lw rat F 1 (m 3 /s) and cncntratin C B1 (ml/m 3 ) and anthr dilut stram with lw rat F 2 (m 3 /s) and cncntratin C B2 (ml/m 3 ). Th lunt has lw rat F (m 3 /s) and cncntratin C B (ml/m 3 ). Th ractant A is assumd t b in xcss. F 1 C B1 F 2 C B2 V F C B Figur 2-5 Ractin in a CSTR Th ractin rat is assumd t b: k1c B 3 r = ( ml / m. s) 2 (1 + k C ) 2 B (2.49) 41
whr k 1 is th ractin rat cnstant and k 2 is th adsrptin quilibrium cnstant. Assuming th pratin t b isthrmal and th dnsity is cnstant, and llwing th sam prcdur th prvius xampl w gt th llwing mdl: Ttal mass balanc: dl A F + F = 1 2 F (2.50) Cmpnnt B balanc: V d( C A ) = F ( C C ) + F ( C C ) rv 1 B1 B 2 B2 B (2.51) Dgrs rdm analysis Paramtr cnstant valus: A, k 1 and k 1 (Frcd variabl): F 1 F 2 C B1 and C B2 Rmaining variabls: V, F, and C B Numbr quatins: 2 (Eq. 2.50 and Eq. 2.51) Th dgr rdm is thrr 3 2 =1. Th xtra rlatin is btwn th lunt lw F and th lvl L as in th prvius xampl. 2.1.4 Gas-Phas Prssurizd CSTR S ar w hav cnsidrd nly liquid-phas ractin whr dnsity can b takn cnstant. T illustrat th ct gas-phas chmical ractin n mass balanc quatin, w cnsidr th llwing lmntary rvrsibl ractin: A 2B (2.52) taking plac in prctly mixd vssl sktchd in igur 2.6. Th inlunt t th vssl has vlumtric rat F (m 3 /s), dnsity ρ (kg/m 3 ), and ml ractin y. Prduct cms ut th ractr with vlumtric rat F, dnsity ρ, and ml ractin y. Th tmpratur 42
and vlum insid th vssl ar cnstant. Th ractr lunt passs thrugh cntrl valv which rgulat th gas prssur at cnstant prssur P g. P, T, V, y F, ρ, y F, ρ, y Figur 2-6 Gas Prssurizd Ractr Writing th macrscpic ttal mass balanc arund th vssl givs: d( ρv ) = ρ F ρ F (2.53) Sinc V is cnstant w hav: V dρ = ρ F ρ F (2.54) Writing th cmpnnt balanc, r ixd V, rsults in: V dc A = F C A F C rv + rv 0 A 1 2 (2.55) Th ractin rats r th rvrsibl ractin ar assumd t b: r 1 = k 1 C A (2.56) 2 2 k2cb r = (2.57) Equatins (2.54) and (2.55) din th variatins dnsity and mlar cncntratin. On can als rwrit th quatin t din th bhavir th prssur (P) and ml 43
ractin (y). Th cncntratin can b xprssd in trm th dnsity thrugh idal gas law: C A = yp/rt (2.58) C B = (1 y)p/rt (2.59) Similarly, th dnsity can b rlatd t th prssur using idal gas law: ρ = MP/RT = [M A y + M B (1 y)]p/rt (2.60) Whr M A and M B ar th mlcular wight A and B rspctivly. Thrr n can substitut quatins (2.58) t (2.60) int quatins (2.54 & 2.55) in rdr t xplicitly writ th lattr tw quatins in trms y and P. Or, altrnativly, n can slv all quatins simultanusly. Dgrs rdm analysis: Paramtrs: V, k 1, k 2, R, T, M A and M B Frcing unctin: F, C A, y Variabls: C A, C B, y, P, ρ, F Numbr quatins: 5 (Eqs. 2.54, 2.55, 2.58, 2.59, 2.60) Th dgr rdm is thrr 6 5 =1. Th xtra rlatin rlats th utlt lw t th prssur as llws: F = C v P P ρ g (2.61) whr C v is th valv-sizing cicint. Rcall als that P g is assumd t b cnstant. 2.1.5 Nn-Isthrmal CSTR W rcnsidr th prvius CSTR xampl (Sc 2.1.3), but r nn-isthrmal cnditins. Th ractin A B is xthrmic and th hat gnratd in th ractr is 44
rmvd via a cling systm as shwn in igur 2.7. Th lunt tmpratur is dirnt rm th inlt tmpratur du t hat gnratin by th xthrmic ractin. F, C A, T Q V F 0, C A, T Figur 2-7 Nn-isthrmal CSTR Assuming cnstant dnsity, th macrscpic ttal mass balanc (Eq. 2.6) and mass cmpnnt balanc (Eq. 2.43) rmain th sam as br. Hwvr, n mr ODE will b prducd rm th applying th cnsrvatin law (quatin 2.23) r ttal nrgy balanc. Th dpndnc th rat cnstant n th tmpratur: k = k -E/RT (2.62) shuld b mphasizd. Th gnral nrgy balanc (Eq. 2.23) r macrscpic systms applid t th CSTR yilds, assuming cnstant dnsity and avrag hat capacity: ( V ( T T )) d r ρ Cp = ρf Cp( T Tr ) ρfc p( T Tr ) + Q r Q (2.63) whr Q r (J/s) is th hat gnratd by th ractin, and Q (J/s) th rat hat rmvd by th cling systm. Assuming T r = 0 r simplicity and using th dirntiatin principls, quatin 2.63 can b writtn as llws: dt dv (2.64) ρc pv + ρc pt = ρf C pt ρfc pt + Qr Q 45
Substituting Equatin 2.6 int th last quatin and rarranging yilds: dt (2.65) ρ CpV = ρf C p( T T ) + Qr Q Th rat hat xchangd Q r du t ractin is givn by: Q r = ( H r )Vr (2.66) whr H r (J/ml) is th hat ractin (has ngativ valu r xthrmic ractin and psitiv valu r nhrmic ractin). Th nn-isthrmal CSTR is thrr mdld by thr ODE's: dv = F F (2.67) V d( CA) (2.68) = F ( C A C A) rv dt ρ CpV = ρf C p( T T ) + ( Hr ) Vr Q (2.69) whr th rat (r) is givn by: r = k -E/RT C A (2.70) Th systm can b slvd i th systm is xactly spciid and i th initial cnditins ar givn: V(t i ) = V i T(t i ) = T i and C A (t i ) = C Ai (2.71) Dgrs rdm analysis Paramtr cnstant valus: ρ, E, R, C p, H r and k (Frcd variabl): F, C A and T 46
Rmaining variabls: V, F, T, C A and Q Numbr quatins: 3 (Eq. 2.67. 2.68 and 2.69) Th dgr rdm is 5 3 = 2. Fllwing th analysis xampl 2.1.3, th tw xtra rlatins ar btwn th lunt stram (F ) and th vlum (V) n n hand and btwn th rat hat xchangd (Q ) and tmpratur (T) n th thr hand, in ithr pn lp r clsd lp pratins. A mr labrat mdl th CSTR wuld includ th dynamic th cling jackt (Fig. 2.8). Assuming th jackt t b prctly mixd with cnstant vlum V j, dnsity ρ j and cnstant avrag thrmal capacity Cp j, th dynamic th cling jackt tmpratur can b mdld by simply applying th macrscpic nrgy balanc n th whl jackt: dtj (2.72) ρ jcp V j = ρ jfjcp ( Tj Tj ) + Q j j Sinc V j, ρ j, Cp j and T j ar cnstant r knwn, th additin this quatin intrducs nly n variabl (T j ). Th systm is still xactly spciid. V j ρ j C pj T j Figur 2-8 Jacktd Nn-isthrmal CSTR 2.1.6 Mixing Prcss Cnsidr th tank igur 2.9 whr tw slutins 1 and 2 cntaining matrials A and B ar bing mixd. Stram 1 has lw rat F 1 (m 3 /s), dnsity ρ 1 (kg/m 3 ), T 1 (K), 47