NONISOTHERMAL OPERATION OF IDEAL REACTORS Continuous Flow Stirred Tank Reactor (CSTR)

Transcription

1 he 47 Fall 2005 LEURE 7 NONISOHERML OPERION OF IDEL REORS ntinuu Flw Stirred ank Reactr (SR) F, Q V F r F, Q V F Figure : Scheatic f SR with acket and cil uptin: Hgeneu yte a) Single Reactin υ 0 b) Steady tate SR i alway aued perfectly ied that the cncentratin f every pecie i unifr thrughut the reactr and equal t the cncentratin in the utflw. Due t auptin f perfect iing the teperature,, thrughut the reactr i unifr and equal t the teperature f the utflw. he nly difference between an itheral SR treated previuly and the general cae treated nw, i that nw we d nt necearily aue that reactr teperature,, and feed teperature,, are equal. SR can be acketed r equipped with a cling (heating) cil. w baic type f prble arie:. Given feed cpitin and teperature, rate fr and deired eit cnverin and teperature find the neceary reactr ize t get the deired prductin rate and find the neceary heat duty fr the reactr. 2. Given feed cnditin and flw rate and reactr ize tgether with cling r heating rate, deterine the cpitin and teperature f the effluent trea. lve either f the abve tw prble we need t ue bth a pecie a balance n the yte and the energy balance. nider a ingle reactin r υ 0 () a + bb pp (a)

2 he 47 Fall 2005 LEURE 7 Suppe the reactin i practically irreverible and the rate f reactin, which i a functin f cpitin and teperature i given by: E / R α β kl r ke B 3 (2) rrheniu cncentratin dependence eperature n-th rder reactin n α + β Dependence f the rate Fr a ingle reactin we can alway eliinate all cncentratin in ter f cnverin f the liiting reactant (Lecture ). Fr liquid υ M / ; M / υ Fr gae υ M / υ P + ε P where υ ε y ( υ ) In a SR we aue P P cnt i.e cntant preure. he rate nw bece: Fr liquid ( ) E / R ( α + β ) α r ke M B / β b (3a) a Fr gae r k e E / R α + β α ( ) M B / α + ( + β ) b a ( + ε ) α β β (3b) Recall that at teady tate the baic cnervatin equatin i: ( Rate f input) ( Rate f utput) + ( Rate f generatin by reactin) 0 (4) 2

3 he 47 Fall 2005 LEURE 7 pply equatin (4) t a f pecie : (r t any pecie ) F F + υ rv 0; F F + υ rv 0 F arv 0 arτ (5) X rv (5a) where τ V Q he energy balance f cure cannt cntain a generatin ter (in abence f nuclear reactin) and hence can be written a: F H F H + q 0 (6) kl F - lar flw rate f pecie in the feed kl F - lar flw rate f pecie in the utflw H J kl - virtual partial lal enthalpy f pecie in the utflw iture J H - virtual partial lal enthalpy f pecie in the feed kl J q - rate f heat additin fr the urrunding i.e fr the acket r cil t the reactin iture in the reactr he energy balance given by equatin (6) i nt general in the ene that the fllwing auptin have already been ade in rder t preent it in that fr: 3

4 he 47 Fall 2005 LEURE 7 uptin invlved in deriving eq (6):. Ptential energy change are negligible with repect t internal energy change. 2. Kinetic energy change are negligible with repect t internal energy change. 3. here i n haft wrk invlved i.e the nly wrk ter i the epanin (flw) wrk. he virtual partial lal enthalpy, H J, i defined by: l tt H F H kg tt - a flw rate f the iture J H - enthalpy per unit a f the iture kg In principle the virtual partial lal enthalpy can be evaluated by the fllwing prcedure (fr gae): H H (, P, y ) f * ΔH f + d + [ H (, P) H, P] + [ H (, P, y ) H (, P)] * * Δ - tandard enthalpy f fratin fr pecie at the preure P f tandard tate and teperature (enthalpy f kle f pure at, P ) * f d - change in enthalpy f pecie, if it behaved a an ideal ga, due t change in teperature fr tandard tate teperature ga tate. In reality t the teperature f interet. p * i the pecific heat f in ideal p data fr real gae btained at atpheric r lwer preure can be ued. Δ H + * * d H f f (,P ) enthalpy per l f fr pure, if it behaved like an ideal ga, at and P. Since enthalpy f an ideal ga de nt depend n preure thi i al the enthalpy per kl f, if it were an ideal ga, at and P, H H * * (,P). (, P) H (, P) preure crrectin factr i.e the difference between the enthalpy f being a real ga at, P, ( H (, P) and enthalpy f being an ideal ga at, P ( H (, P) ). * 4

5 he 47 Fall 2005 LEURE 7 Frequently the preure crrectin i read ff apprpriate chart and i given by: Where H (,P) H * (,P) c H * H c r,p r c critical teperature f pecie. * H H c i the crrectin read ff the chart at apprpriate reduced teperature r c and reduced preure P r P p c P c critical preure f pecie. Finally ( ) H H,P, y (, P) π i the crrectin factr which accunt fr the nnideality f the iture. Fr liquid (in a firt appriatin) H Δ H + f p d + π Here we will aue: Gae: a. ideal iture π 0 * b. ideal ga behavir H H Liquid: a. ideal iture π 0 Nw the energy balance f eq (6) baed n the abve auptin can be written a 5

6 he 47 Fall 2005 LEURE 7 F p d + Δ H r X q (7a) where F p d + Δ H r X q ΔH r υ Δ H + υ f p d (7b) i the heat f reactin at teperature. Finally, in preliinary reactr deign we aue that the heat f reactin de nt vary uch with teperature Δ H r cnt ΔH ΔH r r and that e ean value f the pecific heat can be ued p d p ( ) Equatin (7) can then be written a: r where F p ( ) + Δ H r X q ( ) + ( ΔH ) X + q 0 ρ Q (8) p F Q X υ ) ( υ ) ( r Δ H r ΔH r ( υ ) ΔH r - heat f reactin fr the tichietry a written ΔH r - heat f reactin per le f. r ρ p Q( ) + ( ΔH r ) Q + q 0 (9) ρ p Q( ) + ( ΔH r )rv + q 0 (9a) 6

7 he 47 Fall 2005 LEURE 7 hi final fr f the energy balance reulting fr all f the abve auptin can be interpreted a a heat balance i.e a rate f input f enible heat by the flwing trea inu rate f utput f enible heat by eit trea plu heat generated by reactin plu heat added fr the urrunding ut add t zer. In additin we nw need an energy balance n the acket r cil and a cntitutive relatinhip fr heat tranfer rate, q. Fr a acket at teady tate (auing that the acket i well ied t) ρ Q p ( ) q 0 (0) ρ,q - are denity, vluetric flw rate and ean pecific heat f the fluid flwing thrugh the acket. p - inlet acket teperature. eit acket teperature. Let q U( ) () J U 3 - verall heat tranfer cefficient. Fr eq (0) ( 2 ) area fr heat tranfer between reactr and acket. where κ + κ + κ ρ U p Q (2) Fr a cil with plug flw f heating/cling ediu: d κ ( ) (3) dz z 0 (3a) z fractinal length f the cil. 7

8 he 47 Fall 2005 LEURE 7 κ z ( z) + e ( ) (4) Fr an n-th rder irreverible reactin G() alway ha a igidal hape and at high teperature tend t a hrizntal ayptte. Finally, in dieninalized fr the tw equatin that have t be lved iultaneuly are: arτ (5) + β arτ ω (6) ( ) 0 where r i given by equatin (3) and β ΔH r ρ p κ U ρ ω p Q +κ ρ p Q +U ρ p Q fr a acketed reactr ω κ κ κ e κ ( ) fr a reactr with cil U κ U ρ p Q ρ p Q When we deal with prble f type, t find reactr ize fr given feed and prduct trea cnditin we ue directly eq (5). τ V Q ar R hen calculate the deired heating r cling rate fr eq (6) q ρ p Q( ) + ( ΔH r )arv When we deal with prble f type 2 and try t find the perating cnditin fr a given reactr, then we ut lve eq (5) and (6) iultaneuly by trial and errr fr and. Setie we can lve eplicitly fr fr eq (5) in ter f teperature (). Subtituting thi relatinhip int eq (6) we get 8

9 he 47 Fall 2005 LEURE 7 + β ar τ ω( ) 0 r indicate that the rate i nw a functin f teperature nly ince r (,) r ( (),) + ω + ω β ar β τ (5) G( ) L( ) Reactr perating teperature i given by the interectin f the traight line L() and curvilinear functin G(). Fr an n-th rder irreverible reactin G() alway ha a igidal hape and at high teperature tend t a hrizntal ayptte. Fr an endtheric reactin ΔH r > 0 β < 0 and line L ha a negative lpe. G L G L lpe + ω β + ω + ω Figure 2: Operating pint fr an endtheric reactin in a SR. Several cncluin can be reached fr endtheric reactin: i) L & G can interect at t nce and fr a given et f paraeter nly ne teady tate eit. ii) Fr a given reactr and flw rate, fied τ, given feed,, and heating ediu feed teperature,, and given heat tranfer prpertie, ω, the re endtheric the reactin, the larger β and iii) the aller the lpe f the L line, therefre the lwer the perating teperature and cnverin. Fr a fied reactin, feed flw rate and cpitin and given reactr, fied G curve, an increae in feed teperature ve the L-line t the right while it lpe reain unchanged. Hence, the perating and are increaed. Fr an etheric reactin ΔH r < 0 and β > 0. he L-line ha a pitive lpe. 9

10 he 47 Fall 2005 LEURE 7 G L 2 3 lpe + ω β + ω + ω Figure 3: Operating pint() fr an etheric reactin in a SR. Several cncluin can be reached fr etheric reactin: i) L & G can interect etie at re than ne place and fr a given et f cnditin, re than ne teady tate ay be pible. ii) Fr fied τ and reactin, fied G, an increae in the feed teperature ve the L line t the right increaing the perating and. n increae in clant ha the ae effect. iii) Fr a fied τ and reactin, fied G, β, an increae in heat reval, increae in ω, rtate the L-line in the cunterclckwie directin and ve the intercept at the abcia t the left if > r t the right if >. Fr etheric reactin it i iprtant t calculate the adiabatic teperature rie and the aiu adiabatic Δ. Fr eq (6) with ω 0 ( ) ad β (8) Δ ad. a β (9) Maiu fractinal teperature rie i al called the Prater nuber, β. β Δ ad. a β (20) Nw let u cnider a nuber f iple illutrative eaple. Eaple. he fllwing infratin i given. Irreverible reactin R 0

11 he 47 Fall 2005 LEURE 7 SR l R e 25 e 20,000/ R Lin ΔH ρ p l L R 00,000 cal 500( ) L 350 K 77 cal l a. Itheral. Find reactr pace tie needed fr 0.9 at 350 K and heat t be reved. τ R R e τ in q Qρ ( ) + X ΔH X F p Qρ 25 R ( ΔH 0.9 e R 20, ) cal q Qρ in p ( ) + Q ( ΔH R ) q cal Q L ( 00,00) 0 90,000cal L ρ p ( ) + ( ΔH ) 0.9 / Nte that the reactin rate at 350K i nly R l L in t be reved. b. diabatic Find τ fr 0.9. Qρ p ( ) (ΔH R )Q ( + ΔH R ) ρ p , K(257 ) 500

12 he 47 Fall 2005 LEURE 7 τ R τ R e in e ,000 / Nte that at 530K the reactin rate ha increaed by rder f agnitude t R l 407 L in Nnitheral Find τ, q / Q given deired 0. 9 and deired 400K. τ R e 25 e , ( in) q ρ Q p ( ) + ( ΔH 500( ) + ( 90,000) 25,000 90,000 65,000cal / L Nte the value f the rate which i R ) R kl c in Eaple 2 well tirred bench cale reactr (SR) i ued fr a firt rder etheric reactin R (practically irreverible) under the fllwing cnditin: τ.0(in).0(l / liter) ; 350K 350K ; ω.0 β 200 K lit ( ΔH R ) l ρ p r e 25 e 20,000 l R lit in U ρ p Q + U ρ Q p ρ p Q 2

13 he 47 Fall 2005 LEURE 7 a) Find the perating teperature and eit cnverin. b) Hw any teady tate are pible under thee cnditin? c) What i the aiu adiabatic teperature rie? d) Hw wuld yu change e f the perating cnditin (,, ω ) in rder t perate at a unique teady tate f high cnverin. e) What tart up prgra huld ne ue in rder t have the reactr ettle in the teady tate f high cnverin? Slutin Slve iultaneuly eq (5) & (6) E / R arτ k e ( )τ k 0 e 25 E 20,000 cal / l ω β 0 + ω β + E / R arτ k0e ( ) τ (5) (6) Eliinate fr (5). k e τ E / R 0 E / R + k0e Subtitute int eq (6) τ E / R + ω + ω ke τ E / R β k e τ β + 0 L Subtitute in the value f paraeter: G 20,000 ep ,000 + ep ,000 ep (*) 20,000 + ep Slve * by trial and errr fr. hen btain the crrepnding cnverin by: 20,000 ep ,000 + ep (**)

14 he 47 Fall 2005 LEURE 7 We can al repreent equatin (*) graphically a hwn belw: Eaple X eperature, K Figure 4: Operating pint fr the SR f Eaple 2. Figure 5 n net page hw the draatic teperature ecurin that the reactr can eperience during tart-up befre it ettle t a teady tate. 4

15 he 47 Figure 5. ranient peratin f a SR fr eaple 2

16 he 47 Fall 2005 LEURE 7 a) here are three interectin f the L and G line indicating 3 pible teady tate. () (2) (3) 353K K K b) hree teady tate are pible. c) Δ ada β K ada + Δ ada 550K! d) Increae r t bypa the lwer bup in the curve. Increae ω if pible. e) See attached Figure 5. Reverible Reactin Fr reverible reactin the effect f teperature n equilibriu ut be cnidered. Fr endtheric reactin, ΔH r > 0, equilibriu cnverin increae with increaed teperature. I Figure 6: Equilibriu cnverin a functin f teperature fr endtheric reactin. In regin III the reactin can be cnidered practically irreverible. In regin II e rie with increae in. Fr endtheric reactin the aiu periible teperature i alway the ptial teperature fr aiizing cnverin r prductin rate fr a given SR. 6

17 he 47 Fall 2005 LEURE 7 he net rate f an endtheric reactin at fied cpitin alway increae with increaed teperature. r E / R α E2 / R k0e k20e ince E > E 2 β Fr etheric reactin, ΔH r < 0, equilibriu cnverin decreae with increaed teperature. I Figure 7: Equilibriu cnverin a functin f teperature fr etheric reactin. gain in regin I the reactin i practically irreverible. In regin II it i reverible. r 0.0 r 0.02 r 0 r Figure 8: nverin teperature relatin at fied rate fr etheric reactin. Nw we ntice that at fied cpitin, fied, the net rate f reactin ha a aiu at a certain teperature,. Belw that teperature the rate i lwer and abve it, it i lwer. he rean fr thi i that E < E 2. On the abve diagra we can pa a line called the lcu f aiu rate r a line. Fr a given cnverin the line define a teperature at which the rate i aiu and vice vera at every the line define an a which the rate i aiu. 7

18 he 47 Fall 2005 LEURE 7 We huld alway elect,,ω in uch a anner a t ake ure that we perate n the line. he equatin fr the line i btained by r d 0 fr 0 d r Fr eaple if we have a reverible reactin a+bbpp and the rate i given by r k 0 e E / R α β B k 20 e E 2 / R γ p uing further that we deal with liquid and that and B are given while p 0 K c p a b e uing an ideal lutin ( p a e ) p ( ) a B b a e b () ΔG r R K c e (B) Uing () and (B) we can calculate e a a functin f teperature. he given rate fr, if it i t be viable in the vicinity f equilibriu, ut atify the fllwing cntraint: γ p α β B q eq p p a b B he lcu f aiu rate i.e the -line can be btained by r 0 which reult in 8

19 he 47 Fall 2005 LEURE 7 ln k 20E 2 k 0 E γ α β ( (E 2 E ) /R γ p a ( ) α B b a β ) () Fr a deired cnverin eq () give the teperature at which the rate i aiu. Subtituting that cnverin and teperature int the equatin fr the rate prduce the aiu rate Eaple 3 - SR R ΔG ΔH ρ k 2,500cal / l 20,000cal / l cnt 2,000( cal / lit) cnt 8 2, e (in R r 298 r 298 p k0 l 2 lit ) f) Find the ptial ize SR neceary t achieve a prductin rate f F R 00 (l/in) at 0.9. If the reactr i t be perated adiabatically find the neceary feed teperature. g) If the feed i available nly a 298K hw huld ne perate? an ne aintain the deired prductin rate and cnverin? h) Hw huld 2 SR in erie be perated t iniize the ttal reactr vlue and keep F R and at deired level. Feed i at 298K, 350Ki nt t be eceeded. Slutin Find equilibriu cntant at 298K ΔG r 298 2,500 K ep ep R Find equilibriu cntant a a functin f teperature uing Van t Hff equatin d ln K d ΔH r R 2 ΔH 298 r R K K 298 e.460 ep Find equilibriu cnverin variatin with teperature: 9

20 he 47 Fall 2005 LEURE 7 e K ep 0,065 + K ep 0,065 e Find the lcu f aiu rate k ln k ( E E ) E E 2 R 0, ln Plt the equilibriu line, e, and the lcu f aiu rate,, in rder t graphically interpret e f the later reult e a. t K he rate at the perating pint i: 2,500 r e ( ) e l r 0.85 lit in Fr the deign equatin (balance n R) 32,

21 he 47 Fall 2005 LEURE 7 V F R F R R R r ( lit) V.23 3 Fr adiabatic peratin + β ( V ) rτ 0 β Δ H r 0 ρ p β K Fr adiabatic peratin the feed teperature wuld have t be 290K. he ae reult i btained graphically (--- adiabatic line). b. If the feed i available nly at 298K we culd d ne f the fllwing: i) l the feed fr 298K t 290K and perate adiabatically 298K 290K 308K q Qρ p ( ) F R q ( ) ρ p in ( ) cal ii) Intrduce the feed at 298K and cl the reactr 298K 308K q Qρ p q ( ) + ( Δ H r ) Q F R ρ p ( ) + ( Δ H r ) F R q ( ) + 20, cal in 2

22 he 47 Fall 2005 LEURE 7 he cling rate requireent i the ae, a it huld be a there i n wrk ter in the energy balance. But it i eaier t reve heat fr the higher reactr teperature. iii) Operate adiabatically with the new feed teperature while aintaining the previu feed rate F 00 ; F F R 0.9 Q F 00.8 Nw we ut find the new perating pint by a iultaneu lutin f the pecie and energy balance. r τ + β τ r 0 Since we need adiabatic peratin by cbining the abve tw equatin we get the relatinhip between cnverin and teperature + β r τ β 20 [ ] Subtituting int the firt equatin we get: e ( )e By trial and errr 2,500 [ ] ( ) e e K ( ) 32, he purity (cnverin) ha been reduced ewhat fr the required 90%. he prductin rate nw i F R and rate i r

23 he 47 Fall 2005 LEURE 7 2,500 r e ( 0.88) e l r lit in F R F l in and prductin rate ha drpped very little. 32, iii) We culd perate the reactr adiabatically with the new feed teperature and aintain (via a cntrller) fied eit cpitin at 0. 9 which wuld require aduting the feed rate. Nw β K he new rate i: 2,500 r e ( ) e l r lit in Which i uch aller than the aiu rate. he prductin rate nw drp t 32, F R r V l in hi i t high a penalty t pay fr aintaining purity i.e. keeping 0. 9 cntant!. Fr iniu ttal reactr ize we ut find the iniu f the fllwing eprein 23

24 he 47 Fall 2005 LEURE 7 τ + τ 2 r r f r 2, e.987 ( ) e 32, With 0, ln r f By taking ( l lit in) d τ + τ 2 d 0 ne get by trial and errr K l r 0.5 lit in he required vlue are: Fr the firt reactr V F 00 r liter 0.5 Fr the ecnd reactr V 2 F ( f ) 00 r f 0.9 ( ) lit tal vlue V V + V 405 liter 2 he graphical repreentatin f the abve yte i hwn belw 24

25 he 47 Fall 2005 LEURE K V adiabatic V 2 adiabatic Preheat the feed fr 298K t 305.6K 306K in q Q ρ p ( ) ( ) 8.44 cal 05 l the trea fr t reactr t 304.4K 304 q 2 Q ρ p Δ cal in ( ).73 0 It huld be nted that the abve calculatin can be ade eaier with a little analytical wrk t btain an eprein fr the aiu rate a a functin f cnverin. hi i dne by ubtituting the value k ln k ( E E ) E E 2 R fr teperature in the rate eprein ( R ) r k 0 e E R ( ) k 20 e E 2 R If we recall that e a ln b a b e ln b a We get: E k r k 0 E E 2 E k 0 k 0 E 20 k 20 E 2 k 20 E 2 E 2 E 2 E ( ) E 2 E 2 E E ( E 2 E ) 25

26 he 47 Fall 2005 LEURE 7 In ur particular eaple the abve eprein reduce t: r 3.22 ( ) nther variatin n the abve prble i t have tw equal ize SR in erie. aiu prductin rate then under cnditin τ τ 2 we ut find that atifie Fr r 2 r 2 hi happen at V V liter In uary the fllwing type f prble ay be encuntered: I. Given V, Q,,, Q, β, q, find and. hi i equivalent t deterining the perfrance f an eiting reactr r f ne that i cnteplated by deign engineer. In thi cae pecie a balance and energy balance are lved iultaneuly. II. Given,, β, FR and deterine the required V and deired perating. hi i the typical deign prble. III. Prductin rate i pecified. Eit cnverin and teperature can be elected tgether with V in rder t ptiize e prfit functin. hi i al a typical deign prble. We have nt tuched here n tw very iprtant prble. - tranient in a SR - cntrl f a SR arund an untable teady tate. 26

27 he 47 Fall 2005 LEURE 7 Etenin t Multiple Reactin One huld keep in ind that yield and electivitie can be affected draatically by the chice f perating teperature when activatin energie f variu reactin are different. he prble cnit f a. Finding the ptiu teperature fr a deired prduct ditributin irrepective f pace tie requireent. b. Finding an ptial teperature fr a given reactr (given τ ) which aiize the prductin f the deired prduct. he equatin t be lved are given R independent reactin ang S pecie vi 0 i, 2,... R r F F + v i r i V 0 (), 2,... R R i v i [ X i r i V ] 0 and X r i V 0 fr i, 2,3...R (a) ( F H F H ) + q 0 (2) Equatin () repreent independent R pecie balance and eq (2) i the energy balance baed n auptin f a) n wrk ther than flw wrk b) negligible change in ptential and kinetic energy If we aue further c) ideal ga behavir d) ideal iture 27

28 he 47 Fall 2005 LEURE 7 F Δ H + d f F Δ H + p d f + q 0 p F p 0 R d v i Δ H i f 0 + d r i V + q 0 p F p R d v i Δ H r ri i V + q 0 (2a) i Where Δ H r i the heat f reactin f reactin i at teperature. i If we further aue e) cntant heat f reactin and f) cntant ean pecific heat we get ρ p Q ( ) + Δ H ri R ( ) r i V + q 0 (2b) i 28

29 he 47 Fall 2005 LEURE 7 PROBLEMS. nider the fllwing iple reactin R he reactin rate i given by: π k k 2 R ( l lit ) k ep 7 83, R k 2 ep 8 67, R ( ) ( ) Nte: the activatin energie are given in (J/l) ue the value f the ga cntant accrdingly. he heat f reactin i: Δ H r 80, 000 ( J l) Specific heat f the reactin iture i: 40 ( J l K) p ue that thee are cntant.. he abve reactin ccur in the liquid phae. he preure i high enugh that the liquid will nt vlatize in the periible teperature perating range which i: 300 t 900K. he feed cnditin are: Q 00 ( lit ); ( l lit) ; 300K Yu have a V 00 liter SR reactr. Hw wuld yu perate thi SR in rder t aiize the prductin rate f R?. a) What i ( l ) F R -aiu prductin rate? b) What are & -perating cnditin? c) What i the heat duty fr the yte? d) Sketch yur yte and lcatin and heat duty f all heat echanger, if any. 29