Non-isothermal experimental conditions are achieved:

Transcription

1 HPTER 0: Basic reacor models and evaluaion o rae exressions rom exerimenal daa In sie o he advances made by absolue ineic rae heories, he ulimae deerminaion o he ineic rae orm and he evaluaion o he aroriae ineic arameers has o be based on exerimenal resuls. Only when he rae orm has been conirmed in he laboraory and he rae arameers evaluaed, reerably a condiions close o hose conemlaed or he large scale rocess, can an engineer use wih some conidence he rae and is arameers or design uroses or or redicions o evens in he amoshere. The quesion hen arises, i he reacion raes have o be deermined by exerimens o wha hel are he ineic heories o chemical or environmenal engineers? The nowledge o reacion mechanism leads o osulaed rae orms. I is always much easier o chec a osulaed rae orm, ind ou wheher he exerimenal daa conirm i in is enirey or indicae ha a limiing case is suicien, han o ind wha rae orm exerimenal daa conorm o wihou he rior nowledge o ha orm. In oher words, i is clear ha i one nows he execed rae orm one can lan well he exerimens, and minimize he number o necessary exerimens, in order o conirm he osulaed rae exression and deermine is arameers. Wihou "a riori" nowledge o he rae exression more exerimenaion and more wor is necessary in order o exrac all he inormaion. he same ime redicion o ineic consans and acivaion energies rom ransiion sae heory hels in seing u he execed uer limis on he ineic consans and in assessing he emeraure sensiiviy o reacion. This also hels in lanning roerly he exerimens. In order o undersand how rae exressions are evaluaed rom laboraory exerimenal measuremens i is insrucive o consider irs wha yes o exerimens are usually ossible and in wha environmens and under wha condiions are hey done. We will resric ourselves here o exerimens erormed in homogeneous sysems. 0. Reacion environmens and condiions or deerminaion o reacion raes in homogeneous sysems Tyically laboraory exerimens can be erormed in one o he reacor yes described below. Each o he reacor yes can be oeraed (more or less successully) under isohermal, adiabaic, or nonisohermal condiions. Under isohermal exerimenal condiions he emeraure o any oin o he reacion mixure in he reacor should be he same and consan (equal o he desired exerimenal emeraure) a all imes during he run. diabaic exerimenal condiions are achieved when he reacion mixure does no exchange any hea wih he surroundings (oher han sensible hea o he inlow and oulow sream in low reacors) and no wor is done on or by he reacion mixure (oher han PV wor o he luid enering and leaving he vessel in low reacors). Non-isohermal exerimenal condiions are achieved: a. eiher when a emeraure rogram wih resec o ime or osiion in he reacor is esablished b. or when he reacor esablishes is own emeraure roile in ime and sace while exchanging hea wih he surroundings. From he oin o view o deerminaion o ineic raes, isohermal condiions are reerred. Kineic daa or simle reacions wih well deined soichiomery can also be obained rom adiabaic runs, bu inerreaion o non-isohermal runs is usually exremely edious and is o be avoided.

2 The yical reacor yes in which exerimens are erormed are: 0.. BTH RETOR P samler samler V cons P cons ( yical bach reacor V cons) is a vessel o consan volume ( i.e., a las, auoclave, ec.) ino which he reacans are charged a he beginning o he run. The reacor is equied eiher wih a cooling/heaing coil or jace, or is well insulaed, and can be run isohermally (or close o i) or adiabaically. mixer rovides or vigorous agiaion o he reacion sysem. The rogress o reacion can be moniored by aing samles o he reacion mixure in seciied ime inervals and analyzing heir comosiion, i.e. concenraions o cerain comonens are observed as a uncion o ime. In case o gas hase reacions, which roceed wih he change in he number o moles, he change o he overall ressure in he sysem can be moniored in ime and ied o reacion rogress in case o single reacions. noher ye o bach reacor, which is inrequenly used in racice o generae rae daa bu which illusraes an imoran conce o be used laer, is he consan ressure bach sysem where he volume o he reacion mixure may change in ime (i.e. in case o gas hase reacions which roceed wih he change in he oal number o moles). 0.. SEMIBTH RETOR: samling In he case o a semibach reacor some o he reacans are charged a he beginning o he run while one or more reacans are added coninuously hroughou he run. gain, he reacor can be run isohermally or adiabaically, and samling o he reacion mixure is erormed in ime in order o monior he rogress o reacion. This reacor ye may be quie useul when one is rying o deermine he reacion order wih resec o say reacan B and consanly ees adding reacan in large excess. I is also a convenien device in comlex reacion sysems o sudy he eec o he order o reacan addiion on seleciviy and yield ec.

3 0..3 ONTINUOUS FLOW RETOR: PFR FSTR Reacion raes and rae exressions can also be deermined in coninuous low sysems run a seady sae. Two basic yes o coninuous low reacors are: he ideal lug low reacor (PFR) and he ideal coninuous low sirred an reacion (FSTR) PLUG FLOW RETOR (PFR): The main assumion o he ideal lug low reacor is ha he luid is erecly mixed in he direcion erendicular o main low and ha here is absoluely no mixing in he axial direcion, i.e. in he direcion o low. Thus, i is visualized ha all luid molecules move a a uniorm velociy u, i.e. he molecules ha ener a 0 orm a ron (lug) ha moves a velociy u all he way o he exi. Thereore, here are no variaions in comosiion and emeraure in he direcion erendicular o low, while concenraion changes in he axial or z direcion as one roceeds downsream. The assumions o he lug low reacor are requenly me in indusrial racice and in he laboraory. High Reynolds number low in suicienly long ubes, i.e. ubes o high Ld (lengh/diameer) raio, will usually aroximae well he lug low reacor. Flow in aced beds can also be reaed as lug low. This reacor can be oeraed isohermally or adiabaically ONTINUOUS FLOW STIRRED TNK RETOR (FSTR): The FSTR is also requenly called he bacmixed reacor, ideal sirred an reacor (ISTR), ec. The main assumion is ha he reacion mixure in a FSTR is ideally or erecly mixed. Thus, when he reacor oeraes a seady sae every oin (i.e. every orion) in he reacor has he same comosiion and emeraure. Since here is nohing o disinguish he oins in he exi line rom he oins in he reacor, his imlies ha he comosiion and emeraure in he exi sream is idenical o he comosiion and emeraure o he reacion mixure in he reacor. This ideal reacor in racice is aroached by devices ha rovide a very vigorous mixing o he reacion mixure. I can be oeraed isohermally or adiabaically. Oher low ye reacors which do no aroach a PFR or FSTR behavior are no useul or evaluaion o rae exressions. We have menioned beore ha i is desirable o erorm ineic runs a consan emeraure. O he above reacor yes FSTR is he easies o run a isohermal condiions. When oeraed a seady sae, he comosiion in he reacor is consan and he hea released (or aen) er uni ime is consan and can be readily removed or rovided. In conras, in a bach reacor, since he comosiion changes in ime so will he hea released or absorbed; hus, one mus have suicien lexibiliy o mee varying hea requiremens. Similarly, in a PFR luid comosiion changes along he reacor, and hus he hea released or aen er uni reacor lengh changes along he reacor. ooling or heaing mus mee his varying requiremens.

4 I also should be oined ou ha low reacors (esecially laboraory ones) oerae essenially a consan ressure condiions since he ressure dros are usually negligible. In maing energy balances, ricion losses, or he wor done on he luid by he imeller, can generally be negleced in comarison o heas o reacion. he same ime, since he volume o he reacion mixure is ixed in low reacors and also P cons, his means ha in he case o gas hase reacions which roceed wih he changes in he oal number o moles here would be a change in he volumeric low rae o he mixure as i moves hrough he PFR. For such gas hase reacions he exi and enrance volumeric low raes in a FSTR are dieren. 0. Deerminaion o he Rae When we run any o he above reacor yes we deermine he rae o reacion by maing a maerial balance on he reacor. Le us or simliciy consider a single reacion or a + bb dd; () S j ν j j 0 (a) Since or a single reacion or No N NBo NB ND NDo ; () a b d X N j ν j N jo (a) i is suicien o monior only one variable o he sysem, i.e. one concenraion, or any roery o he sysem such as is ressure, viscosiy, reacive index, ec. ha is direcly and uniquely relaed o is r T,,, r T, is a uncion o emeraure and comosiion. The rae o reacion ( B D) ( j) comosiion. NOTE: r r a r is he inrinsic or equivalen rae i.e. r b r d B D (3) Suose we run he reacion in a bach reacor rom ime 0 o ime and monior concenraion as a uncion o ime. The overall maerial balance on gives:

5 moun o moun o moun o le unreaced reaced rom 0 charged in a ime ime 0 o ime (4) moun o charged in he reacor N o ( moles ) (5) moun o le unreaced a ime By deiniion N ()( moles) evaluaed a (6) ar moles o reaced uni volume uni ime Then (, j ) a r T dv V44443 moles o reaced uni ime (moles reaced er uni volume and uni ime summed u over he whole volume o he mixure) moun o reaced rom ime 0 o ime o V (, j ) a r T dv (7) Thus, subsiuing eqs. (5-7) ino eq. (4) we ge: () ( j) No N a r T, dv 0 o V (8) Similarly an energy balance gives: Energy o Energy o Hea generaed Hea removed he sysem he sysem + by reacion rom rom ime 0 a ime 0 a ime ime 0 o ime 0 o ime (9) ( ρ v ) ( ρ v ) + (, o r j) VT VT a U r T dv o V Uh T Tc 0 (9a) o

6 3 ρ - densiy o he reacion mixure ( g m ) - seciic hea o he reacion mixure ( J g ) v 3 V - reacion mixure volume ( m ) - hea o reacion ( J mol ) U r U - overall hea ranser coeicien ( J m s ) h - area or hea ranser ( m ) learly he equaions simliy i he emeraure is e consan ( T T c ) in an isohermal sysem and i he volume o he reacion mixure is consan. () No N av r Tc, j 0 (0) o Diereniaing eq. (0) we ge dn av r ( Tc, j) 0 () or ar ( j, Tc) 443 r ( j, Tc) V dn () Now i he volume o he reacion mixure is consan N V and he concenraions o all oher comonens can be relaed o N N bx (3) B Bo X N a N o (3a) I V cons an ξ X o V a (4) b B Bo ( o ) a (5) d D Do ( o ) a (6) d r (, Tc) (7)

7 d Since we monior as a uncion o ime, he derivaive () aricular ime, i.e. a he concenraion resen a ime. gives us he value o he rae a ha o sloe rae a 3 sloe sloe sloe 3 sloe rae a ec. In his way we could exrac he overall order o reacion, n. In order o ge he reacion order wih resec o reacan we can use he mehod o excess, i.e. ae iniial NBo N o >> so ha B says racically consan during reacion. In order o be able o neglec he eec o roduc D on he rae we can use he mehod o iniial raes, i.e. measure he raes only while very lile roduc is resen. Le us briely consider he maerial balance or a coninuous low reacor or he same reacion a isohermal condiions and a seady sae Rae o Rae o molar Rae o molar disaearance inu o ouu o by reacion in he reacor Fo F a r, Tc dv 0 mol V mol hr hr mol ( li) li hr In erms o conversion: F F ( x ) o 0 (8) (8a) (9) F x a r, T dv 0 (0) o c V The inegral is necessary in he PFR since he concenraion o reacor lengh and, hereore, so does he rae. changes as one moves down he

8 However, in a FSTR, by assumion, he mixure is erecly mixed and a uniorm comosiion. Thereore, every elemen o he reacion mixure reacs a he same rae and we have ( q - volumeric low rae o reacion mixure a he exi, - volumeric low rae o enrance): q o r (, T c ) F o x V concenraion in he reacor and a he same ime in he exi o q o q V () learly FSTR is a convenien conacing device or direc measuremen o reacion raes (no derivaives needed). rae a a dieren can be ound by increasing or decreasing q. From he above we have seen ha PFR oeraes as an inegral reacor, i.e. overall conversion is given by he inegral o he rae over he whole reacor volume. Someimes he PFR is used as a dierenial reacor, i.e. overall conversion is e very low so ha he change in concenraion is very small rom enrance o exi. Then he mean value heorem can be used. F x ar, T V 0 () o m c o m mean value o concenraion exi < m < o. Now he rae is obained direcly bu he small changes in concenraion are diicul o measure accuraely. The FSTR in conras can give very large changes in obained direcly. beween inle and oule and sill he rae is 0.3 Evaluaions o rae orms using bach (Homogeneous Sysems) We will concenrae now on evaluaion o rae exressions in homogeneous sysems using bach reacors. The roblem, i we sar rom a simle single reacion, can be saed as ollows: For a reacion a + bb dd + gg ind he reacion rae, i.e. ind he rae's deendence on concenraion and emeraure. The exerimens are seleced o be erormed a isohermal condiions so ha he rae's deendence on concenraions is esablished irs. Once he rae orm is nown, isohermal runs can be reeaed a dieren emeraure levels so ha he deendence o he seciic rae consan(s) on emeraure can be esablished. Le us assume urher ha hese hyoheical exerimens are erormed in an enclosed bach vessel so ha he volume o he reacion mixure is consan V cons. Since V cons he soichiomeric relaionshi No N NBo NB ND NDo NG NGo X () a b d g

9 or or a general single reacion ν j j 0 (a) N j ν j N jo X (a) N o N X can be exended by dividing i by V and since o, ec., while ξ is he V V V exen er uni volume we ge o Bo B D Do G Go ξ (3) a b d g or j ν j jo ξ (3a) Le us ha suose we have charged he reacor wih No, NBo moles o and B and no D and G. balance on gives ar ( j ) dn (4) V Bu N V and V cons so ha d r ar ( j) (5) The rae is in general a uncion o concenraion o all secies. Le us assume ha we exec o ind he rae o he orm α β γ δ B b D G r (6) where E RT e Eb RT and e. We have assumed he rae consans o obey he o b bo rrhenius orm bu since we oerae a consan emeraure (isohermal runs) boh, are consans o be deermined rom he run as well as he owers α, β, γ, δ. Using he soichiomeric relaionshi we could eliminae all oher concenraions in erms o. b In General: B Bo ( o ) a d ν j D Do + ( o ) j jo + ( l lo) (7) a ν l g G Go + ( o ) a b

10 Thus we would have β γ + δ α b dg d a Bo ( o ) b ( o ) a a his ollows nice since Do Go 0 were assumed 0 o (8) (8a) learly, we could monior he concenraion o discreely. We can ry o evaluae he sloe o in ime coninuously, or as is more oen he case d vs. line,, a various values o.,, o i sloe a i d Plugging he and corresonding d values in he above equaion we would ge a large se o nonlinear equaions rom which we would have o deermine a grea number o arameers, namely six (6) all ogeher: α, β, γ, δ,,. b This mehod o aac due o large errors hidden in exerimenal daa does no seem oo ruiul. Insead o we can decide o run he exerimen only u o low conversions x. In his siuaion we o can neglec he rae o he reverse reacion - we are using he mehod o iniial raes. α b d a Bo o a β (9) Now we would have o deermine only hree arameers rom he exerimenal run: α, β, and. In addiion, we can decide, i exerimenally easible, o mae he job even easier and use reacan Bo b B in large excess over he soichiomerically required amoun i.e. >>. a o

11 The mass balance can be rewrien in he ollowing orm: α β b d a Bo a o x o β (30) bu since x << comarison o Bo o (low conversion) and Bo o b b b o >> we can saely ignore x in a a a o Thus, we ge wih β Bo : β α d α Bo a a (3) and we have used he mehod o excess in addiion o he mehod o iniial raes. Now we have only wo arameers o deermine: α and. This seems a reasonable as. We can aerwards roceed o reea he runs a dieren emeraures, sill measuring raes a small conversions and using B in excess, in order o deermine rom he rrhenius lo. er ha, exerimens can be erormed in excess o E o deermine β and, and hen runs saring wih only and D can be erormed o deermine o he reverse rae's arameers γ, δ, bo, Eb. The revious discussion indicaes ha i we wan o deermine a rae exression or a single reacion, and we monior he concenraion o one comonen (one secies) as a uncion o ime (say o limiing reacan ), we can almos always, by roer selecion o exerimenal condiions, reduce he roblem o one o he ye: d α ; 0 o (3) where rom a se o vs., i.e. vs., daa arameers α and have o be deermined. i i For he analysis o his ye we can emloy wo basic mehods: - dierenial mehod or - inegral mehod. Le us discuss each o hem.

12 0.3.. Dierenial Mehod: Objecive: given a se o daa vs., i.e. i vs. i, deermine α and in he hyohesized rae orm, α.. Tabulae he daa and evaluae he dierences i i + i and i i + i. d. Evaluae he aroximaion o he derivaive. This can be done eiher direcly rom he able o dierences o he daa i.e. d ( % + i ) i i i i i + i ; i < < i + % (33) or by numerical diereniaion o he daa, or by loing a smooh curve hrough he vs. daa and evaluaing is sloe grahically, or by maching he vs. daa by some uncion and evaluaing is derivaive. Using he raio o he dierences rom he able o daa leads o very oor esimaes o he derivaive when i are uneven and large and i are large. he same ime he quesion arises o wha value o % ( i < % < ) i +, i.e. o wha value o % i > % > i + does he derivaive i aroximaion corresond. The qualiy (i.e., oor qualiy) o error rone ineic daa usually does no warran he use o more sohisicaed numerical diereniaion mehods. The bes racical rocedure is o lo i i was evaluaed by using, and, i i verses as a sewise curve shown below. Since i + i i + i i is valid or he inerval i + i

13 i i ime or las deenden N+ i The area under every se is he value o he ordinae imes he value o he corresonding ime i i inerval i + i, i.e., i i which measures he variaion in concenraion i + i during one ime inerval. The area under he whole sewise curve is: N+ N i i O N + (34) i i 0 and measures he overall concenraion change o during he exerimen. Ye we now ha he d overall concenraion change we can obain by inegraing he derivaive wih resec o ime: O n + di N + (35) o This ells us o lo a smooh curve on our se-wise lo in such a way ha he area under he smooh curve is equal o he area under he sewise curves. Since boh areas are he smooh O N + d curve gives us our bes esimae o. This was done on he igure in he revious age. Now d or every i we can read he value o he derivaive ( i ) by going o he smooh curve a ha

14 value o and reading o he corresonding value o he ordinae. Since or every we have a i d measured i value, we can now air he values ( i ) Now lo ( ) d means evaluaed a i and i. d i versus i on log-log aer. By doing his d log log { + α log (36) a x y i we should be able o ass a sraigh line hrough he daa oins rovided ha he rae indeed was o n-h order orm and no o a more comlex orm. The sloe rom he log-log lo gives our α. From any oin hen we can deermine log and. I is advisable o deermine rom a number o oins and o ae an average value. The srengh o he dierenial mehod is ha all ha we have o assume is ha he rae is o a aricular orm (n-h order orm). Boh he reacion rae order α and he ineic consan are hen exraced by he mehod. The main weaness o he mehod is oor accuracy Inegral Mehod: Objecive: given a se o daa vs., i.e. i vs. i, deermine α and in he hyohesized rae orm α. Procedure:. Tabulae he vs. daa. i. ssume a aricular order α, i.e. α, or α or α, ec. 3. Inegrae he bach mass balance. d α o 443 F (, o) (37) o Noe: The orm o F(, ) deends on he chosen value o α: or o α, F( ) α, F( ) or o, o ln (37a), o ec. (37b) o

15 4. Form he airs o values F, vs, i.e. evaluae he nown uncion F, a all i o i measured values and air hem wih corresonding by exending he revious able o daa. 5. Plo F, vs on a linear lo i i o i i o (, o) { F 443 y x (38) I he assumed order α was correc one will be able o ass a sraigh line hrough he above loed oins, he sloe o he lines gives and he assumed order α has been conirmed. I he assumed order α was incorrec, a sraigh line lo canno be obained, and one has o go bac o se and assume anoher α and reea all oher ses. The srengh o he inegral mehod is ha he inegraion rocedure smoohes he error o exerimenal daa and allows more accurae evaluaion o 's. The main weaness is ha he mehod is very edious o use unless one has a good indicaion o wha α may be. learly he nowledge o he mechanism would hel. In general, i one does no have any idea wha α may be, i is advisable o ge an esimae o α and using he dierenial mehod, and hen rees he ound α and ugrade he accuracy o he value by he inegral mehod. In he above discussion we have assumed ha he concenraion o a reacan was moniored in ime. Naurally, all he above rocedures hold i insead a roduc concenraion is moniored, or a hysical quaniy roorional o he comosiion o he sysem, or oal ressure o he sysem (in case o gas hase reacions which roceed wih he change in he oal number o moles). Le us consider his las case: a + bb dd + gg or n d + g a b S ν j j 0 (a) j S n ν (39) j j Le us assume a rae o he orm: α β r B (40) ssuming ideal gases o P RT o Bo Bo P (4) RT The oal iniial ressure, i here were no iners and roducs resen, is given by: Po + PBo PTo o + Bo To RT RT (4)

16 oncenraions are relaed o exen er uni volume ξ, as shown below: P N No + ax o RT V v aξ PB B Bo RT bξ PD D RT dξ PG G RT gξ ( P + PB + PD + PG) PT T ( + B + D + G) To RT RT + ( n) ξ PT PTo PT PTo T + ( n) ξ ξ RT RT n RT T To j j or in general S PT PT + ν ξ ξ o (44) ν RT Now le us ge an equaion in erms o oal ressure: d dp α β P PB RT RT j α β B (45) dp + α + β ( α β) α β RT P P B a P Po aξ Po ( PT PTo ) n (47) Po yo PTo Mole racion o iniially (48) dp a dpt n (49) a dpt a b y P P P y P P P n n n o To ( T To) Bo To ( T To) β α dp n T a b a n n yo PTo PT PTo ybo PTo PT PTo α This equaion relaes now he change o oal ressure T wih ime and rom he PT i vs. i daa α, β, can be deermined by he reviously oulined mehods. Mehod o excess can be again used in order o urher simliy he aaren rae orm. P β (43) (46) (50) (5)

17 learly, boh he inegral and dierenial mehod have one hing in common - hey aem o ind a relaionshi beween roerly arranged exerimenal daa which is linear. The ey is o ind and lo a sraigh line relaionshi. Dierenial mehod (n-h order reacion) d log log + n log x y (5) Inegral mehod (n-h order reacion) ( F F + n) (53) o n n + n o 3 y ( ) { x (53a) For rae orms oher han n-h order sraigh line relaionshis are also sough. For examle or Michaelis-Menen rae orm: d (54) K + r Dierenial nalysis: K + d or { x 443 Inegral nalysis: y K + d 443 y { x (55) o ln (56) o K o K 4443 x 4443 y Thus any combinaion ha yields a sraigh line relaionshi is sough.

18 0.4 Examles o evaluaion o rae orms rom ineic daa in bach sysems EXMPLE : Deermine he reacion order and he rae consan or a single reacion o he ye roducs based on he ollowing exerimenal inormaion obained a isohermal condiions a V cons. ( min) mol li The las daa oin simly indicaes ha aer a very long ime (several hours as comared o minues) racically no is ound. Thus, a he exerimenal condiions used he reacion is racically irreversible.. Dierenial nalysis: i We can orm i and i i i + rom he able, and i i i We can lo now so ha i vs. i as a sewise curve shown below. Now we have o ass a smooh curve mol li min (min) he area under he sewise curve and he smooh curve are aroximaely equal. From he smooh d curve we now read o he corresonding values o a desired measured concenraions i.

19 i d Now we can lo d log log + α log i.e. we lo d vs. on a log-log lo. From he sloe o he sraigh line ha we managed o ass hrough he daa oins we ind ha α.05. From any oin o he line now we could deermine. For examle: log log +.05 log 0. log log log 0. li mol min However, we should quicly realize ha he esimaed order is only.5% rom nd order and, hus, mos liely he reacion is o order wo. α. d. We can now evaluae or every daa oin and hen average hem ou. d

20 d log sloe log li The mean value o urns ou o be Thus a he emeraure o he mol min exerimen we have deermined mol r 0.50 li min Noe: lhough he values o he rae consans calculaed rom various daa oins vary considerably, he variaion is random and shows no rend wih concenraion level indicaing ha he seleced order is correc. Inegral Mehod: Suose ha we have aemed o solve he same roblem by he inegral mehod. Since0 he soichiomery is roducs, we will ry irs wheher a irs order rae orm can i he daa (hoing or an elemenary reacion).

21 ssume α : d ln d o o o log.306 o log log.306 o We should hen lo he exerimenal daa on a semi log lo ( on he log scale and on he linear scale). I he assumed order o one is correc we should be able o obain a sraigh line hrough he daa oins. I is clear rom he enclosed igure ha a sraigh line canno be obained since he daa show a deinie curve here - convex owards he boom. I we connec he irs and he las daa oin by a sraigh line all he oher daa are below he line indicaing ha he concenraion dro is aser han rediced by irs order behavior. mol li (min) Try nd order α d + o Plo vs. on a linear lo.

22 This ime a sraigh line is obained which roerly inersecs he ordinae a o. From he li sloe o he line we ge 0.50 mol min. I is insrucive again o evaluae 's rom he individual daa oins. o sloe (min) li The mean value o urns o be 0.46 he dierence beween his mean value and mol min he one obained by "eye iing" he line hrough daa oins is.7% and is negligible as ar as engineering alicaions are concerned. Noice ha he dierence beween he larges 0.54 and smalles 0.4 is only 0.0 or less han 8.5% based on he smalles -value. For he same daa he 's evaluaed by he dierenial mehod varied beween a low o 0.34 and a high o 0.68, he dierence being or 5% based on he smalles. Thus, or he same qualiy daa he inegral mehod ends o smooh ou he errors and give more consisen esimaes or he rae consan. Here by inegral mehod we have deermined also: mol r li min 0.50 EXMPLE : Gas hase decomosiion o di--buyl eroxide is moniored in a bach reacor o consan volume a isohermal condiions o 70. The run is sared wih ure di--buyl eroxide and he change o oal

23 ressure o he sysem was recorded in ime. From he daa below ind he rae exression and he rae consan. ime min P mm Hg The reacion is: H O O H H H O H V cons, T cons The hyohesized rae is o he orm: P + Q 3 mol r li min α We have seen beore ha we can also exress he rae in erms o change o he arial ressure: mm Hg r% min P α where RT α In a bach sysem o consan volume: r% dp P α We have shown beore ha P P o ξ RT s + ν ξ j PT PTo Po T To j PT RT here ν j 3 Since in his case Po P To

24 P 3 P To P T dpt 3 PTo PT α dpt ( α ) [ 3 PTo PT] α DIFFERENTIL NLYSIS: PT Evaluae, lo vs. a sewise curve, rom a smooh curve ha has he same area underneah as dpt dpt he sewise curve. Evaluae and calculae he corresonding 3 PTo PT. Plo vs. P P on a log-log scale. The sloe gives α ; rom he daa ind and and heir mean value. 3 To T The augmened able is shown below as well as he wo igures (nex age) PT dpt PT 3 PTo PT min dpt I seems ha more han one sraigh line can be assed hrough he oins on he log vs. ( 3 P To P T ) lo. The maximum sloe seems o be. and he minimum This indicaes ha robably α.0. 3 P dpt To P T These values are given in he las column o he above able.

25 P T (min) 0 - dp T sloe sloe log 8 log 0 8 log 5.0 log P To P T From he above igures and able we ind mm Hg mol r 0.08 P 0.08 P or r P min li min

26 INTEGRL METHOD: Suose we assumed zero-h order dp T " PT PTo + " Daa shows deinie curvaure and reacion is no zero-h order. P T 0 0 ssume nd order: (min) dpt " 3 PTo PT + 3 PTo PT PTo 443 y " { x 0.3 3P To P T (min) gain daa show a deinie curvaure and he reacion is no nd order. ssume s order

27 dpt " ( 3 PTo PT ) P To ln " 3 PTo PT ( ) log 3 P P log P.306 " To T To sloe log P To P T (mm Hg) sloe log ( 5 3. ) (min) Now we do ge a sraigh line. ".306 sloe min Direcly rom daa P ln To PTo PT min mol r 0.08 li min 0.00 mm Hg r% 0.08 P min P

28 Noice again ha he variaion in he rae consan based on inegral mehod is much less han i dierenial mehod is used. Examle 3: onsider he reacion beween suluric acid and diehylsulae in aqueous soluion H SO4 + H 5 SO 4 H5 SO4 H + B P ) o isohermal condiions ( T.9 saring wih equimolar mixure o he reacans and wih no roduc he daa resened below were obained. Iniial reacan concenraion was each o hem. Find he rae exression. min mol li min several days mol li ( mol li ) or Since or each mole o reaced one ges wo moles o P i he reacion wen o comleion one would ind o 5.80 mol li o P are ound his indicaes ha he reacion is reversible. ( mol li) P. Since only P eq P eq eq B eq o o eq P eq x e o i e P eq o o 0.57 Le us assume ha he reacion is nd order in boh direcions d R B b Since we sar in soichiomeric raio

29 hereore o Bo B o d o b ; 0 0 Le us use inegral analysis. From he above rae exression a equilibrium 4 x e K B b eq ( x ) e 4.98 We can searae he variables in he above dierenial equaion: o o d b d o The inegral on he le hand side can be romly evaluaed by using a se o inegraion ables. For an exercise we will inegrae i here: b K + 4 K o o 0 Find he roos o he denominaor:, o 4 m K The inegrand can now be wrien as: B + o o o o K 4 4 K K + K K Using arial racions and evaluaing or and B we ge:

30 B 4 o K Thus: d d d o o o 0 + o o 4 K K K K o 4 K + 4 K ln 4 o o K + 4 K 4 K er some reorganizaion we ge: ln ( ) K + K o K K o o K Using he reviously esablished relaionshi K 4 x e ( x ) e We can rewrie his in he orm: ln ( ) xe xe x xe o xe x xe Using he already evaluaed value o K and nown o we should lo ln i.e. y vs on a semi-log lo.

31 y y y y Suose ha we have assumed a irs order reversible reacion: d b b Peq K eq xe x e uon inegraion K.3 ln + o + K o K P ln P or in alernaive orm: ln x e x x x e e Now i we exec his rae orm o hold we should also ge a sraigh line on a semi-log lo o y 0.7 vs.. The values o y are also calculaed in he able on he revious age. Boh orms are loed on semi-log aer on he nex age and boh yield a reasonable sraigh line!? arenly rom he exerimenal daa given we are unable o disinguish beween a reversible nd order reacion in boh direcions and a reversible irs order reacion in boh direcions. I we consider again he wo inegraed orms wrien in erms o conversion we can readily see ha when x 0.5 he wo orms become idenical and indisinguishable rom each oher. Since under he e condiions o he exerimen equilibrium conversion was x e 0.57, which is close o 0.5, due o exerimenal scaer we canno disinguish beween he wo orms. I we erormed he exerimens a dieren T so ha x e is ar rom 0.5; or i we used nonsoichiomeric raio o reacans, we would ind ha he rae indeed is nd order in each direcion. I is imoran o deermine he roer order since when designing a larger reacor we may be oeraing a

32 condiions when x e >> 0.5 and he redicions o he reacor size or a desired roducion rae will dier vasly based on he rae orm. Plo or nd order reversible: y sloe log (min) Plo or s order reversible: y sloe log (min) From he sloe or he nd order rae orm we ge he value o he rae consan sloe li 4.99 mol min

33 Then b K b li.7 0 K 4.98 mol min o The rae in ( mol li min) is given by he above exression a T.9. The emeraure deendence o he consans would have o be ound by erorming exerimens a dieren emeraures. auion: noe o cauion is here in order. The inegraed exression ha we used or a reversible nd order reacion o he ye + B P was: ln ( ) x x x x xe x xe e e e o Noe : Noe : Quesion: nswer: This exression is only valid when he exerimen is erormed wih soichiomeric raio o and B. I is no valid when o Bo. The exression or he same reacion ye is reored by Levensiel (age 63, equaion 56) and i loos almos exacly he same as he one above exce ha i has an exra acor o on he righ hand side. Is here a misae? Noe ha and in he boo (which are rae consans or he reacion orward) will dier by a acor o wo!!? There is no misae bu is based on roducion o P while is based on r disaearance o. Since due o soichiomery r his imlies which indeed is he case. Poenial Trouble: The choice o he subscri or ec. does no indicae on which comonen he rae consan is based. Thus, i one has only an inegraed orm o wor wih, one has no way o nowing wheher is based on reacan or roduc, ec. lariy ha whenever ossible. Noe 3: Since he above menioned ambiguiy abou rae consans always exiss ry o: a. use inegraed orms only when you now wha he 's are based on; b. develo your own inegraed orms by he hel o inegral ables. This las choice aer all is no ha diicul. In he roblem ha we jus solved we had o inegrae

34 d 0 0 o d o + o K 4 K quic loo in he R Mahemaical Tables shows ha we have a roblem o he ye dx y where x y a + bx + cx a o b c o 4 K The answer is: dx cx + b q ln y q cx + b + q where q 4 ac b Thus in our case q 4 o. K Subsiue roer erms or q, c, a, b and x in he above exression. auion: Do no orge o evaluae he above exression a he uer and lower limi o inegraion (exression a uer limi - exression o lower limi) since you sared wih a deinie inegral and R Tables gives you he answer or an indeinie one. You should ge he exression used in he roblem. 0.5 Precision o Kineic Measuremens Random errors can never be comleely avoided in ineic measuremens bu we can oen esimae he maximum ossible error in our measuremens o concenraions, emeraure lucuaions, ec. In general, i we are ineresed in obaining values o a deenden variable rom values o indeenden variables xi, i,,... where ( x, x, x,...) i 3 he relaive error in can be relaed o he relaive errors in x i by

35 n ln x j j ln x (57) j xj Suose we are ineresed in he accuracy o he rae consan which we have obained rom he ollowing exression: r (58) B ln ln - r ln ln B ln ln ln (60) ln r ln ln ( ) ( r ) ( ) B + + r B B (59) (6) I we measured each concenraion wih recision beer han % and he recision o he rae measuremens is 0% we ge (6) (63) The error in is 0.4% From he rrhenius relaionshi we see d E dt RT T The larger he error in T and he larger he acivaion energy, E, he larger he error in. We usually esimae E R E ( ) ln 0 T T 0 rom E ln ln ln ( 0) ln R (66) T0 T E 0 T T 0 T E ln 0 0 T0 T T0 T T The larger he inerval T T0 he smaller he error in E rovided ha classical rrhenius orm wih E cons holds over such a emeraure range. Useul guidelines can be ound in Benson, Foundaions o hemical Kineics. (64) (65) (67)

36 To obain he rae consan wih an error o. To measure he concenraion wih an accuracy ± hange in concenraion ε Larges concenraion.4 % ± ε i is necessary:. To measure ime (or equivalen) wih accuracy o ± Time inerval ε Larges ime.4 % 3. To measure emeraure wih an accuracy o ε ε T ± % ± T in K Esimaion o ineic arameers in more comlex reacion sysems So ar we have deal wih single reacions only, ye an engineer will almos always encouner a roblem involving mulile-mixed reacions. Neverheless he mehods described so ar can oen be uilized, i adequae amoun o exerimenaion can be obained, o esimae a leas some o he ineic arameers. The mehod o hal-ime is oen uilized o: a. evaluae he overall order o reacion (single reacion) by using all reacans in soichiomeric raio; b. evaluae he order wih resec o when oher reacans are used in excess. The assumion is ha he reacion is irreversible and ollows n-h order behavior o d α d α o (68) α ( α ) α o (69) α log log + ( α ) log o 3 ( α ) 443 y x (70) Saring wih dieren 's and measuring he ime ha i aes or o dro o / and by o loing vs. on a log-log lo one can deermine overall order α and rae consan. o When invesigaing mulile reacions in bach sysems o consan volume we have o monior as many variables as here are indeenden reacions in he sysem. For examle, i we are sudying he raes in a sysem o he ye + B D desired roduc U unwaned roduc o o

37 we could monior and D in ime. I he exerimens are erormed in a consan volume bach sysem and i we assume n-h order rae orms we have o evaluae he arameers,,,, α, α, β, γ, δ which aear in he ollowing descriion o he sysems comonen b b mass balance: d + d D α β δ α γ B b D b U B b D (7) α β δ (7) We could again erorm he exerimens a low conversions o and B so as o measure he iniial raes in he region where reverse raes can be negleced. Then: d α β α B + (73) d D α β (74) B By orming he exression or a oin yield D y d d D + The soichiomery o he reacion dicaes: D y we ge: αα β B o ξ ξ D Do + ξ ξ + ξ B Bo U Uo Eliminaing he exens ξ, ξ in erms o measured concenraions we ge: ξ D Do D since Do 0 was seleced ξ Thus B Bo D ( ) o D (75) dd d αα β + Bo D (76) I in addiion we erorm he exerimen wih large excess o B, hen Bo >> D dd d + β Bo α α (77)

38 d d β D Bo αα (78) From d α α d β log log + log D Bo vs. D daa we can ge esimaes o d d indicaed above on a semi-log lo we ge a sloe o α α. Woring now a moderae levels o exerimens Bo d d so ha D log log β log αα y (79) as shown beore, and by loing he quaniies D Bo D Bo we can lo or he new se o ( ) Bo D 443 x From he sloe one can ge β and rom all oins an average or. Now one can use vs. daa vs. d D D α β α B ( Bo D) β (8) dd log log α log b + ( ) Bo D x y (80) ( { ) (8) From he sloe α is deermined and rom he daa oins an average value o and are already nown, also and α can be evaluaed.. Now since α α, The above served o demonsrae ha he dierenial mehod o analysis, as illusraed or single reacions, can also be used in more comlex reacion schemes. In order o do so one needs o lan he exerimens in a roer way and maniulae he daa roerly. Similarly or many racical reacion orders o ineres he exression or he yield can be inegraed and inegral analysis can be alied. For examle i we guess α α β dd d + + Bo B ( ) D (83)

39 D o + Bo D dd d o (84) + ln Bo D o Bo D (85) Thus, i he hyohesized orders α α β are correc, hen a lo o ln Bo ( o D ( Bo D ) x y ) (86) should yield a sraigh line wih a sloe. I we guessed α, α β we would have o inegrae: d d D which uon inegraion yields Bo D + Bo D Bo Bo o Bo ( ) D (87) (88) Now i is diicul o rewrie he resul in a orm which would yield a sraigh line lo in a roerly seleced coordinae sysem. learly when dealing wih mulile reacions a oin is reached very as when convenional mehods o inegral analysis become edious and oen imossible o use. In such a siuaion one mus erorm a suicien number o exerimens so ha some saisical arameer esimaion mehods can be used. The iniial esimaes may requenly be obained by dierenial analysis as resened above Parameers in sysems o monomolecular reacions In many comlex reacion schemes here oen is some indicaion ha mos or even all reacions can be considered o be irs order. (The same assumion is requenly arbirarily made, since his simliies remendously he reamen o he sysem, and his is o be avoided). For M indeenden irs order reversible reacions beween S secies in a bach sysem o consan volume one can wrie: d S j s s (89a) j

40 S d j (89b) j s s j S ds s + s + s sj j j s where is he rae consan or he rae or ormaion o comonen j in reacion. For ji examle, is he rae consan or he rae o ormaion o secies in reacion. The reversible rae is viewed or each secies as rae o ormaion - rae o disaearance. Thus, o obain d he accumulaion o say secies,, we have o sum over he raes o ormaion o in all reacions where aears. Then is he sum o he rae consans or raes o ormaion o secies j rom secies summed u over all secies o disaearance o secies. We could arbirarily deine ii S j j i ji. S j j s (89c) j. This roerly aears wih he negaive sign since i is equal j j S i j or in general The above scheme considers he ossibiliy ha every secies in he sysem can roduce all he ohers or be roduced rom hem. For examle: ( + + ) ( ) ( ) Naurally he above orm is he mos general one and in seciic siuaions many consans can be a riori se o be zero. The roblem in marix orm is: where d K; 0 o (90)

41 .. s K... s s s... s s s wih ii S j j i ji Mos oen in order o deal wih uni vecors every row is divided by and he equaions are wrien in erms o mole racions: y j j (9) ToT K is ( s s) marix o rae consans. dy K y; 0 y y (9) The roblem is as ollows: o ToT, oal molar concenraion, From a coninuous record o y vs. (or rom a se o N discree measuremens o y a, l,... N where N >> S ) deermine he consans comrising he marix K. n elegan and useul heory o monomolecular reacions o his ye was develoed by Wei and Praer [dvances in aalysis 3, (96)] and is also resened in abbreviaed orm by Boudar, M. "Kineics o hemical Processes", haer 0,. 0, Prenice Hall, N.J., 968. Basically he heory saes ha since K is an ( s s) osiive deinie marix i mus have S real eigenvalues λ m, m 0,,... s. To each eigenvalue corresonds a aricular eigenvecor x m so ha he ollowing marix equaion is saisied: K X X Λ or K x x m (93) where X [ x x x... x ] m x x x x x x... 0 s, 0 s, 0 s... x x x 05 5 s, s λo λ Λ 0 0 λ.. 0 diagonal marix λs (94) (95)

42 Thus he marix K mas every eigenvecor x m ono isel augmened by a "sreching" acor λ m. Eigenvecors x m esablish he characerisic direcions or he marix K, eigenvalues λ m give he scale acors in hese direcions. I one selecs now b0 y X b b b dy bs (96) db X Ky KXb (97) Premulilying boh sides wih X db X X X K X b (98) and remembering X X I Ideniy marix (99) KX X Λ (00) we ge db Λ b 0 b b X y (0) This is a decouled marix equaion db0 db db s λ b b b e λ b b b e 0 λ s s s s0 λo λ b b b e 0 0 λs (0) The above marix maniulaion shows ha in rincile one can ind a se o S iciious comonens B so ha he equaions or quaniies o B, b, are uncouled and can be solved one by one. Furhermore a equilibrium: dy eq 0 K y (03) eq Remember λ m x K x, m 0,,,... s (04) m m Thus he irs eigenvecor can be seleced as he equilibrium comosiion: x y and b b cons (05) 0 eq 0 00 and he irs eigenvalue is zero λ 0 0

43 Thus y x b b x + b x + b x + b x y y b x + b x + b x... + b x eq 3 3 s s s b0 e λ x b0 e λ λ x... bs e + xs (06) This describes he decay o deviaions rom equilibrium. Each eigenvecor x m is a direcion in sace, and he righ hand side o he equaion reresens all conribuions ha mae u he reacion ahs. good choice o iniial comosiions y 0, so ha say b 0 0 bu b 0 b 30 b , would yield a sraigh line reacion ah in he direcion o x. Thus when one nows he rae consans b vecor and deermine all he sraigh line reacions ahs. ji However, his heory is helul even when one can comue all λ m 's and corresonding x m 's, ind he ji are no nown and are sough since i rovides he guidelines or searching or sraigh line reacion ahs. Once hese are exerimenally esablished semi bi log los o b vs. give all he eigenvalues λ m. ln b b 0 ln b b 0 io λ (07) λ (08) From he eigenvalues λ m, by simle marix maniulaion, ji are deermined. Since ym, m,... s are measured as a uncion o ime and bm, m 0,,... s, are obainable rom y 's only i 's are nown, which hey are no, his means ha one has o use a rial and error m ji mehod and search or suiable combinaions o which λ and bs can be deermined leading o evaluaion o 's. n i y io 's giving he desired sraigh line reacion ahs rom One o he bes examles is he classical analysis o he isomerizaion o buane on alumina which was sudied exensively by Haag & Pines and Lago & Haag and was used by Wei & Praer o illusrae he ower o heir mehod. - Buene ji 5 - Buene 3 Trans - Buene

44 Le us describe now a se o exerimens and he use o Wei-Praer echnique. We will ollow he reacion ah or every exerimen on he riangular comosiion diagram. In he irs exerimen (ah ) sar wih ure and le he sysem reach equilibrium. The equilibrium comosiion o oin e esablishes he zeroh eigenvecor x 0 or λ 0 0. Since ah is no sraigh we now ha ure is no a comonen o b. 4a 3 We can now erorm an exerimen saring wih ure (ah ). Since his does no yield a sraigh line we now ha ure is no a comonen o b vecor. However, Wei and Praer mehod shows how o use he daa rom ah close o equilibrium in order o exraolae o he side o he riangle and selec a new saring comosiion or run 3. Run 3 (ah 3) does no give a sraigh line bu wih one more correcion leads o saring oin or run 4. Pah 4a is a sraigh line, hus, he comosiion on he side o he riangle deermines b0 and λ can be ound rom ln b λ. However, a run 4b rom b ure 3 o veriy he sraigh line ah is necessary. Then marix algebra allows he redicion o he las sraigh line, ah 5, and calculaion o all λ 's and b 's and 's by marix maniulaion as shown in Boudar's boo. Sraigh reacion ah 5 can naurally be veriied by exerimens. erainly a owerul mehod o cu down on exerimenal wor bu resriced o irs order reacions. heng, Fizgerald and ar [Ind. Eng. hem. Process Design and Develomen Vol. 9, No., 59, 977] roosed o ollowing mehod or deermining ij 's in a sysem o unimolecular reacions. Le [ ],, 3... N be a row vecor o concenraion o all he secies and le hese concenraions be deermined a equal ime incremens. Then K ( ) ( 0 ) K ( ) ( ) ( 0 ) K ( ) ( ) e e e m m e or in marix noaion ( ) ( ) ( 0) ( ) K K e m ( m ) i 5 e 4b ij 3 0 (09) (0)

45 ( 0) ( ) ( ) ( ) ( m ) m K e () ( 0) ( ) K ln ( ) ( ( m ) ) ( m ) () ll o he above comuaions (marix inversion, logarihm o a marix ec.) are readily rogrammable or available on modern comuers. Thus he above comuaions yield he marix o rae consans direcly. n-h order reacions: more general rocedure valid or a se o n-h order reacions o nown order (or even or comlex now rae orms) was develoed by Himmelbaum, Jones & Bischo [Ind. Eng. hem. Fundamenals Vol. 6, No. 4, 539 (967)] or evaluaion o he rae consans. In a bach consan volume sysem he equaions or maerials balance or all comonens can be cas in he ollowing orm: S M r ji i d j M i i rji ; j,,... S (3) oal number o comonens oal number o reacions i.e. number o ems in he rae exressions concenraion deenden erm in rae o roducion i > negaive sign) o comonen j in reacion ah i rae consan or ah i vecor o S concenraions 0 or deleion (i wih d The derivaives j canno be measured direcly bu he dierences in concenraions a various imes can. By inegraing he above equaions one ges: M () j n j o i rji i direcly rom measured daa o n by inegraion (since concenraion deendence o he raes is nown) o measured daa (4) ( ) ( ) X j n j o i nji i M (5) The sandard leas squares mehod can now be used o minimize he ollowing exression N S ( ˆ nj nj ynj ) S w y n j (6)

46 n indicaes n-h daa oin a n N oal number o daa oins S oal number o secies y exerimenally measured concenraion dierence o comonen j a ime nj j n j o o and a ime yˆ M nj i nji i deenden variable X - rediced valued o he deenden variable ynj a oin n where X nji are calculaed rom exerimenal daa and w nj - any desired weighing uncion i are o be ound. These weighs can be iced all o be leading o he usual leas square is, or hey can be seleced o be roorional o he inverse o measured concenraion dierence, or o be inversely roorional o he var iance o concenraion values or each secies j. Seing hen S i 0 or all i,,... M (7) leads o a se o linear equaions rom which i can be romly deermined. The above mehod alhough quie owerul in many siuaions also leads in general only o rough esimaes o reacion consans. n