Distributions of Residence Times for Chemical Reactors

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1 Disribuions of Residence Times for Chemical Reacors DVD 13 Nohing in life is o be feared. I is only o be undersood. Marie Curie Overview In his chaper we learn abou nonideal reacors, ha is, reacors ha do no follow he models we have developed for ideal CSTRs, PFRs, and PBRs. In Par 1 we describe how o characerize hese nonideal reacors using he residence ime disribuion funcion E(), he mean residence ime m, he cumulaive disribuion funcion F(), and he variance s 2. Nex we evaluae E(), F(), m, and s 2 for ideal reacors, so ha we have a reference poin as o how far our real (i.e., nonideal) reacor is off he norm from an ideal reacor. The funcions E() and F() will be developed for ideal PPRs, CSTRs and laminar flow reacors. Examples are given for diagnosing problems wih real reacors by comparing m and E() wih ideal reacors. We will hen use hese ideal curves o help diagnose and roubleshoo bypassing and dead volume in real reacors. In Par 2 we will learn how o use he residence ime daa and funcions o make predicions of conversion and exi concenraions. Because he residence ime disribuion is no unique for a given reacion sysem, we mus use new models if we wan o predic he conversion in our nonideal reacor. We presen he five mos common models o predic conversion and hen close he chaper by applying wo of hese models, he segregaion model and he maximum mixedness model, o single and o muliple reacions. Afer sudying his chaper he reader will be able o describe he cumulaive F() and exernal age E() and residence-ime disribuion funcions, and o recognize hese funcions for PFR, CSTR, and laminar flow reacors. The reader will also be able o apply hese funcions o calculae he conversion and concenraions exiing a reacor using he segregaion model and he maximum mixedness model for boh single and muliple reacions Pearson Educaion, Inc.

2 868 Disribuions of Residence Times for Chemical Reacors Chap. 13 We wan o analyze and characerize nonideal reacor behavior General Characerisics The reacors reaed in he book hus far he perfecly mixed bach, he plug-flow ubular, he packed bed, and he perfecly mixed coninuous ank reacors have been modeled as ideal reacors. Unforunaely, in he real world we ofen observe behavior very differen from ha expeced from he exemplar; his behavior is rue of sudens, engineers, college professors, and chemical reacors. Jus as we mus learn o work wih people who are no perfec, so he reacor analys mus learn o diagnose and handle chemical reacors whose performance deviaes from he ideal. Nonideal reacors and he principles behind heir analysis form he subjec of his chaper and he nex. Par 1 Characerizaion and Diagnosics The basic ideas ha are used in he disribuion of residence imes o characerize and model nonideal reacions are really few in number. The wo major uses of he residence ime disribuion o characerize nonideal reacors are 1. To diagnose problems of reacors in operaion 2. To predic conversion or effluen concenraions in exising/available reacors when a new reacion is used in he reacor Sysem 1 In a gas liquid coninuous-sirred ank reacor (Figure 13-1), he gaseous reacan was bubbled ino he reacor while he liquid reacan was fed hrough an inle ube in he reacor s side. The reacion ook place a he gas liquid inerface of he bubbles, and he produc was a liquid. The coninuous liquid phase could be regarded as perfecly mixed, and he reacion rae was proporional o he oal bubble surface area. The surface area of a paricular bubble depended on he ime i had spen in he reacor. Because of heir differen sizes, some gas bubbles escaped from he reacor almos immediaely, while ohers spen so much ime in he reacor ha hey were almos com- Figure 13-1 Gas liquid reacor. 26 Pearson Educaion, Inc.

3 Sec General Characerisics 869 No all molecules are spending he same ime in he reacor. pleely consumed. The ime he bubble spends in he reacor is ermed he bubble residence ime. Wha was imporan in he analysis of his reacor was no he average residence ime of he bubbles bu raher he residence ime of each bubble (i.e., he residence ime disribuion). The oal reacion rae was found by summing over all he bubbles in he reacor. For his sum, he disribuion of residence imes of he bubbles leaving he reacor was required. An undersanding of residence-ime disribuions (RTDs) and heir effecs on chemical reacor performance is hus one of he necessiies of he echnically compeen reacor analys. Sysem 2 A packed-bed reacor is shown in Figure When a reacor is packed wih caalys, he reacing fluid usually does no flow hrough he reacor uniformly. Raher, here may be secions in he packed bed ha offer lile resisance o flow, and as a resul a major porion of he fluid may channel hrough his pahway. Consequenly, he molecules following his pahway do no spend as much ime in he reacor as hose flowing hrough he regions of high resisance o flow. We see ha here is a disribuion of imes ha molecules spend in he reacor in conac wih he caalys. Figure 13-2 Packed-bed reacor. Sysem 3 In many coninuous-sirred ank reacors, he inle and oule pipes are close ogeher (Figure 13-3). In one operaion i was desired o scale up pilo plan resuls o a much larger sysem. I was realized ha some shor circuiing occurred, so he anks were modeled as perfecly mixed CSTRs wih a bypass sream. In addiion o shor circuiing, sagnan regions (dead zones) are ofen encounered. In hese regions here is lile or no exchange of maerial wih he well-mixed regions, and, consequenly, virually no reacion occurs We wan o find ways of deermining he dead volume and amoun of bypassing. Dead zone Bypassing Figure 13-3 CSTR. 26 Pearson Educaion, Inc.

4 87 Disribuions of Residence Times for Chemical Reacors Chap. 13 The hree conceps RTD Mixing Model here. Experimens were carried ou o deermine he amoun of he maerial effecively bypassed and he volume of he dead zone. A simple modificaion of an ideal reacor successfully modeled he essenial physical characerisics of he sysem and he equaions were readily solvable. Three conceps were used o describe nonideal reacors in hese examples: he disribuion of residence imes in he sysem, he qualiy of mixing, and he model used o describe he sysem. All hree of hese conceps are considered when describing deviaions from he mixing paerns assumed in ideal reacors. The hree conceps can be regarded as characerisics of he mixing in nonideal reacors. One way o order our hinking on nonideal reacors is o consider modeling he flow paerns in our reacors as eiher CSTRs or PFRs as a firs approximaion. In real reacors, however, nonideal flow paerns exis, resuling in ineffecive conacing and lower conversions han in he case of ideal reacors. We mus have a mehod of accouning for his nonidealiy, and o achieve his goal we use he nex-higher level of approximaion, which involves he use of macromixing informaion (RTD) (Secions 13.1 o 13.4). The nex level uses microscale (micromixing) informaion o make predicions abou he conversion in nonideal reacors. We address his hird level of approximaion in Secions 13.6 o 13.9 and in Chaper Residence-Time Disribuion (RTD) Funcion The idea of using he disribuion of residence imes in he analysis of chemical reacor performance was apparenly firs proposed in a pioneering paper by MacMullin and Weber. 1 However, he concep did no appear o be used exensively unil he early 195s, when Prof. P. V. Danckwers 2 gave organizaional srucure o he subjec of RTD by defining mos of he disribuions of ineres. The ever-increasing amoun of lieraure on his opic since hen has generally followed he nomenclaure of Danckwers, and his will be done here as well. In an ideal plug-flow reacor, all he aoms of maerial leaving he reacor have been inside i for exacly he same amoun of ime. Similarly, in an ideal bach reacor, all he aoms of maerials wihin he reacor have been inside i for an idenical lengh of ime. The ime he aoms have spen in he reacor is called he residence ime of he aoms in he reacor. The idealized plug-flow and bach reacors are he only wo classes of reacors in which all he aoms in he reacors have he same residence ime. In all oher reacor ypes, he various aoms in he feed spend differen imes inside he reacor; ha is, here is a disribuion of residence imes of he maerial wihin he reacor. For example, consider he CSTR; he feed inroduced ino a CSTR a any given ime becomes compleely mixed wih he maerial already in he reacor. In oher words, some of he aoms enering he CSTR 1 R. B. MacMullin and M. Weber, Jr., Trans. Am. Ins. Chem. Eng., 31, 49 (1935). 2 P. V. Danckwers, Chem. Eng. Sci., 2, 1 (1953). 26 Pearson Educaion, Inc.

5 Sec Measuremen of he RTD 871 The RTD : Some molecules leave quickly, ohers oversay heir welcome. We will use he RTD o characerize nonideal reacors. leave i almos immediaely because maerial is being coninuously wihdrawn from he reacor; oher aoms remain in he reacor almos forever because all he maerial is never removed from he reacor a one ime. Many of he aoms, of course, leave he reacor afer spending a period of ime somewhere in he viciniy of he mean residence ime. In any reacor, he disribuion of residence imes can significanly affec is performance. The residence-ime disribuion (RTD) of a reacor is a characerisic of he mixing ha occurs in he chemical reacor. There is no axial mixing in a plug-flow reacor, and his omission is refleced in he RTD. The CSTR is horoughly mixed and possesses a far differen kind of RTD han he plug-flow reacor. As will be illusraed laer, no all RTDs are unique o a paricular reacor ype; markedly differen reacors can display idenical RTDs. Neverheless, he RTD exhibied by a given reacor yields disincive clues o he ype of mixing occurring wihin i and is one of he mos informaive characerizaions of he reacor. Use of racers o deermine he RTD 13.2 Measuremen of he RTD The RTD is deermined experimenally by injecing an iner chemical, molecule, or aom, called a racer, ino he reacor a some ime and hen measuring he racer concenraion, C, in he effluen sream as a funcion of ime. In addiion o being a nonreacive species ha is easily deecable, he racer should have physical properies similar o hose of he reacing mixure and be compleely soluble in he mixure. I also should no adsorb on he walls or oher surfaces in he reacor. The laer requiremens are needed so ha he racer s behavior will honesly reflec ha of he maerial flowing hrough he reacor. Colored and radioacive maerials along wih iner gases are he mos common ypes of racers. The wo mos used mehods of injecion are pulse inpu and sep inpu Pulse Inpu Experimen The C curve In a pulse inpu, an amoun of racer N is suddenly injeced in one sho ino he feedsream enering he reacor in as shor a ime as possible. The oule concenraion is hen measured as a funcion of ime. Typical concenraion ime curves a he inle and oule of an arbirary reacor are shown in Figure The effluen concenraion ime curve is referred o as he C curve in RTD analysis. We shall analyze he injecion of a racer pulse for a single-inpu and single-oupu sysem in which only flow (i.e., no dispersion) carries he racer maerial across sysem boundaries. Firs, we choose an incremen of ime sufficienly small ha he concenraion of racer, C(), exiing beween ime and is essenially he same. The amoun of racer maerial, N, leaving he reacor beween ime and is hen N C() v (13-1) 26 Pearson Educaion, Inc.

6 872 Disribuions of Residence Times for Chemical Reacors Chap. 13 Inerpreaion of E() d where v is he effluen volumeric flow rae. In oher words, N is he amoun of maerial exiing he reacor ha has spen an amoun of ime beween and in he reacor. If we now divide by he oal amoun of maerial ha was injeced ino he reacor, N, we obain (13-2) which represens he fracion of maerial ha has a residence ime in he reacor beween ime and. For pulse injecion we define so ha N N vc ( ) vc ( ) E() (13-3) N N E() (13-4) The quaniy E() is called he residence-ime disribuion funcion. I is he funcion ha describes in a quaniaive manner how much ime differen fluid elemens have spen in he reacor. The quaniy E()d is he fracion of fluid exiing he reacor ha has spen beween ime and + d inside he reacor. N N Feed Reacor Effluen Injecion Deecion Pulse injecion Pulse response C C The C curve τ τ Sep injecion Sep response C C Figure 13-4 RTD measuremens. 26 Pearson Educaion, Inc.

7 Sec Measuremen of he RTD 873 C() The C curve Area = C()d C() We find he RTD funcion, E(), from he racer concenraion C() E() The E curve If N is no known direcly, i can be obained from he oule concenraion measuremens by summing up all he amouns of maerials, N, beween ime equal o zero and infiniy. Wriing Equaion (13-1) in differenial form yields and hen inegraing, we obain dn vc() d (13-5) N vc ( ) d The volumeric flow rae v is usually consan, so we can define E() as E ( ) C ( ) (13-6) (13-7) The inegral in he denominaor is he area under he C curve. An alernaive way of inerpreing he residence-ime funcion is in is inegral form: C ( ) d Fracion of maerial leaving he reacor ha has resided in he reacor for imes beween 1 and E ( ) d We know ha he fracion of all he maerial ha has resided for a ime in he reacor beween and is 1; herefore, mus leave E ( ) d 1 Evenually all (13-8) The following example will show how we can calculae and inerpre E() from he effluen concenraions from he response o a pulse racer inpu o a real (i.e., nonideal) reacor. Example 13 1 Consrucing he C() and E() Curves A sample of he racer hyane a 32 K was injeced as a pulse o a reacor, and he effluen concenraion was measured as a funcion of ime, resuling in he daa shown in Table E Pulse Inpu TABLE E TRACER DATA (min) C (g/m 3 ) The measuremens represen he exac concenraions a he imes lised and no average values beween he various sampling ess. (a) Consruc figures showing C() and E() as funcions of ime. (b) Deermine boh he fracion of maerial leaving 26 Pearson Educaion, Inc.

8 874 Disribuions of Residence Times for Chemical Reacors Chap. 13 he reacor ha has spen beween 3 and 6 min in he reacor and he fracion of maerial leaving ha has spen beween 7.75 and 8.25 min in he reacor, and (c) deermine he fracion of maerial leaving he reacor ha has spen 3 min or less in he reacor. Soluion (a) By ploing C as a funcion of ime, using he daa in Table E13-1.1, he curve shown in Figure E is obained. The C curve Figure E The C curve. To obain he E() curve from he C() curve, we jus divide C() by he inegral C ( ) d, which is jus he area under he C curve. Because one quadraure (inegraion) formula will no suffice over he enire range of daa in Table E13-1.1, we break he daa ino wo regions, -1 minues and 1 o 14 minues. The area under he C curve can now be found using he numerical inegraion formulas (A-21) and (A-25) in Appendix A.4: C ( ) d C ( ) d C ( ) d (E13-1.1) 1 1 C ( ) d -- [1( ) 41 ( ) 25 ( ) 48 ( ) 3 2( 1) 48 ( ) 26 ( ) 44 ( ) 23. ( ) 42.2 ( ) 11.5 ( )] 47.4 g min m 3 (A-25) C ( ) d -- [ 1.54(.6) ] 2.6 gmin m 3 3 C ( ) d 5. g minm 3 (A-21) (E13-1.2) 26 Pearson Educaion, Inc.

9 Sec Measuremen of he RTD 875 We now calculae E() wih he following resuls: C ( ) C ( ) g minm C ( ) d 3 (E13-1.3) TABLE E C() AND E() (min) C() (g/m 3 ) E() (min 1 ) (b) These daa are ploed in Figure E The shaded area represens he fracion of maerial leaving he reacor ha has resided in he reacor beween 3 and 6 min. The E curve Figure E Analyzing he E curve. Using Equaion (A-22) in Appendix A.4: 3 6 E ( ) d shaded area -- ( f 1 3f 2 3f 3 f 4 ) ( )[.163(.2) 3(.16).12].51 (A-22) Evaluaing his area, we find ha 51% of he maerial leaving he reacor spends beween 3 and 6 min in he reacor. Because he ime beween 7.75 and 8.25 min is very small relaive o a ime scale of 14 min, we shall use an alernaive echnique o deermine his fracion o reinforce he inerpreaion of he quaniy E() d. The average value of E() beween hese imes is.6 min 1 : E() d (.6 min 1 )(.5 min).3 The ail Consequenly, 3.% of he fluid leaving he reacor has been in he reacor beween 7.75 and 8.25 min. The long-ime porion of he E() curve is called he ail. In his example he ail is ha porion of he curve beween say 1 and 14 min. 26 Pearson Educaion, Inc.

10 876 Disribuions of Residence Times for Chemical Reacors Chap. 13 (c) Finally, we shall consider he fracion of maerial ha has been in he reacor for a ime or less, ha is, he fracion ha has spen beween and minues in he reacor. This fracion is jus he shaded area under he curve up o minues. This area is shown in Figure E for 3 min. Calculaing he area under he curve, we see ha 2% of he maerial has spen 3 min or less in he reacor (min) Figure E Analyzing he E curve. Drawbacks o he pulse injecion o obain he RTD The principal difficulies wih he pulse echnique lie in he problems conneced wih obaining a reasonable pulse a a reacor s enrance. The injecion mus ake place over a period which is very shor compared wih residence imes in various segmens of he reacor or reacor sysem, and here mus be a negligible amoun of dispersion beween he poin of injecion and he enrance o he reacor sysem. If hese condiions can be fulfilled, his echnique represens a simple and direc way of obaining he RTD. There are problems when he concenraion ime curve has a long ail because he analysis can be subjec o large inaccuracies. This problem principally affecs he denominaor of he righ-hand side of Equaion (13-7) [i.e., he inegraion of he C() curve]. I is desirable o exrapolae he ail and analyically coninue he calculaion. The ail of he curve may someimes be approximaed as an exponenial decay. The inaccuracies inroduced by his assumpion are very likely o be much less han hose resuling from eiher runcaion or numerical imprecision in his region. Mehods of fiing he ail are described in he Professional Reference Shelf 13 R Sep Tracer Experimen Now ha we have an undersanding of he meaning of he RTD curve from a pulse inpu, we will formulae a more general relaionship beween a ime-varying racer injecion and he corresponding concenraion in he effluen. We shall sae wihou developmen ha he oupu concenraion from a vessel is relaed o he inpu concenraion by he convoluion inegral: 3 3 A developmen can be found in O. Levenspiel, Chemical Reacion Engineering, 2nd ed. (New York: Wiley, 1972), p Pearson Educaion, Inc.

11 Sec Measuremen of he RTD 877 C in C ou Sep Inpu C ou () C in ( )E() d (13-9) The inle concenraion mos ofen akes he form of eiher a perfec pulse inpu (Dirac dela funcion), imperfec pulse injecion (see Figure 13-4), or a sep inpu. Jus as he RTD funcion E() can be deermined direcly from a pulse inpu, he cumulaive disribuion F() can be deermined direcly from a sep inpu. We will now analyze a sep inpu in he racer concenraion for a sysem wih a consan volumeric flow rae. Consider a consan rae of racer addiion o a feed ha is iniiaed a ime. Before his ime no racer was added o he feed. Saed symbolically, we have C Ï ( ) Ì Ó( C ) consan The concenraion of racer in he feed o he reacor is kep a his level unil he concenraion in he effluen is indisinguishable from ha in he feed; he es may hen be disconinued. A ypical oule concenraion curve for his ype of inpu is shown in Figure Because he inle concenraion is a consan wih ime, C, we can ake i ouside he inegral sign, ha is, C ou C E() d Dividing by C yields C ou C sep F ( ) E() d F() C ou C sep (13-1) Advanages and drawbacks o he sep injecion We differeniae his expression o obain he RTD funcion E(): E() ---- d C ( ) (13-11) d The posiive sep is usually easier o carry ou experimenally han he pulse es, and i has he addiional advanage ha he oal amoun of racer in he feed over he period of he es does no have o be known as i does in he pulse es. One possible drawback in his echnique is ha i is someimes difficul o mainain a consan racer concenraion in he feed. Obaining he RTD from his es also involves differeniaion of he daa and presens an addiional and probably more serious drawback o he echnique, because differeniaion of daa can, on occasion, lead o large errors. A hird problem lies wih he large amoun of racer required for his es. If he racer is very expensive, a pulse es is almos always used o minimize he cos. C sep 26 Pearson Educaion, Inc.

12 878 Disribuions of Residence Times for Chemical Reacors Chap. 13 Oher racer echniques exis, such as negaive sep (i.e., eluion), frequency-response mehods, and mehods ha use inpus oher han seps or pulses. These mehods are usually much more difficul o carry ou han he ones presened and are no encounered as ofen. For his reason hey will no be reaed here, and he lieraure should be consuled for heir virues, defecs, and he deails of implemening hem and analyzing he resuls. A good source for his informaion is Wen and Fan. 4 From E() we can learn how long differen molecules have been in he reacor Characerisics of he RTD Someimes E() is called he exi-age disribuion funcion. If we regard he age of an aom as he ime i has resided in he reacion environmen, hen E() concerns he age disribuion of he effluen sream. I is he mos used of he disribuion funcions conneced wih reacor analysis because i characerizes he lenghs of ime various aoms spend a reacion condiions Inegral Relaionships The fracion of he exi sream ha has resided in he reacor for a period of ime shorer han a given value is equal o he sum over all imes less han of E(), or expressed coninuously, The cumulaive RTD funcion F() E ( ) d Fracion of effluen ha has been in reacor for less han ime F ( ) (13-12) Analogously, we have E ( ) d Fracion of effluen ha has been in reacor for longer han ime 1 F ( ) (13-13) Because appears in he inegraion limis of hese wo expressions, Equaions (13-12) and (13-13) are boh funcions of ime. Danckwers 5 defined Equaion (13-12) as a cumulaive disribuion funcion and called i F(). We can calculae F() a various imes from he area under he curve of an E() versus plo. For example, in Figure E we saw ha F() a 3 min was.2, meaning ha 2% of he molecules spen 3 min or less in he reacor. Similarly, using Figure E we calculae F() =.4 a 4 minues. We can coninue in his manner o consruc F(). The shape of he F() curve is shown in Figure One noes from his curve ha 8% [F()] of he molecules spend 8 min or less in he reacor, and 2% of he molecules [1 F()] spend longer han 8 min in he reacor. 4 C. Y. Wen and L. T. Fan, Models for Flow Sysems and Chemical Reacors (New York: Marcel Dekker, 1975). 5 P. V. Danckwers, Chem. Eng. Sci., 2, 1 (1953). 26 Pearson Educaion, Inc.

13 Sec Characerisics of he RTD 879 The F curve 1..8 F() (min) Figure 13-5 Cumulaive disribuion curve, F(). The F curve is anoher funcion ha has been defined as he normalized response o a paricular inpu. Alernaively, Equaion (13-12) has been used as a definiion of F(), and i has been saed ha as a resul i can be obained as he response o a posiive-sep racer es. Someimes he F curve is used in he same manner as he RTD in he modeling of chemical reacors. An excellen example is he sudy of Wolf and Whie, 6 who invesigaed he behavior of screw exruders in polymerizaion processes Mean Residence Time In previous chapers reaing ideal reacors, a parameer frequenly used was he space ime or average residence ime, which was defined as being equal o V/v. I will be shown ha, in he absence of dispersion, and for consan volumeric flow (v = v ) no maer wha RTD exiss for a paricular reacor, ideal or nonideal, his nominal space ime,, is equal o he mean residence ime, m. As is he case wih oher variables described by disribuion funcions, he mean value of he variable is equal o he firs momen of he RTD funcion, E(). Thus he mean residence ime is The firs momen gives he average ime he effluen molecules spen in he reacor. m E ( ) d E ( ) d E ( ) d (13-14) We now wish o show how we can deermine he oal reacor volume using he cumulaive disribuion funcion. 6 D. Wolf and D. H. Whie, AIChE J., 22, 122 (1976). 26 Pearson Educaion, Inc.

14 88 Disribuions of Residence Times for Chemical Reacors Chap. 13 Wha we are going o do now is prove m = for consan volumeric flow, v = v. You can skip wha follows and go direcly o Equaion (13 21) if you can accep his resul. All we are doing here is proving ha he space ime and mean residence ime are equal. 1 Consider he following siuaion: We have a reacor compleely filled wih maize molecules. A ime we sar o injec blue molecules o replace he maize molecules ha currenly fill he reacor. Iniially, he reacor volume V is equal o he volume occupied by he maize molecules. Now, in a ime d, he volume of molecules ha will leave he reacor is ( vd). The fracion of hese molecules ha have been in he reacor a ime or greaer is [1 F()]. Because only he maize molecules have been in he reacor a ime or greaer, he volume of maize molecules, dv, leaving he reacor in a ime d is dv ( vd)[1 F()] (13-15) If we now sum up all of he maize molecules ha have lef he reacor in ime, we have Because he volumeric flow rae is consan, Using he inegraion-by-pars relaionship gives V v [1 F()] d (13-16) V v [1 F()] d (13-17) x dy xy and dividing by he volumeric flow rae gives y dx (13-18) A, F() ; and as Æ, hen [1 F()]. The firs erm on he righ-hand side is zero, and he second erm becomes However, df E() d; herefore, V -- [ 1 F ( ) ] v (13-19) E() d (13-2) The righ-hand side is jus he mean residence ime, and we see ha he mean residence ime is jus he space ime : V -- df v 1 1 df Noe: For gas-phase reacions a consan emperaure and no pressure drop m = /(1 + ex). 26 Pearson Educaion, Inc.

15 Sec Characerisics of he RTD 881 m, Q.E.D. m (13-21) End of proof! and no change in volumeric flow rae. For gas-phase reacions, his means no pressure drop, isohermal operaion, and no change in he oal number of moles (i.e., e, as a resul of reacion). This resul is rue only for a closed sysem (i.e., no dispersion across boundaries; see Chaper 14). The exac reacor volume is deermined from he equaion V v m (13-22) Oher Momens of he RTD I is very common o compare RTDs by using heir momens insead of rying o compare heir enire disribuions (e.g., Wen and Fan 7 ). For his purpose, hree momens are normally used. The firs is he mean residence ime. The second momen commonly used is aken abou he mean and is called he variance, or square of he sandard deviaion. I is defined by The second momen abou he mean is he variance. The wo parameers mos commonly used o characerize he RTD are and 2 2 ( m ) 2 E ( ) d (13-23) The magniude of his momen is an indicaion of he spread of he disribuion; he greaer he value of his momen is, he greaer a disribuion s spread will be. The hird momen is also aken abou he mean and is relaed o he skewness. The skewness is defined by s ( m ) 3 E ( ) d (13-24) 32 The magniude of his momen measures he exen ha a disribuion is skewed in one direcion or anoher in reference o he mean. Rigorously, for complee descripion of a disribuion, all momens mus be deermined. Pracically, hese hree are usually sufficien for a reasonable characerizaion of an RTD. Example 13 2 Mean Residence Time and Variance Calculaions Calculae he mean residence ime and he variance for he reacor characerized in Example 13-1 by he RTD obained from a pulse inpu a 32 K. Soluion Firs, he mean residence ime will be calculaed from Equaion (13-14): 7 C. Y. Wen and L. T. Fan, Models for Flow Sysems and Chemical Reacors (New York: Decker, 1975), Chap Pearson Educaion, Inc.

16 882 Disribuions of Residence Times for Chemical Reacors Chap. 13 m E() d (E13-2.1) The area under he curve of a plo of E() as a funcion of will yield m. Once he mean residence ime is deermined, he variance can be calculaed from Equaion (13-23): 2 ( m ) 2 E ( ) d (E13-2.2) To calculae m and 2, Table E was consruced from he daa given and inerpreed in Example One quadraure formula will no suffice over he enire range. Therefore, we break he inegral up ino wo regions, o 1 min and 1 o 14 (minues), i.e., infiniy ( ). m E() d E() d E() d 1 Saring wih Table E in Example 13-1, we can proceed o calculae E(), ( m ) and ( m ) 2 E() and 2 E() shown in Table E TABLE E CALCULATING E(), m, AND 2 1 C() E() E() ( m ) a ( m ) 2 E() a 2 E() a a The las wo columns are compleed afer he mean residence ime ( m ) is found. Again, using he numerical inegraion formulas (A-25) and (A-21) in Appendix A.4, we have m h h fx ( ) dx ( f 3 1 4f 2 2f 3 4f 4 4f n1 f n ) h ( f 3 n1 4f n2 f n3 ) (A-25) (A-21) Numerical inegraion o find he mean residence ime, m 1 m 3 -- [1( ) 4(.2) 2.2 ( ) 4(.48) 2.8 ( ) 4.8 ( ) 2(.72) 4(.56) 2(.48) 4(.4) 1.3 ( )] -- [.34(.14) ] min 26 Pearson Educaion, Inc.

17 Sec Characerisics of he RTD 883 Calculaing he mean residence ime, m E( ) d Noe: One could also use he spreadshees in Polymah or Excel o formulae Table E and o calculae he mean residence ime m and variance. Figure E Calculaing he mean residence ime. Ploing E() versus we obain Figure E The area under he curve is 5.15 min. m 5.15 min Calculaing he variance, 2 ( m ) 2 E ( )d = E ( ) d m Now ha he mean residence ime has been deermined, we can calculae he variance by calculaing he area under he curve of a plo of ( m ) 2 E() as a funcion of (Figure E13-2.2[a]). The area under he curve(s) is 6.11 min Area = 2 E()d = min E() 3 2 One could also use Polymah or Excel o make hese calculaions (min) (a) Figure E Calculaing he variance (min) (b) Expanding he square erm in Equaion (13-23) E ( ) d2 m E( ) d m E ( ) d (E13-2.2) = E ( ) d 2 m m E ( ) d m (E13-2.3) We will use quadraure formulas o evaluae he inegral using he daa (columns 1 and 7) in Table E Inegraing beween 1 and 1 minues and 1 and 14 minues using he same form as Equaion (E13-2.3) 26 Pearson Educaion, Inc.

18 884 Disribuions of Residence Times for Chemical Reacors Chap E ( ) d 2 E ( ) d 2 E ( ) d 1 = -- [ 4(.2) 2.4 ( ) 4( 1.44) 23.2 ( ) 3 +4(4.) 2( 4.32) 4( 3.92) 2( 3.84) 2 +4(3.56) 3.] -- [ 3.4( 1.73) ] min 2 3 = min 2 This value is also he shaded area under he curve in Figure E13-2.2(b). 2 The square of he sandard deviaion is min 2, so 2.49 min E ( ) d m min 2 ( 5.15 min) min Normalized RTD Funcion, E() Frequenly, a normalized RTD is used insead of he funcion E(). If he parameer is defined as - (13-25) E() Why we use a normalized RTD E() for a CSTR v 1 v 2 a dimensionless funcion E() can be defined as E() E() (13-26) and ploed as a funcion of. The quaniy represens he number of reacor volumes of fluid based on enrance condiions ha have flowed hrough he reacor in ime. The purpose of creaing his normalized disribuion funcion is ha he flow performance inside reacors of differen sizes can be compared direcly. For example, if he normalized funcion E() is used, all perfecly mixed CSTRs have numerically he same RTD. If he simple funcion E() is used, numerical values of E() can differ subsanially for differen CSTRs. As will be shown laer, for a perfecly mixed CSTR, and herefore 1 E ( ) -- e (13-27) E() E() e (13-28) v 1 > v 2 From hese equaions i can be seen ha he value of E() a idenical imes can be quie differen for wo differen volumeric flow raes, say v 1 and v 2. Bu for 26 Pearson Educaion, Inc.

19 Sec RTD in Ideal Reacors 885 v 1, v 2 he same value of, he value of E() is he same irrespecive of he size of a perfecly mixed CSTR. I is a relaively easy exercise o show ha and is recommended as a 93-s diverissemen Inernal-Age Disribuion, I() E() d 1 (13-29) Tombsone jail How long have you been here? I() When do you expec o ge ou? Alhough his secion is no a prerequisie o he remaining secions, he inernal-age disribuion is inroduced here because of is close analogy o he exernal-age disribuion. We shall le represen he age of a molecule inside he reacor. The inernal-age disribuion funcion I() is a funcion such ha I() is he fracion of maerial inside he reacor ha has been inside he reacor for a period of ime beween and. I may be conrased wih E(), which is used o represen he maerial leaving he reacor ha has spen a ime beween and in he reacion zone; I() characerizes he ime he maerial has been (and sill is) in he reacor a a paricular ime. The funcion E() is viewed ouside he reacor and I() is viewed inside he reacor. In unseady-sae problems i can be imporan o know wha he paricular sae of a reacion mixure is, and I() supplies his informaion. For example, in a caalyic reacion using a caalys whose aciviy decays wih ime, he inernal age disribuion of he caalys in he reacor I(a) is of imporance and can be of use in modeling he reacor. The inernal-age disribuion is discussed furher on he Professional Reference Shelf where he following relaionships beween he cumulaive inernal age disribuion I(a) and he cumulaive exernal age disribuion F(a) and beween E() and I() I(a) = (1 F(a))/ (13-3) E(a) = d [ I( ) ] (13-31) d are derived. For a CSTR i is shown ha he inernal age disribuion funcion is I(a) = 1 -- e 13.4 RTD in Ideal Reacors RTDs in Bach and Plug-Flow Reacors The RTDs in plug-flow reacors and ideal bach reacors are he simples o consider. All he aoms leaving such reacors have spen precisely he same 26 Pearson Educaion, Inc.

20 886 Disribuions of Residence Times for Chemical Reacors Chap. 13 E() for a plugflow reacor amoun of ime wihin he reacors. The disribuion funcion in such a case is a spike of infinie heigh and zero widh, whose area is equal o 1; he spike occurs a V/ v, or 1. Mahemaically, his spike is represened by he Dirac dela funcion: E ( ) ( ) (13-32) The Dirac dela funcion has he following properies: Properies of he ( x) Ï when x Ì Ó when x Dirac dela funcion ( x) d x 1 (13-33) (13-34) g ( x) ( x ) dx To calculae he mean residence ime, we se g(x) (13-35) m E() d ( ) d (13-36) Bu we already knew his resul. To calculae he variance we se, g() = ( ) 2, and he variance, s 2, is 2 () 2 ( ) d (13-37) All maerial spends exacly a ime in he reacor, here is no variance! The cumulaive disribuion funcion F() is F() E ( )d g ( ) ( )d The E() funcion is shown in Figure 13-6(a), and F() is shown in Figure 13-6(b). In Ou E() F() 1. (a) (b) Figure 13-6 Ideal plug-flow response o a pulse racer inpu. 26 Pearson Educaion, Inc.

21 Sec RTD in Ideal Reacors 887 From a racer balance we can deermine E(). E() and E(Q) for a CSTR Single-CSTR RTD In an ideal CSTR he concenraion of any subsance in he effluen sream is idenical o he concenraion hroughou he reacor. Consequenly, i is possible o obain he RTD from concepual consideraions in a fairly sraighforward manner. A maerial balance on an iner racer ha has been injeced as a pulse a ime ino a CSTR yields for In Ou = Accumulaion (13-38) vc V dc d Because he reacor is perfecly mixed, C in his equaion is he concenraion of he racer eiher in he effluen or wihin he reacor. Separaing he variables and inegraing wih C C a yields C() C e / (13-39) This relaionship gives he concenraion of racer in he effluen a any ime. To find E() for an ideal CSTR, we firs recall Equaion (13-7) and hen subsiue for C() using Equaion (13-39). Tha is, } C ( ) C E() e e (13-4) C ( ) d Evaluaing he inegral in he denominaor complees he derivaion of he RTD for an ideal CSTR given by Equaions (13-27) and (13-28): C e d e E() (13-27) E() e (13-28) Recall and E() = E(). Response of an ideal CSTR E(Q) = e Q F(Q) = 1 e Q (a) Figure 13-7 (b) E(Q) and F(Q) for an Ideal CSTR. 26 Pearson Educaion, Inc.

22 888 Disribuions of Residence Times for Chemical Reacors Chap. 13 The cumulaive disribuion F() is F() E()d =1e The E() and F() funcions for an Ideal CSTR are shown in Figure 13-7 (a) and (b), respecively. Earlier i was shown ha for a consan volumeric flow rae, he mean residence ime in a reacor is equal o V/ v, or. This relaionship can be shown in a simpler fashion for he CSTR. Applying he definiion of he mean residence ime o he RTD for a CSTR, we obain m E() d - e / d (13-2) Thus he nominal holding ime (space ime) V/ v is also he mean residence ime ha he maerial spends in he reacor. The second momen abou he mean is a measure of he spread of he disribuion abou he mean. The variance of residence imes in a perfecly mixed ank reacor is (le x /) For a perfecly mixed CSTR: m = and. 2 ( ) e / d 2 (x 1) 2 e x dx 2 (13-41) Then. The sandard deviaion is he square roo of he variance. For a CSTR, he sandard deviaion of he residence-ime disribuion is as large as he mean iself!! Laminar Flow Reacor (LFR) Molecules near he cener spend a shorer ime in he reacor han hose close o he wall. Before proceeding o show how he RTD can be used o esimae conversion in a reacor, we shall derive E() for a laminar flow reacor. For laminar flow in a ubular reacor, he velociy profile is parabolic, wih he fluid in he cener of he ube spending he shores ime in he reacor. A schemaic diagram of he fluid movemen afer a ime is shown in Figure The figure a he lef shows how far down he reacor each concenric fluid elemen has raveled afer a ime. R r r + dr dr R r R U Parabolic Velociy Profile Figure 13-8 The velociy profile in a pipe of ouer radius R is Schemaic diagram of fluid elemens in a laminar flow reacor. Êr ˆ2 U U max 1 Á ËR -- Êr ˆ2 2U avg 1 Á ËR -- 2 v Êr Á R 2 ËR -- ˆ2 (13-42) 26 Pearson Educaion, Inc.

23 Sec RTD in Ideal Reacors 889 where U max is he cenerline velociy and U avg is he average velociy hrough he ube. U avg is jus he volumeric flow rae divided by he cross-secional area. The ime of passage of an elemen of fluid a a radius r is ( r) L U ( r) R 2 L v [ ( r R) 2 ] [ ( r R) 2 ] (13-43) The volumeric flow rae of fluid ou beween r and (r + dr), dv, is dv = U(r) 2prdr The fracion of oal fluid passing beween r and (r + dr) is dv/v, i.e. dv v Ur ( )2( rdr) (13-44) The fracion of fluid beween r and (r + dr) ha has a flow rae beween v and (v + dv) spends a ime beween and ( + d) in he reacor is v E ( )d dv v (13-45) We now need o relae he fluid fracion [Equaion (13-45)] o he fracion of fluid spending beween ime and d in he reacor. Firs we differeniae Equaion (13-43): 2r dr 4 Ï 2 d R 2 [ 1 ( r R) 2 ] 2 R 2 Ì Ó[ 1 ( r R) 2 ] and hen subsiue for using Equaion (13-43) o yield 2 r dr d r dr R 2 (13-46) Combining Equaions (13-44) and (13-46), and hen using Equaion (13-43) for U(r), we now have he fracion of fluid spending beween ime and d in he reacor: E ( )d dv L Ê2r drˆ L Ê -- Á ˆ R 2 Á d d v Ë v Ëv E ( ) The minimum ime he fluid may spend in he reacor is 2 26 Pearson Educaion, Inc.

24 89 Disribuions of Residence Times for Chemical Reacors Chap. 13 L L Ê R2 ˆ V Á U max 2U avg ËR 2 2v 2 Consequenly, he complee RTD funcion for a laminar flow reacor is E() for a laminar flow reacor E ( ) Ï Ô Ì Ô Ó (13-47) The cumulaive disribuion funcion for /2 is F ( ) E ( ) d + E ( ) d d 2 d (13-48) The mean residence ime m is For LFR m = m E ( ) d This resul was shown previously o be rue for any reacor. The mean residence ime is jus he space ime. The dimensionless form of he RTD funcion is d Normalized RTD funcion for a laminar flow reacor E ( ) Ï Ô Ì Ô Ó and is ploed in Figure The dimensionless cumulaive disribuion, F(Q) for Q > 1/2, is F( ) E( ) d d Ë Ê 4 2 ˆ 2 2 (13-49) Ï Ô F( ) ÌÊ1 1 Ë ˆ Ô 4 2 Ó Ô Ô 26 Pearson Educaion, Inc.

25 Sec Diagnosics and Troubleshooing PFR E(Θ) 2 F(Θ) CSTR LFR Θ Θ Figure 13-9 (a) E(Q) for an LFR; (b) F(Q) for a PFR, CSTR, and LFR. Figure 13-9(a) shows E(Q) for a laminar flow reacor (LFR), while Figure 9-13(b) compares F(Q) for a PFR, CSTR, and LFR. Experimenally injecing and measuring he racer in a laminar flow reacor can be a difficul ask if no a nighmare. For example, if one uses as a racer chemicals ha are phoo-acivaed as hey ener he reacor, he analysis and inerpreaion of E() from he daa become much more involved Diagnosics and Troubleshooing General Commens As discussed in Secion 13.1, he RTD can be used o diagnose problems in exising reacors. As we will see in furher deail in Chaper 14, he RTD funcions E() and F() can be used o model he real reacor as combinaions of ideal reacors. Figure 13-1 illusraes ypical RTDs resuling from differen nonideal reacor siuaions. Figures 13-1(a) and (b) correspond o nearly ideal PFRs and CSTRs, respecively. In Figure 13-1(d) one observes ha a principal peak occurs a a ime smaller han he space ime (= V/v ) (i.e., early exi of fluid) and also ha some fluid exis a a ime greaer han space-ime. This curve could be represenaive of he RTD for a packed-bed reacor wih channeling and dead zones. A schemaic of his siuaion is shown in Figure 13-1(c). Figure 13-1(f) shows he RTD for he nonideal CSTR in Figure 13-1(e), which has dead zones and bypassing. The dead zone serves o reduce he effecive reacor volume, so he acive reacor volume is smaller han expeced. 8 D. Levenspiel, Chemical Reacion Engineering, 3rd ed. (New York: Wiley, 1999), p Pearson Educaion, Inc.

26 892 Disribuions of Residence Times for Chemical Reacors Chap. 13 Ideal RTDs ha are commonly observed E() Acual E() (a) (b) Channeling E() z = Dead Zones z = L (d) (c) Channeling Dead Zones (e) E() Bypassing (f) Long ail dead zone Figure 13-1 (a) RTD for near plug-flow reacor; (b) RTD for near perfecly mixed CSTR; (c) Packed-bed reacor wih dead zones and channeling; (d) RTD for packed-bed reacor in (c); (e) ank reacor wih shor-circuiing flow (bypass); (f) RTD for ank reacor wih channeling (bypassing or shor circuiing) and a dead zone in which he racer slowly diffuses Simple Diagnosics and Troubleshooing Using he RTD for Ideal Reacors A The CSTR We will firs consider a CSTR ha operaes (a) normally, (b) wih bypassing, and (c) wih a dead volume. For a well-mixed CSTR, he mole (mass) balance on he racer is Rearranging, we have VdC v d C dc C d 26 Pearson Educaion, Inc.

27 Sec Diagnosics and Troubleshooing 893 We saw he response o a pulse racer is Concenraion: RTD Funcion: Cumulaive Funcion: C ( ) C T e / 1 E ( ) --e / F ( ) 1 e / V ---- v where is he space ime he case of perfec operaion. a. Perfec Operaion (P) Here we will measure our reacor wih a yardsick o find V and our flow rae wih a flow meer o find v in order o calculae = V/v. We can hen compare he curves shown below for he perfec operaion in Figure wih he subsequen cases, which are for imperfec operaion. v 1. Yardsick v E() e ransien F() Figure Perfec operaion of a CSTR. V ---- v If is large, here will be a slow decay of he oupu ransien, C(), and E() for a pulse inpu. If is small, here will be rapid decay of he ransien, C(), and E() for a pulse inpu. b. Bypassing (BP) A volumeric flow rae v b bypasses he reacor while a volumeric flow rae v SB eners he sysem volume and v = v SB + v b. The reacor sysem volume V S is he well-mixed porion of he reacor, and he volumeric flow rae enering he sysem volume is v SB. The subscrip SB denoes ha par of he flow has bypassed and only v SB eners he sysem. Because some of he fluid bypasses, he flow passing hrough he sysem will be less han he oal volumeric rae, v SB < v, consequenly SB >. Le s say he volumeric flow rae ha bypasses he 26 Pearson Educaion, Inc.

28 894 Disribuions of Residence Times for Chemical Reacors Chap. 13 reacor, v b, is 25% of he oal (e.g., v b =.25 v ). The volumeric flow rae enering he reacor sysem, v SB is 75% of he oal (v SB =.75 v ) and he corresponding rue space ime ( SB ) for he sysem volume wih bypassing is SB V V v SB.75v The space ime, SB, will be greaer han ha if here were no bypassing. Because SB is greaer han here will be a slower decay of he ransiens C() and E() han ha of perfec operaion. An example of a corresponding E() curve for he case of bypassing is E ( ) v b v ----d ( ) e SB Vv The CSTR wih bypassing will have RTD curves similar o hose in Figure v 2 SB v v v b v SB E() 1. v v v2 SB Vv F() v b v Figure Ideal CSTR wih bypass. We see from he F() curve ha we have an iniial jump equal o he fracion by-passed. c. Dead Volume (DV) Consider he CSTR in Figure wihou bypassing bu insead wih a sagnan or dead volume. v 1. Sysem Volume V SD v Dead Volume V D E() F() Figure Ideal CSTR wih dead volume. The oal volume, V, is he same as ha for perfec operaion, V = V D + V SD. 26 Pearson Educaion, Inc.

29 Sec Diagnosics and Troubleshooing 895 We see ha because here is a dead volume which he fluid does no ener, here is less sysem volume, V SD, han in he case of perfec operaion, V SD < V. Consequenly, he fluid will pass hrough he reacor wih he dead volume more quickly han ha of perfec operaion, i.e., SD <. If V D.2V, V SD.8V, hen.8v SD v Also as a resul, he ransiens C() and E() will decay more rapidly han ha for perfec operaion because here is a smaller sysem volume. Summary A summary for ideal CSTR mixing volume is shown in Figure DV P BP DV 1 F() E() P 2 v SB v Vv v BP Figure Comparison of E() and F() for CSTR under perfec operaion, bypassing, and dead volume. (BP = bypassing, P = perfec, and DV = dead volume). Knowing he volume V measured wih a yardsick and he flow rae v enering he reacor measured wih a flow meer, one can calculae and plo E() and F() for he ideal case (P) and hen compare wih he measured RTD E() o see if he RTD suggess eiher bypassing (BP) or dead zones (DV) B Tubular Reacor A similar analysis o ha for a CSTR can be carried ou on a ubular reacor. a. Perfec Operaion of PFR (P) We again measure he volume V wih a yardsick and v wih a flow meer. The E() and F() curves are shown in Figure The space ime for a perfec PFR is = V/v b. PFR wih Channeling (Bypassing, BP) Le s consider channeling (bypassing), as shown in Figure 13-16, similar o ha shown in Figures 13-2 and 13-1(d). The space ime for he reacor sysem wih bypassing (channeling) SB is 26 Pearson Educaion, Inc.

30 896 Disribuions of Residence Times for Chemical Reacors Chap v V v E() F() Yardsick Figure Perfec operaion of a PFR. v v v V v E() 1. F() v v Figure PFR wih bypassing similar o he CSTR. SB V v SB Because v SB < v, he space ime for he case of bypassing is greaer when compared o perfec operaion, i.e., SB > If 25% is bypassing (i.e., v b =.25 v ) and 75% is enering he reacor sysem (i.e., v SB =.75 v ), hen SB = V/(.75v ) = The fluid ha does ener he reacor sysem flows in plug flow. Here we have wo spikes in he E() curve. One spike a he origin and one spike a SB ha comes afer for perfec operaion. Because he volumeric flow rae is reduced, he ime of he second spike will be greaer han for perfec operaion. c. PFR wih Dead Volume (DV) The dead volume, V D, could be manifesed by inernal circulaion a he enrance o he reacor as shown in Figure Dead zones v V SD v E() F() V D Figure PFR wih dead volume. 26 Pearson Educaion, Inc.

31 Sec Diagnosics and Troubleshooing 897 The sysem V SD is where he reacion akes place and he oal reacor volume is (V = V SD + V D ). The space ime, SD, for he reacor sysem wih only dead volume is Compared o perfec operaion, he space ime SD is smaller and he racer spike will occur before for perfec operaion. SD < Here again, he dead volume akes up space ha is no accessible. As a resul, he racer will exi early because he sysem volume, V SD, hrough which i mus pass is smaller han he perfec operaion case. Summary SD V SD v Figure is a summary of hese hree cases. F() DV P BP Figure Comparison of PFR under perfec operaion, bypassing, and dead volume (DV = dead volume, P = perfec PFR, BP = bypassing). In addiion o is use in diagnosis, he RTD can be used o predic conversion in exising reacors when a new reacion is ried in an old reacor. However, as we will see in Secion , he RTD is no unique for a given sysem, and we need o develop models for he RTD o predic conversion PFR/CSTR Series RTD Modeling he real reacor as a CSTR and a PFR in series In some sirred ank reacors, here is a highly agiaed zone in he viciniy of he impeller ha can be modeled as a perfecly mixed CSTR. Depending on he locaion of he inle and oule pipes, he reacing mixure may follow a somewha oruous pah eiher before enering or afer leaving he perfecly mixed zone or even boh. This oruous pah may be modeled as a plug-flow reacor. Thus his ype of ank reacor may be modeled as a CSTR in series wih a plug-flow reacor, and he PFR may eiher precede or follow he CSTR. In his secion we develop he RTD for his ype of reacor arrangemen. Firs consider he CSTR followed by he PFR (Figure 13-19). The residence ime in he CSTR will be denoed by s and he residence ime in he PFR by p. If a pulse of racer is injeced ino he enrance of he CSTR, he 26 Pearson Educaion, Inc.

32 898 Disribuions of Residence Times for Chemical Reacors Chap. 13 Side Noe: Medical Uses of RTD The applicaion of RTD analysis in biomedical engineering is being used a an increasing rae. For example, Professor Bob Langer s * group a MIT used RTD analysis for a novel Taylor-Couee flow device for blood deoxificaion while Lee e al. used an RTD analysis o sudy arerial blood flow in he eye. In his laer sudy, sodium fluorescein was injeced ino he anicubical vein. The cumulaive disribuion funcion F() is shown schemaically in Figure 13.5.N-1. Figure 13.5N-2 shows a laser ophhalmoscope image afer injecion of he sodium fluorescein. The mean residence ime can be calculaed for each arery o esimae he mean circulaion ime (ca s). Changes in he reinal blood flow may provide imporan decision-making informaion for sickle-cell disease and reiniis pigmenosa. Figure 13.5.N-1 Cumulaive RTD funcion for arerial blood flow in he eye. Couresy of Med. Eng. Phys. Figure 13.5.N-2 Image of eye afer racer injecion. Couresy of Med. Eng. Phys. * G. A. Ameer, E. A. Grovender, B. Olradovic, C. L. Clooney, and R. Langer, AIChE J. 45, 633 (1999). E. T. Lee, R. G. Rehkopf, J. W. Warnicki, T. Friberg, D. N. Finegold, and E. G. Cape, Med. Eng. Phys. 19, 125 (1997). Figure Real reacor modeled as a CSTR and PFR in series. CSTR oupu concenraion as a funcion of ime will be C C e s This oupu will be delayed by a ime p a he oule of he plug-flow secion of he reacor sysem. Thus he RTD of he reacor sysem is 26 Pearson Educaion, Inc.

33 Sec Diagnosics and Troubleshooing 899 E ( ) Ï Ô e ( p) Ì s Ô Ó s p p (13-5) See Figure The RTD is no unique o a paricular reacor sequence. E() F() Figure 13-2 RTD curves E() and F() for a CSTR and a PFR in series. Nex he reacor sysem in which he CSTR is preceded by he PFR will be reaed. If he pulse of racer is inroduced ino he enrance of he plug-flow secion, hen he same pulse will appear a he enrance of he perfecly mixed secion p seconds laer, meaning ha he RTD of he reacor sysem will be E() is he same no maer which reacor comes firs. E ( ) Ï Ô e ( p) Ì s Ô Ó s p p (13-51) which is exacly he same as when he CSTR was followed by he PFR. I urns ou ha no maer where he CSTR occurs wihin he PFR/CSTR reacor sequence, he same RTD resuls. Neverheless, his is no he enire sory as we will see in Example Example 13 3 Comparing Second-Order Reacion Sysems Examples of early and lae mixing for a given RTD Consider a second-order reacion being carried ou in a real CSTR ha can be modeled as wo differen reacor sysems: In he firs sysem an ideal CSTR is followed by an ideal PFR; in he second sysem he PFR precedes he CSTR. Le s and p each equal 1 min, le he reacion rae consan equal 1. m 3 /kmolmin, and le he iniial concenraion of liquid reacan, C A, equal 1 kmol/m 3. Find he conversion in each sysem. Soluion Again, consider firs he CSTR followed by he plug-flow secion (Figure E13-3.1). A mole balance on he CSTR secion gives v ( C A C Ai ) kc Ai 2 V (E13-3.1) 26 Pearson Educaion, Inc.

34 9 Disribuions of Residence Times for Chemical Reacors Chap. 13 C A C Ai C A Rearranging, we have Figure E Early mixing scheme. 2 s k C Ai C A C Ai Solving for C Ai gives 14 C s kc A 1 Ai s k (E13-3.2) Then 1 14 C Ai kmol/m 2 3 (E13-3.3) This concenraion will be fed ino he PFR. The PFR mole balance df A v dc A dc dv A r dv d A kc A p p k C A C Ai (E13-3.4) (E13-3.5) Subsiuing C Ai.618, p 1, and k 1 in Equaion (E13-3.5) yields Solving for C A gives ( 1) ( 1).618 C A CSTR Æ PFR X =.618 C A.382 kmol/m 3 as he concenraion of reacan in he effluen from he reacion sysem. Thus, he conversion is 61.8% i.e., X ([1.382]/1).618. When he perfecly mixed secion is preceded by he plug-flow secion (Figure E13-3.2) he oule of he PFR is he inle o he CSTR, C Ai : p k C Ai C A ( 1) ( 1) 1 C Ai (E13-3.6) C Ai.5 kmol/m 3 and a maerial balance on he perfecly mixed secion (CSTR) gives 26 Pearson Educaion, Inc.

35 Sec Diagnosics and Troubleshooing 91 C Ai C A C A Figure E Lae mixing scheme. PFR Æ CSTR X =.634 Early Mixing X =.618 Lae Mixing X = s k C A C A C Ai (E13-3.7) 14 C A s kc Ai s k kmol/m 2 3 (E13-3.8) as he concenraion of reacan in he effluen from he reacion sysem. The corresponding conversion is 63.4%. Tha is X 1 ( C A C A ) % 1. In he firs configuraion, a conversion of 61.8% was obained; in he second, 63.4%. While he difference in he conversions is small for he parameer values chosen, he poin is ha here is a difference. While E() was he same for boh reacion sysems, he conversion was no. The Quesion The conclusion from his example is of exreme imporance in reacor analysis: The RTD is no a complee descripion of srucure for a paricular reacor or sysem of reacors. The RTD is unique for a paricular reacor. However, he reacor or reacion sysem is no unique for a paricular RTD. When analyzing nonideal reacors, he RTD alone is no sufficien o deermine is performance, and more informaion is needed. I will be shown ha in addiion o he RTD, an adequae model of he nonideal reacor flow paern and knowledge of he qualiy of mixing or degree of segregaion are boh required o characerize a reacor properly. There are many siuaions where he fluid in a reacor neiher is well mixed nor approximaes plug flow. The idea is his: We have seen ha he RTD can be used o diagnose or inerpre he ype of mixing, bypassing, ec., ha occurs in an exising reacor ha is currenly on sream and is no yielding he conversion prediced by he ideal reacor models. Now le's envision anoher use of he RTD. Suppose we have a nonideal reacor eiher on line or siing in sorage. We have characerized his reacor and obained he RTD funcion. Wha will be he conversion of a reacion wih a known rae law ha is carried ou in a reacor wih a known RTD? How can we use he RTD o predic conversion in a real reacor? In Par 2 we show how his quesion can be answered in a number of ways. 26 Pearson Educaion, Inc.

36 92 Disribuions of Residence Times for Chemical Reacors Chap. 13 Par 2 Predicing Conversion and Exi Concenraion The Answer 13.6 Reacor Modeling Using he RTD Now ha we have characerized our reacor and have gone o he lab o ake daa o deermine he reacion kineics, we need o choose a model o predic conversion in our real reacor. RTD + MODEL + KINETIC DATA Ï EXIT CONVERSION and fi Ì ÓEXIT CONCENTRATION We now presen he five models shown in Table We shall classify each model according o he number of adjusable parameers. We will discuss he firs wo in his chaper and he oher hree in Chaper 14. TABLE MODELS FOR PREDICTING CONVERSION FROM RTD DATA Ways we use he RTD daa o predic conversion in nonideal reacors 1. Zero adjusable parameers a. Segregaion model b. Maximum mixedness model 2. One adjusable parameer a. Tanks-in-series model b. Dispersion model 3. Two adjusable parameers Real reacors modeled as combinaions of ideal reacors The RTD ells us how long he various fluid elemens have been in he reacor, bu i does no ell us anyhing abou he exchange of maer beween he fluid elemens (i.e., he mixing). The mixing of reacing species is one of he major facors conrolling he behavior of chemical reacors. Forunaely for firs-order reacions, knowledge of he lengh of ime each molecule spends in he reacor is all ha is needed o predic conversion. For firs-order reacions he conversion is independen of concenraion (recall Equaion E9-1.3): dx d k(1 X) (E9-1.3) Consequenly, mixing wih he surrounding molecules is no imporan. Therefore, once he RTD is deermined, we can predic he conversion ha will be achieved in he real reacor provided ha he specific reacion rae for he firs-order reacion is known. However, for reacions oher han firs order, knowledge of he RTD is no sufficien o predic conversion. In hese cases he degree of mixing of molecules mus be known in addiion o how long each molecule spends in he reacor. Consequenly, we mus develop models ha accoun for he mixing of molecules inside he reacor. 26 Pearson Educaion, Inc.

37 Sec Reacor Modeling Using he RTD 93 The more complex models of nonideal reacors necessary o describe reacions oher han firs order mus conain informaion abou micromixing in addiion o ha of macromixing. Macromixing produces a disribuion of residence imes wihou, however, specifying how molecules of differen ages encouner one anoher in he reacor. Micromixing, on he oher hand, describes how molecules of differen ages encouner one anoher in he reacor. There are wo exremes of micromixing: (1) all molecules of he same age group remain ogeher as hey ravel hrough he reacor and are no mixed wih any oher age unil hey exi he reacor (i.e., complee segregaion); (2) molecules of differen age groups are compleely mixed a he molecular level as soon as hey ener he reacor (complee micromixing). For a given sae of macromixing (i.e., a given RTD), hese wo exremes of micromixing will give he upper and lower limis on conversion in a nonideal reacor. For reacion orders greaer han one or less han zero, he segregaion model will predic he highes conversion. For reacion orders beween zero and one, he maximum mixedness model will predic he highes conversion. This concep is discussed furher in Secion We shall define a globule as a fluid paricle conaining millions of molecules all of he same age. A fluid in which he globules of a given age do no mix wih oher globules is called a macrofluid. A macrofluid could be visualized as a noncoalescen globules where all he molecules in a given globule have he same age. A fluid in which molecules are no consrained o remain in he globule and are free o move everywhere is called a microfluid. 9 There are wo exremes of mixing of he macrofluid globules o form a microfluid we shall sudy early mixing and lae mixing. These wo exremes of lae and early mixing are shown in Figure (a) and (b), respecively. These exremes can also be seen by comparing Figures (a) and (a). The exremes of lae and early mixing are referred o as complee segregaion and maximum mixedness, respecively. C A, X C A, X Figure (a) Macrofluid; and (b) microfluid mixing on he molecular level. 9 J. Villermaux, Chemical Reacor Design and Technology (Boson: Marinus Nijhoff, 1986). 26 Pearson Educaion, Inc.

38 94 Disribuions of Residence Times for Chemical Reacors Chap Zero-Parameer Models Segregaion Model In a perfecly mixed CSTR, he enering fluid is assumed o be disribued immediaely and evenly hroughou he reacing mixure. This mixing is assumed o ake place even on he microscale, and elemens of differen ages mix ogeher horoughly o form a compleely micromixed fluid. If fluid elemens of differen ages do no mix ogeher a all, he elemens remain segregaed from each oher, and he fluid is ermed compleely segregaed. The exremes of complee micromixing and complee segregaion are he limis of he micromixing of a reacing mixure. In developing he segregaed mixing model, we firs consider a CSTR because he applicaion of he conceps of mixing qualiy are illusraed mos easily using his reacor ype. In he segregaed flow model we visualize he flow hrough he reacor o consis of a coninuous series of globules (Figure 13-22). In he segregaion model globules behave as bach reacors operaed for differen imes Figure Lile bach reacors (globules) inside a CSTR. The segregaion model has mixing a he laes possible poin. These globules reain heir ideniy; ha is, hey do no inerchange maerial wih oher globules in he fluid during heir period of residence in he reacion environmen, i.e., hey remain segregaed. In addiion, each globule spends a differen amoun of ime in he reacor. In essence, wha we are doing is lumping all he molecules ha have exacly he same residence ime in he reacor ino he same globule. The principles of reacor performance in he presence of compleely segregaed mixing were firs described by Danckwers 1 and Zwieering. 11 Anoher way of looking a he segregaion model for a coninuous flow sysem is he PFR shown in Figures 13-23(a) and (b). Because he fluid flows down he reacor in plug flow, each exi sream corresponds o a specific residence ime in he reacor. Baches of molecules are removed from he reacor a differen locaions along he reacor in such a manner as o duplicae he RTD funcion, E(). The molecules removed near he enrance o he reacor correspond o hose molecules having shor residence imes in he reacor. 1 P. V. Danckwers, Chem. Eng. Sci., 8, 93 (1958). 11 T. N. Zwieering, Chem. Eng. Sci., 11, 1 (1959). 26 Pearson Educaion, Inc.

39 Sec Zero-Parameer Models 95 v v (a) E() maches he removal of he bach reacors E() (b) Figure Mixing a he laes possible poin. Lile bach reacors Physically, his effluen would correspond o he molecules ha channel rapidly hrough he reacor. The farher he molecules ravel along he reacor before being removed, he longer heir residence ime. The poins a which he various groups or baches of molecules are removed correspond o he RTD funcion for he reacor. Because here is no molecular inerchange beween globules, each acs essenially as is own bach reacor. The reacion ime in any one of hese iny bach reacors is equal o he ime ha he paricular globule spends in he reacion environmen. The disribuion of residence imes among he globules is given by he RTD of he paricular reacor. RTD + MODEL + KINETIC DATA Ï EXIT CONVERSION and fi Ì ÓEXIT CONCENTRATION To deermine he mean conversion in he effluen sream, we mus average he conversions of all of he various globules in he exi sream: Mean conversion of hose globules spending beween ime and d in he reacor Conversion achieved in a globule afer spending a ime in he reacor Fracion of globules ha spend beween and d in he reacor hen dx X() E() d 26 Pearson Educaion, Inc.

40 96 Disribuions of Residence Times for Chemical Reacors Chap. 13 dx X ( )E ( ) d (13-52) Mean conversion for he segregaion model Summing over all globules, he mean conversion is X X( )E ( ) d (13-53) Consequenly, if we have he bach reacor equaion for X() and measure he RTD experimenally, we can find he mean conversion in he exi sream. Thus, if we have he RTD, he reacion rae expression, hen for a segregaed flow siuaion (i.e., model), we have sufficien informaion o calculae he conversion. An example ha may help give addiional physical insigh o he segregaion model is given in he Summary Noes on he CD-ROM, click he More back buon jus before secion 4A.2. Consider he following firs-order reacion: For a bach reacor we have A k æææ producs dn A r A V d For consan volume and wih N A N A (1 X), dx N A r A V kc A V kn A kn A (1 X) d dx k ( 1 X) d (13-54) Solving for X(), we have Mean conversion for a firs-order reacion X X( )E ( ) d X() 1 e k (1 e k )E() d E() d e k E() d (13-55) X 1 e k E() d (13-56) We will now deermine he mean conversion prediced by he segregaion model for an ideal PFR, a CSTR, and a laminar flow reacor. 26 Pearson Educaion, Inc.

41 Sec Zero-Parameer Models 97 Example 13 4 Mean Conversion in an Ideal PFR, an Ideal CSTR, and a Laminar Flow Reacor Derive he equaion of a firs-order reacion using he segregaion model when he RTD is equivalen o (a) an ideal PFR, (b) an ideal CSTR, and (c) a laminar flow reacor. Compare hese conversions wih hose obained from he design equaion. Soluion (a) For he PFR, he RTD funcion was given by Equaion (13-32) Recalling Equaion (13-55) E() ( ) (13-32) X X()E() d 1 e k E() d (13-55) Subsiuing for he RTD funcion for a PFR gives X 1 (e k )( ) d (E13-4.1) Using he inegral properies of he Dirac dela funcion, Equaion (13-35) we obain X 1 e k 1e Da (E13-4.2) where for a firs-order reacion he Damköhler number is Da = k. Recall ha for a PFR afer combining he mole balance, rae law, and soichiomeric relaionships (cf. Chaper 4), we had dx d k(1 X) (E13-4.3) Inegraing yields X 1 e k = 1 e Da (E13-4.4) which is idenical o he conversion prediced by he segregaion model X. (b) For he CSTR, he RTD funcion is 1 E() -- e / (13-27) Recalling Equaion (13-56), he mean conversion for a firs-order reacion is X 1 e k E() d (13-56) X 1 e ( 1 k ) d X 1 1 k e k1 ( ) 26 Pearson Educaion, Inc.

42 98 Disribuions of Residence Times for Chemical Reacors Chap. 13 As expeced, using he E() for an ideal PFR and CSTR wih he segregaion model gives a mean conversion X idenical o ha obained by using he algorihm in Ch. 4. X k (E13-4.5) 1 k Da Da Combining he CSTR mole balance, he rae law, and soichiomery, we have F A X r A V v C A X kc A (1 X)V k X (E13-4.6) 1 k which is idenical o he conversion prediced by he segregaion model X. (c) For a laminar flow reacor he RTD funcion is Ï for ( /2) Ô Ô E ( ) Ì 2 Ô for ( /2) Ó2 3 Ô (13-47) The dimensionless form is From Equaion (13-15), we have Inegraing wice by pars Ï for.5 Ô Ô E( ) Ì Ô for.5 Ó2 3 Ô X 1 e k E ( ) d 1 e k E( ) d X 1.5 e k d 2 3 X 1 ( 1.5k)e.5k (.5k) 2 e k d.5 (13-49) (E13-4.7) (E13-4.8) (E13-4.9) The las inegral is he exponenial inegral and can be evaluaed from abulaed values. Forunaely, Hilder 12 developed an approximae formula (k = Da). 12 M. H. Hilder, Trans. I. ChemE, 59, 143 (1979). 26 Pearson Educaion, Inc.

43 Sec Zero-Parameer Models 99 1 X ( 1.25k)e.5k.25k ( 1.25 Da) e.5da.25 Da X ( 4Da)e.5Da Da ( 4Da)e.5Da (E13-4.1) Da A comparison of he exac value along wih Hilder s approximaion is shown in Table E for various values of he Damköhler number, k, along wih he conversion in an ideal PFR and an ideal CSTR. TABLE E COMPARISON OF CONVERSION IN PFR, CSTR, AND LAMINAR FLOW REACTOR FOR DIFFERENT DAMKÖHLER NUMBERS FOR A FIRST-ORDER REACTION Da = k X L.F. Exac X L.F. Approx. X PFR X CSTR where X L.F. Exac = exac soluion o Equaion (E13-4.9) and X L.F. Approx. = Equaion (E13-4.1). For large values of he Damköhler number hen, here is complee conversion along he sreamlines off he cener sreamline so ha he conversion is deermined along he pipe axis such ha X 1 4e k d 14e.5k /k.5 (E ) Figure E shows a comparison of he mean conversion in an LFR, PFR, and CSTR as a funcion of he Damköhler number for a firs-order reacion. X PFR LFR CSTR Figure E Conversion in a PFR, LFR, and CSTR as a funcion of he Damköhler number (Da) for a firs-order reacion (Da = k). Da 26 Pearson Educaion, Inc.

44 91 Disribuions of Residence Times for Chemical Reacors Chap. 13 Imporan Poin: For a firs-order reacion, knowledge of E() is sufficien. We have jus shown for a firs-order reacion ha wheher you assume complee micromixing [Equaion (E13-4.6)] or complee segregaion [Equaion (E13-4.5)] in a CSTR, he same conversion resuls. This phenomenon occurs because he rae of change of conversion for a firs-order reacion does no depend on he concenraion of he reacing molecules [Equaion (13-54)]; i does no maer wha kind of molecule is nex o i or colliding wih i. Thus he exen of micromixing does no affec a firs-order reacion, so he segregaed flow model can be used o calculae he conversion. As a resul, only he RTD is necessary o calculae he conversion for a firs-order reacion in any ype of reacor (see Problem P13-3 c ). Knowledge of neiher he degree of micromixing nor he reacor flow paern is necessary. We now proceed o calculae conversion in a real reacor using RTD daa. Example 13 5 Mean Conversion Calculaions in a Real Reacor Calculae he mean conversion in he reacor we have characerized by RTD measuremens in Examples 13-1 and 13-2 for a firs-order, liquid-phase, irreversible reacion in a compleely segregaed fluid: A æææ producs The specific reacion rae is.1 min 1 a 32 K. These calculaions are easily carried ou wih he aid of a spreadshee such as Excel or Polymah. Soluion Because each globule acs as a bach reacor of consan volume, we use he bach reacor design equaion o arrive a he equaion giving conversion as a funcion of ime: X 1 e k 1 e.1 To calculae he mean conversion we need o evaluae he inegral: X (E13-5.1) X()E() d (13-53) The RTD funcion for his reacor was deermined previously and given in Table E and is repeaed in Table E To evaluae he inegral we make a plo of X()E() as a funcion of as shown in Figure E and deermine he area under he curve. 26 Pearson Educaion, Inc.

45 Sec Zero-Parameer Models For a given RTD, he segregaion model gives he upper bound on conversion for reacion orders less han zero or greaer han 1. X () E (), min Area = X (min) Figure E Plo of columns 1 and 4 from he daa in Table E Using he quadaure formulas in Appendix A.4 The mean conversion is 38.5%. Polymah or Excel will easily give X afer seing up columns 1 and 4 in Table E The area under he curve in Figure E is he mean conversion X. As discussed previously, because he reacion is firs order, he conversion calculaed in Example 13-5 would be valid for a reacor wih complee TABLE E PROCESSED DATA TO FIND THE MEAN CONVERSION (min) E() (min 1 ) X() X()E() (min 1 ) X ( )E ( ) d 14 X ( )E ( ) d X ( )E ( ) d [4(.19) 2(.18) 4(.414) 2(.66) 3 4(.629) 2(.541) 4(.42) 2(.331) 2 4(.261).1896] -- [.18964(.84) ] (.35) (.35).385 X area Pearson Educaion, Inc. X

46 912 Disribuions of Residence Times for Chemical Reacors Chap. 13 mixing, complee segregaion, or any degree of mixing beween he wo. Alhough early or lae mixing does no affec a firs-order reacion, micromixing or complee segregaion can modify he resuls of a second-order sysem significanly. Example 13 6 Mean Conversion for a Second-Order Reacion in a Laminar Flow Reacor The liquid-phase reacion beween cyidine and aceic anhydride NH 2 NHAc OH N O N O + (CH 3 CO) 2 O NMP OH N O N O + CH 3 COOH OH OH cyidine aceic anhydride OH OH (A) (B) (C) (D) A + B Æ C + D is carried ou isohermally in an iner soluion of N-mehyl-2-pyrrolidone (NMP) wih Q NMP = The reacion follows an elemenary rae law. The feed is equal molar in A and B wih C A =.75 mol/dm 3, a volumeric flow rae of.1 dm 3 /s and a reacor volume of 1 dm 3. Calculae he conversion in (a) a PFR, (b) a bach reacor, and (c) a laminar flow reacor. Addiional informaion: 13 k = dm 3 /mol s a 5 C wih E = 13.3 kcal/mol, DH RX = 1.5 kcal/mol F Hea of mixing for Q NMP = NMP 28.9, DH mix =.44 kcal/mol F A Soluion The reacion will be carried ou isohermally a 5 C. The space ime is (a) For a PFR Mole Balance V dm s v.1 dm 3 s dx r A dv F A (E13-6.1) 13 J. J. Shaynski and D. Hanesian, Ind. Eng. Chem. Res., 32, 594 (1993). 26 Pearson Educaion, Inc.

47 Sec Zero-Parameer Models 913 PFR Calculaion Rae Law r A = kc A C B (E13-6.2) Soichiomery, Q B = 1 C A = C A (1 X) (E13-6.3) C B = C A (E13-6.4) Combining dx kc A ( 1X) dv Solving wih = V/v and X = for V = gives v (E13-6.5) kc X A Da kc A 1Da 2 (E13-6.6) where Da 2 is he Damköhler number for a second-order reacion. Da 2 kc A ( 1s) ( dm 3 /smol)(.75 mol/dm 3 ) = 3.7 (b) Bach Reacor X X =.787 Bach Calculaion dx d r A C A dx kc A ( 1X) 2 d kc X ( ) A kc A (E13-6.7) (E13-6.8) (E13-6.9) If he bach reacion ime is he same ime as he space ime he bach conversion is he same as he PFR conversion X =.787. (c) Laminar Flow Reacor The differenial form for he mean conversion is obained from Equaion (13-52) dx X ( ) E ( ) (13-52) d We use Equaion (E13-6.9) o subsiue for X() in Equaion (13-52). Because E() for he LFR consiss of wo pars, we need o incorporae he IF saemen in our ODE solver program. For he laminar flow reacion, we wrie E 1 = for < /2 (E13-6.1) 26 Pearson Educaion, Inc.

48 914 Disribuions of Residence Times for Chemical Reacors Chap. 13 LFR Calculaion E for /2 2 3 Le 1 = /2 so ha he IF saemen now becomes E = If ( < 1 ) hen (E 1 ) else (E 2 ) (E ) (E ) One oher hing is ha he ODE solver will recognize ha E 2 = a = and refuse o run. So we mus add a very small number o he denominaor such as (.1); for example, E ( 2 3.1) (E ) The inegraion ime should be carried ou o 1 or more imes he reacor space ime. The Polymah Program for his example is shown below. We see ha he mean conversion Xbar ( X ) for he LFR is 74.1%. In summary, X PFR =.786 X LFR =.741 Compare his resul wih he exac analyical formula 14 for he laminar flow reacor wih a second-order reacion Analyical Soluion X = Da[1 (Da/2) ln(1+2/da)] where Da = kc A. For Da = 3.7 we ge X = K. G. Denbigh, J. Appl. Chem., 1, 227 (1951). 26 Pearson Educaion, Inc.

49 Sec Zero-Parameer Models Maximum Mixedness Model Segregaion model mixing occurs a he laes possible poin. In a reacor wih a segregaed fluid, mixing beween paricles of fluid does no occur unil he fluid leaves he reacor. The reacor exi is, of course, he laes possible poin ha mixing can occur, and any effec of mixing is posponed unil afer all reacion has aken place as shown in Figure We can also hink of compleely segregaed flow as being in a sae of minimum mixedness. We now wan o consider he oher exreme, ha of maximum mixedness consisen wih a given residence-ime disribuion. We reurn again o he plug-flow reacor wih side enrances, only his ime he fluid eners he reacor along is lengh (Figure 13-24). As soon as he fluid eners he reacor, i is compleely mixed radially (bu no longiudinally) wih he oher fluid already in he reacor. The enering fluid is fed ino he reacor hrough he side enrances in such a manner ha he RTD of he plug-flow reacor wih side enrances is idenical o he RTD of he real reacor. v v Maximum mixedness: mixing occurs a he earlies possible poin (a) E() Figure (b) Mixing a he earlies possible poin. The globules a he far lef of Figure correspond o he molecules ha spend a long ime in he reacor while hose a he far righ correspond o he molecules ha channel hrough he reacor. In he reacor wih side enrances, mixing occurs a he earlies possible momen consisen wih he RTD. Thus he effec of mixing occurs as early as possible hroughou he reacor, and his siuaion is ermed he condiion of maximum mixedness. 15 The approach o calculaing conversion for a reacor in a condiion of maximum mixedness will now be developed. In a reacor wih side enrances, le be he ime i akes for he fluid o move from a paricular poin o he end of he reacor. In oher words, is he life expecancy of he fluid in he reacor a ha poin (Figure 13-25). 15 T. N. Zwieering, Chem. Eng. Sci., 11, 1 (1959). 26 Pearson Educaion, Inc.

50 916 Disribuions of Residence Times for Chemical Reacors Chap. 13 v v v v v V = V = V Figure enrances. Modeling maximum mixedness by a plug-flow reacor wih side Moving down he reacor from lef o righ, decreases and becomes zero a he exi. A he lef end of he reacor, approaches infiniy or he maximum residence ime if i is oher han infinie. Consider he fluid ha eners he reacor hrough he sides of volume DV in Figure The fluid ha eners here will have a life expecancy beween l and l+dl. The fracion of fluid ha will have his life expecancy beween l and l+dl is E(l)Dl. The corresponding volumeric flow rae IN hrough he sides is [v E(l)Dl]. v v v The volumeric flow rae a l, v l, is he flow rae ha enered a l+dl, v l+dl plus wha enered hrough he sides v E(l)Dl, i.e., v v v E( ) Rearranging and aking he limi as Dl Æ dv v d E( ) (13-57) 26 Pearson Educaion, Inc.

51 Sec Zero-Parameer Models 917 The volumeric flow rae v a he enrance o he reacor (X = ) is zero because he fluid only eners hrough he sides along he lengh. Inegraing equaion (13-57) wih limis v l = a l = and v l = v l a l = l, we obain v v E( ) d v [ 1 F( ) ] The volume of fluid wih a life expecancy beween and is V v [ 1 F ( ) ] (13-58) (13-59) The rae of generaion of he subsance A in his volume is r A V r A v [ 1 F ( ) ] (13-6) We can now carry ou a mole balance on subsance A beween and : Mole balance In a In hrough side Ou a Generaion by reacion v [1 F()]C A v C A E() v [1 F()]C A r A v [1 F()] (13-61) Dividing Equaion (13-61) by v and aking he limi as Æ gives d {[ 1 F ( ) ]C E()C A A ( ) } r A [1 F()] d Taking he derivaive of he erm in brackes or dc C A E() [1 F()] A C A E() r A [1 F()] d dc A r d A ( C A C A ) E ( ) F ( ) (13-62) or We can rewrie Equaion (13-62) in erms of conversion as dx C A r A C A X E ( ) (13-63) d 1 F ( ) r A dx E ( ) d ( F ( ) X ) C A (13-64) 26 Pearson Educaion, Inc.

52 918 Disribuions of Residence Times for Chemical Reacors Chap. 13 MM gives he lower bound on X. The boundary condiion is as Æ, hen C A C A for Equaion (13-62) [or X for Equaion (13-64)]. To obain a soluion, he equaion is inegraed backwards numerically, saring a a very large value of and ending wih he final conversion a. For a given RTD and reacion orders greaer han one, he maximum mixedness model gives he lower bound on conversion. Example 13 7 Conversion Bounds for a Nonideal Reacor The liquid-phase, second-order dimerizaion 2 2A æææ B r A kc A for which k.1 dm 3 /molmin is carried ou a a reacion emperaure of 32 K. The feed is pure A wih C A 8 mol/dm 3. The reacor is nonideal and perhaps could be modeled as wo CSTRs wih inerchange. The reacor volume is 1 dm 3, and he feed rae for our dimerizaion is going o be 25 dm 3 /min. We have run a racer es on his reacor, and he resuls are given in columns 1 and 2 of Table E We wish o know he bounds on he conversion for differen possible degrees of micromixing for he RTD of his reacor. Wha are hese bounds? Tracer es on ank reacor: N 1 g, v 25 dm 3 /min. TABLE E RAW AND PROCESSED DATA (min) C (mg/dm 3 ) E() (min 1 ) 1 F() E()/[1 F()] (min 1 ) l (min) Columns 3 hrough 5 are calculaed from columns 1 and Soluion The bounds on he conversion are found by calculaing conversions under condiions of complee segregaion and maximum mixedness. Conversion if fluid is compleely segregaed. The bach reacor equaion for a second-order reacion of his ype is kc X A kc A 26 Pearson Educaion, Inc.

53 Sec Zero-Parameer Models 919 The conversion for a compleely segregaed fluid in a reacor is Spreadshees work quie well here. X X()E() d The calculaions for his inegraion are carried ou in Table E The numerical inegraion uses he simple rapezoid rule. The conversion for his sysem if he fluid were compleely segregaed is.61 or 61%. TABLE E SEGREGATION MODEL (min) X() X()E() (min 1 ) X()E().1 X.E.5 Area = X = (min) For he firs poin we have X()E()D = ( +.686) (5/2) =.172 Conversion for maximum mixedness. The Euler mehod will be used for numerical inegraion: X i1 X i () E ( i ) 1F ( i ) X kc ( 1 X ) 2 i A i Inegraing his equaion presens some ineresing resuls. If he equaion is inegraed from he exi side of he reacor, saring wih, he soluion is unsable and soon approaches large negaive or posiive values, depending on wha he saring value of X is. We wan o find he conversion a he exi o he reacor l =. Consequenly, we need o inegrae backwards. X i E ( i )X i F ( i ) k C A ( 1 X i ) 2 X i 1 If inegraed from he poin where ææ, oscillaions may occur bu are damped ou, and he equaion approaches he same final value no maer wha iniial value of X beween and 1 is used. We shall sar he inegraion a 2 and le X a his poin. If we se oo large, he soluion will blow up, so we will sar ou wih 25 and use he average of he measured values of E()/[(1 F()] where necessary. We will now use he daa in column 5 of Table E o carry ou he inegraion. 26 Pearson Educaion, Inc.

54 92 Disribuions of Residence Times for Chemical Reacors Chap. 13 A l = 2, X = 175: X( = 175) X( = 2) E ( 2 )X ( 2 ) 1F( 2) kc ( 1 X ( 2 )) 2 A X (25)[(.75)() ((.1)(8)(1)) 2 ] 2 15: X( = 15) X( = 175) E ( 175 )X ( ) 1F( 175) kc ( 1 X ( 175 )) 2 A We need o ake an average of E / (1 F) beween l = 2 and l = 15. X ( = 15) 2 (25) Ê ˆ Á ( 2) (.1) ( 8) ( 12) Ë 2 125: X ( = 125) 1.46 (25)[(.266)(1.46) (.1)(8)(1 1.46) 2 ].912 1: X ( = 1).912 (25) Ê ˆ Á (.912) (.1) ( 8) ( 1.912) 2 Ë 2 Summary PFR 76% Segregaion 61% CSTR 58% Max. mix 56% 7: 5: 4:.372 X.372 (3)[(.221)(.372) (.1)(8)(1.372) 2 ] 1.71 X 1.71 (2)[(.226)(1.71) (.1)(8)(1 1.71) 2 ].595 X.595 (1)[(.237)(.595) (.1)(8)(1.595) 2 ].585 Running down he values of X along he righ-hand side of he preceding equaion shows ha he oscillaions have now damped ou. Carrying ou he remaining calculaions down o he end of he reacor complees Table E The conversion for a condiion of maximum mixedness in his reacor is.56 or 56%. I is ineresing o noe ha here is lile difference in he conversions for he wo condiions of complee segregaion (61%) and maximum mixedness (56%). Wih bounds his narrow, here may no be much poin in modeling he reacor o improve he predicabiliy of conversion. For comparison i is lef for he reader o show ha he conversion for a PFR of his size would be.76, and he conversion in a perfecly mixed CSTR wih complee micromixing would be Pearson Educaion, Inc.

55 Sec Zero-Parameer Models 921 TABLE E MAXIMUM MIXEDNESS MODEL (min) X Calculae backwards o reacor exi The Inensiy Funcion, L() can be hough of as he probabiliy of a paricle escaping he sysem beween a ime and ( + d) provided he paricle is sill in he sysem. Equaions (13-62) and (13-64) can be wrien in a slighly more compac form by making use of he inensiy funcion. 16 The inensiy funcion ( ) is he fracion of fluid in he vessel wih age ha will leave beween and d. We can relae () o I() and E() in he following manner: Volume of fluid leaving beween imes and d Volume of fluid remaining a ime Fracion of he fluid wih age ha will leave beween ime and d v [ E() d] [V I()][() d] (13-65) Then ( ) E ( ) d ln [ I ( ) ] E ( ) I ( ) d F ( ) Combining Equaions (13-64) and (13-66) gives (13-66) dx r A ( ) ()X() (13-67) d C A We also noe ha he exi age,, is jus he sum of he inernal age,, and he life expecancy, : (13-68) 16 D. M. Himmelblau and K. B. Bischoff, Process Analysis and Simulaion (New York: Wiley, 1968). 26 Pearson Educaion, Inc.

56 922 Disribuions of Residence Times for Chemical Reacors Chap. 13 In addiion o defining maximum mixedness discussed above, Zwieering 17 also generalized a measure of micromixing proposed by Danckwers 18 and defined he degree of segregaion, J, as variance of ages beween fluid poins J variance of ages of all molecules in sysem A fluid poin conains many molecules bu is small compared o he scale of mixing. The wo exremes of he degree of segregaion are J 1: complee segregaion J : maximum mixedness Equaions for he variance and J for he inermediae cases can be found in Zwieering Comparing Segregaion and Maximum Mixedness Predicions In he previous example we saw ha he conversion prediced by he segregaion model, X seg, was greaer han ha by he maximum mixedness model X max. Will his always be he case? No. To learn he answer we ake he second derivaive of he rae law as shown in he Professional Reference Shelf R13.3 on he CD-ROM. 2 ( r A ) If hen X seg > X mm Comparing 2 ( r A ) X seg and X mm If hen X mm > X seg 2 C A 2 C A 2 ( r A ) If hen X mm = X seg C A 2 For example, if he rae law is a power law model r A kc A n From he produc [(n)(n 1)], we see ( r If n > 1, hen A ) and X seg > X mm ( r If n <, hen A ) and X seg > X mm ( r A ) n nkc A C A 2 ( r A ) n nn ( 1)kC A 2 2 C A 2 C A 2 C A 2 17 T. N. Zwieering, Chem. Eng. Sci., 11, 1 (1959). 18 P. V. Danckwers, Chem. Eng. Sci., 8, 93 (1958). 26 Pearson Educaion, Inc.

57 Sec Using Sofware Packages 923 ( r If < n < 1, hen A ) and X mm > X seg 2 C A 2 Imporan poin We noe ha in some cases X seg is no oo differen from X mm. However, when one is considering he desrucion of oxic wase where X >.99 is desired, hen even a small difference is significan!! In his secion we have addressed he case where all we have is he RTD and no oher knowledge abou he flow paern exiss. Perhaps he flow paern canno be assumed because of a lack of informaion or oher possible causes. Perhaps we wish o know he exen of possible error from assuming an incorrec flow paern. We have shown how o obain he conversion, using only he RTD, for wo limiing mixing siuaions: he earlies possible mixing consisen wih he RTD, or maximum mixedness, and mixing only a he reacor exi, or complee segregaion. Calculaing conversions for hese wo cases gives bounds on he conversions ha migh be expeced for differen flow pahs consisen wih he observed RTD Using Sofware Packages Example 13-7 could have been solved wih an ODE solver afer fiing E() o a polynomial. Fiing he E() Curve o a Polynomial Some forms of he equaion for he conversion as a funcion of ime muliplied by E() will no be easily inegraed analyically. Consequenly, i may be easies o use ODE sofware packages. The procedure is sraighforward. We recall Equaion (13-52) dx X ( ) E ( ) (13-52) d where X is he mean conversion and X() is he bach reacor conversion a ime. The mean conversion X is found by inegraing beween = and = or a very large ime. Nex we obain he mole balance on X() from a bach reacor dx d and would wrie he rae law in erms of conversion, e.g., r A The ODE solver will combine hese equaions o obain X() which will be used in Equaion (13-52). Finally we have o specify E(). This equaion can be an analyical funcion such as hose for an ideal CSTR, r A kc A C A ( 1 X) 2 E ( ) e / Pearson Educaion, Inc.

58 924 Disribuions of Residence Times for Chemical Reacors Chap. 13 or i can be polynomial or a combinaion of polynomials ha have been used o fi he experimenal RTD daa E ( ) a a 1 a 2 2 (13-69) or F ( ) b b 1 b 2 2 (13-7) We now simply combine Equaions (13-52), (13-69), and (13-7) and use an ODE solver. There are hree cauions one mus be aware of when fiing E() o a polynomial. Firs, you use one polynomial E 1 () as E() increases wih ime o he op of he curve shown in Figure A second polynomial E 2 () is used from he op as E() decreases wih ime. One needs o mach he wo curves a he op. E Mach E 1 E 2 Figure Maching E 1 () and E 2 (). Polymah Tuorial Second, one should be cerain ha he polynomial used for E 2 () does no become negaive when exrapolaed o long imes. If i does, hen consrains mus be placed on he fi using IF saemens in he fiing program. Finally, one should check ha he area under he E() curve is virually one and ha he cumulaive disribuion F() a long imes is never greaer han 1. A uorial on how o fi he C() and E() daa o a polynomial is given in he Summary Noes for Chaper 5 on he CD-ROM and on he web. Segregaion Model Here we simply use he coupled se of differenial equaions for he mean or exi conversion, X, and he conversion X() inside a globule a any ime,. dx X ( )E ( ) d (13-52) dx r A d C A (13-71) The rae of reacion is expressed as a funcion of conversion: for example, 26 Pearson Educaion, Inc.

59 Sec Using Sofware Packages 925 r A and he equaions are hen solved numerically wih an ODE solver. Maximum Mixedness Model 2 k A C A ( 1 X) 2 Because mos sofware packages won inegrae backwards, we need o change he variable such ha he inegraion proceeds forward as decreases from some large value o zero. We do his by forming a new variable, z, which is he difference beween he longes ime measured in he E() curve, T, and. In he case of Example 13-7, he longes ime a which he racer concenraion was measured was 2 minues (Table E13-7.1). Therefore we will se T 2. Then, z T T 2 z 2 z dx dz r A C A ET ( z) X 1 FT ( z) (13-72) One now inegraes beween he limi z and z 2 o find he exi conversion a z 2 which corresponds o. In fiing E() o a polynomial, one has o make sure ha he polynomial does no become negaive a large imes. Anoher concern in he maximum mixedness calculaions is ha he erm 1 F ( ) does no go o zero. Seing he maximum value of F() a.999 raher han 1. will eliminae his problem. I can also be circumvened by inegraing he polynomial for E() o ge F() and hen seing he maximum value of F() a.999. If F() is ever greaer han one when fiing a polynomial, he soluion will blow up when inegraing Equaion (13-72) numerically. Example 13 8 Using Sofware o Make Maximum Mixedness Model Calculaions Use an ODE solver o deermine he conversion prediced by he maximum mixedness model for he E() curve given in Example E13-7. Soluion Because of he naure of he E() curve, i is necessary o use wo polynomials, a hird order and a fourh order, each for a differen par of he curve o express he RTD, E(), as a funcion of ime. The resuling E() curve is shown in Figure E To use Polymah o carry ou he inegraion, we change our variable from o z using he larges ime measuremens ha were aken from E() in Table E13-7.1, which is 2 min: 26 Pearson Educaion, Inc.

60 926 Disribuions of Residence Times for Chemical Reacors Chap. 13 Firs, we fi E(). Figure E Polynomial fi of E(). z 2 The equaions o be solved are Maximum mixedness model 2 z r A dx E ( 2z) dz F ( 2z) X C A (E13-8.1) (E13-8.2) For values of less han 7, we use he polynomial E 1 ()4.447e e e e 4.28 For values of greaer han 7, we use he polynomial E 2 () 2.64e e e 4.15 df E ( ) d (E13-8.3) (E13-8.4) (E13-8.5) wih z ( 2), X, F 1 [i.e., F().999]. Cauion: Because [ 1 F( ) ] 1 ends o infiniy a F 1, (z ), we se he maximum value of F a.999 a z. The Polymah equaions are shown in Table E The soluion is a z 2 X.563 The conversion prediced by he maximum mixedness model is 56.3%. 26 Pearson Educaion, Inc.

61 Sec RTD and Muliple Reacions 927 TABLE E POLYMATH PROGRAM FOR MAXIMUM MIXEDNESS MODEL Polynomials used o fi E() and F() Hea Effecs If racer ess are carried ou isohermally and hen used o predic nonisohermal condiions, one mus couple he segregaion and maximum mixedness models wih he energy balance o accoun for variaions in he specific reacion rae. This approach will only be valid for liquid phase reacions because he volumeric flow rae remains consan. For adiabaic operaion and C P, ( H T T Rx ) X (T8-1.B) Â i C Pi As before, he specific reacion rae is (T8-2.3) Assuming ha E() is unaffeced by emperaure variaions in he reacor, one simply solves he segregaion and maximum mixedness models, accouning for he variaion of k wih emperaure [i.e., conversion; see Problem P13-2 A (i)] RTD and Muliple Reacions As discussed in Chaper 6, when muliple reacions occur in reacing sysems, i is bes o work in concenraions, moles, or molar flow raes raher han conversion Segregaion Model k k 1 exp E R -- Ê1 1ˆ Á ËT 1 T In he segregaion model we consider each of he globules in he reacor o have differen concenraions of reacans, C A, and producs, C P. These globules 26 Pearson Educaion, Inc.

62 928 Disribuions of Residence Times for Chemical Reacors Chap. 13 are mixed ogeher immediaely upon exiing o yield he exi concenraion of A,, which is he average of all he globules exiing: C A C A C A ()E() d (13-73) C B C B ()E() d (13-74) The concenraions of he individual species, C A () and C B (), in he differen globules are deermined from bach reacor calculaions. For a consan-volume bach reacor, where q reacions are aking place, he coupled mole balance equaions are iq dc A r r d A ia i1 iq dc B r r d B i B i1 These equaions are solved simulaneously wih (13-75) (13-76) dc A d C A ()E() (13-77) dc B d C B ()E() (13-78) o give he exi concenraion. The RTDs, E(), in Equaions (13-77) and (13-78) are deermined from experimenal measuremens and hen fi o a polynomial Maximum Mixedness For he maximum mixedness model, we wrie Equaion (13-62) for each species and replace r A by he ne rae of formaion dc A Â r d ia ( C A C A ) E ( ) F ( ) (13-79) dc B Â r d i B ( C B C B ) E ( ) F ( ) (13-8) Afer subsiuion for he rae laws for each reacion (e.g., r 1A k 1 C A ), hese equaions are solved numerically by saring a a very large value of, say T 2, and inegraing backwards o o yield he exi concenraions C A, C B,. We will now show how differen RTDs wih he same mean residence ime can produce differen produc disribuions for muliple reacions. 26 Pearson Educaion, Inc.

63 Sec RTD and Muliple Reacions 929 Example 13 9 RTD and Complex Reacions Consider he following se of liquid-phase reacions: A B A k 1 æææ k 2 æææ C D B D k 3 æææ which are occurring in wo differen reacors wih he same mean residence ime m 1.26 min. However, he RTD is very differen for each of he reacors, as can be seen in Figures E and E E E() (min 1 ) (min) Figure E E 1 (): asymmeric disribuion E() (min 1 ) (min) Figure E E 2 (): bimodal disribuion. 26 Pearson Educaion, Inc.

64 93 Disribuions of Residence Times for Chemical Reacors Chap. 13 (a) Fi a polynomial o he RTDs. (b) Deermine he produc disribuion (e.g., S C/D, S D/E ) for 1. The segregaion model 2. The maximum mixedness model Addiional Informaion Soluion k 1 = k 2 = k 3 = 1 in appropriae unis a 35K. Segregaion Model Combining he mole balance and rae laws for a consan-volume bach reacor (i.e., globules), we have dc A d r A r 1A r 2A k 1 C A C B k 2 C A (E13-9.1) dc B d r B r 1B r 3B k 1 C A C B k 3 C B C D (E13-9.2) dc C d r C r 1C k 1 C A C B (E13-9.3) dc D d r D r 2D r 3D k 2 C A k 3 C B C D (E13-9.4) dc E d r E r 3E k 3 C B C D (E13-9.5) and he concenraion for each species exiing he reacor is found by inegraing he equaion dc i C d i E ( ) (E13-9.6) over he life of he E() curve. For his example he life of he E 1 () is 2.42 minues (Figure E13-9.1), and he life of E 2 () is 6 minues (Figure E13-9.2). The iniial condiions are, C A C B 1, and C C C D C E. The Polymah program used o solve hese equaions is shown in Table E for he asymmeric RTD, E 1 (). Wih he excepion of he polynomial for E 2 (), an idenical program o ha in Table E for he bimodal disribuion is given on he CD-ROM. A comparison of he exi concenraion and seleciviies of he wo RTD curves is shown in Table E Pearson Educaion, Inc.

65 Sec RTD and Muliple Reacions 931 TABLE E POLYMATH PROGRAM FOR SEGREGATION MODEL WITH ASYMMETRIC RTD (MULTIPLE REACTIONS) TABLE E SEGREGATION MODEL RESULTS Asymmeric Disribuion The soluion for E 1 () is: Bimodal Disribuion The soluion for E 2 () is: C A C A C B.454 C E X 84.9% C B.51 C E X 75.5% C C.357 S C/D 1.18 C C.321 S C/D 1.21 C D.33 S D/E 1.7 C D.265 S D/E 1.63 Maximum Mixedness Model The equaions for each species are dc A d dc B d dc C d dc D d E ( ) k 1 C A C B k 2 C A (C A C A ) F ( ) E ( ) k 1 C A C B k 3 C B C D (C B C B ) F ( ) E ( ) k 1 C A C B (C C C C ) F ( ) E ( ) k 2 C A k 3 C B C D (C D C D ) F ( ) (E13-9.7) (E13-9.8) (E13-9.9) (E13-9.1) 26 Pearson Educaion, Inc.

66 932 Disribuions of Residence Times for Chemical Reacors Chap. 13 dc E d E ( ) k 3 C B C D (C E C E ) F ( ) (E ) The Polymah program for he bimodal disribuion, E 2 (), is shown in Table E The Polymah program for he asymmeric disribuion is idenical wih he excepion of he polynomial fi for E 1 () and is given on he CD-ROM. A comparison of he exi concenraion and seleciviies of he wo RTD disribuions is shown in Table E TABLE E POLYMATH PROGRAM FOR MAXIMUM MIXEDNESS MODEL WITH BIMODAL DISTRIBUTION (MULTIPLE REACTIONS) TABLE E MAXIMUM MIXEDNESS MODEL RESULTS Asymmeric Disribuion The soluion for E 1 () (1) is: Bimodal Disribuion The soluion for E 2 () (2) is: C A C A C B.467 C E X 83.9% C B.535 C E X 73.4% C C.341 S C/D 1.11 C C.275 S C/D 1.2 C D.36 S D/E 1.59 C D.269 S D/E 1.41 Calculaions similar o hose in Example 13-9 are given in an example on he CD-ROM for he series reacion k A æææ 1 k B æææ 2 C In addiion, he effec of he variance of he RTD on he parallel reacions in Example 13-9 and on he series reacion in he CD-ROM is shown on he CD-ROM. 26 Pearson Educaion, Inc.

67 Chap. 13 Summary 933 Closure Afer compleing his chaper he reader will use he racer concenraion ime daa o calculae he exernal age disribuion funcion E(), he cumulaive disribuion funcion F(), he mean residence ime, m, and he variance, s 2. The reader will be able o skech E() for ideal reacors, and by comparing E() from experimen wih E() for ideal reacors (PFR, PBR, CSTR, laminar flow reacor) he reader will be able o diagnose problems in real reacors. The reader will also be able o couple RTD daa wih reacion kineics o predic he conversion and exi concenraions using he segregaion and he maximum mixedness models wihou using any adjusable parameers. By analyzing he second derivaive of he reacion rae wih respec o concenraion, he reader will be able o deermine wheher he segregaion model or maximum mixedness model will give he greaer conversion. SUMMARY 1. The quaniy E() d is he fracion of maerial exiing he reacor ha has spen beween ime and d in he reacor. 2. The mean residence ime m E() d (S13-1) is equal o he space ime for consan volumeric flow, v = v. 3. The variance abou he mean residence ime is 2 ( m ) 2 E() d (S13-2) 4. The cumulaive disribuion funcion F() gives he fracion of effluen maerial ha has been in he reacor a ime or less: F() E() d 1 F() fracion of effluen maerial ha has been in (S13-3) he reacor a ime or longer 5. The RTD funcions for an ideal reacor are Plug-flow: E() ( ) (S13-4) e CSTR: E() (S13-5) Laminar flow: E() -- (S13-6) 2 2 E() (S13-7) Pearson Educaion, Inc.

68 934 Disribuions of Residence Times for Chemical Reacors Chap The dimensionless residence ime is E() E() (S13-8) (S13-9) 7. The inernal-age disribuion, [I() d], gives he fracion of maerial inside he reacor ha has been inside beween a ime and a ime ( d). 8. Segregaion model: The conversion is - and for muliple reacions X X ( )E ( ) d (S13-1) C A C A ( )E ( ) d 9. Maximum mixedness: Conversion can be calculaed by solving he following equaions: dx d r A E ( ) ( F ( ) X ) C A (S13-11) and for muliple reacions dc A r d Ane ( C A C A ) dc B r d Bne ( C B C B ) E ( ) F ( ) E ( ) F ( ) from = max o. To use an ODE solver le z max. (S13-12) (S13-13) CD-ROM MATERIAL Learning Resources 1. Summary Noes 2. Web Maerial Links A. The Aainable Region Analysis and 4. Solved Problems A. Example CD13-1 Calculae he exi concenraions for he series reacion A æææ B æææ C B. Example CD13-2 Deerminaion of he effec of variance on he exi concenraions for he series reacion A æææ B æææ C 26 Pearson Educaion, Inc.

69 Chap. 13 CD-ROM Maerial 935 Living Example Problems 1. Example 13 6 Laminar Flow Reacor 2. Example 13 8 Using Sofware o Make Maximum Mixedness Model Calculaions 3. Example 13 9 RTD and Complex Reacions 4. Example CD13-1 A Æ B Æ C Effec of RTD 5. Example CD13-2 A Æ B Æ C Effec of Variance Professional Reference Shelf 13R.1. Fiing he Tail Whenever here are dead zones ino which he maerial diffuses in and ou, he C and E curves may exhibi long ails. This secion shows how o analyically describe fiing hese ails o he curves. E ( ) ae b b slope of ln E vs. a be b1 [ 1F ( 1 )] 13R.2. Inernal-Age Disribuion The inernal-age disribuion currenly in he reacor is given by he disribuion of ages wih respec o how long he molecules have been in he reacor. The equaion for he inernal-age disribuion is derived and an example is given showing how i is applied o caalys deacivaion in a fluidized CSTR. d [τ Example 13R2.1 Mean Caalys Aciviy in a Fluidized Bed Reacor. 13R.3.Comparing X seg wih X mm The derivaion of equaions using he second derivaive crieria is carried ou. d E( ) [ I( ) ] d 2 r A 2 C A ( ) ? 26 Pearson Educaion, Inc.

70 936 Disribuions of Residence Times for Chemical Reacors Chap. 13 QUESTIONS AND PROBLEMS The subscrip o each of he problem numbers indicaes he level of difficuly: A, leas difficul; D, mos difficul. P13-1 A Read over he problems of his chaper. Make up an original problem ha uses he conceps presened in his chaper. The guidelines are given in Problem P4-1 A. RTDs from real reacors can be found in Ind. Eng. Chem., 49, 1 (1957); Ind. Eng. Chem. Process Des. Dev., 3, 381 (1964); Can. J. Chem. Eng., 37, 17 (1959); Ind. Eng. Chem., 44, 218 (1952); Chem. Eng. Sci., 3, 26 (1954); and Ind. Eng. Chem., 53, 381 (1961). P13-2 A Wha if... (a) Example Wha fracion of he fluid spends nine minues or longer in he reacor? (b) The combinaions of ideal reacors are used o model he following real reacors, given E(Q), F(Q), or 1 F(Q). 1. In (3) (4) (5) (6) Area = A 1 A 2 A3 A 4 F() (7) (8) (9) (1) F() (11) (12) Sugges a model for each figure. (c) Example How would he E() change if p as reduced by 5% and s was increased by 5%? (d) Example For 75% conversion, wha are he relaive sizes of he CSTR, PFR, and LFR? (e) Example How does he mean conversion compare wih he conversion calculaed wih he same m applied o an ideal PFR and CSTR? Can you give examples of E() where his calculaion would and would no be a good esimae of X? (f) Example Load he Living Example Problem. How would your resuls change if T = 4 C? How would your answer change if he reacion was pseudo firs order wih kc A = /s? Wha if he reacion were carried ou adiabaically where C PA = C PB = 2 cal/mol/k, H Rx = 1 kcal/mol k =.1 dm 3 /mol/min a 25 C wih E = 8 kcal/mol 26 Pearson Educaion, Inc.

71 Chap. 13 Quesions and Problems 937 (g) Example Load he Living Example Problem. How does he X seg and X MM compare wih he conversion calculaed for a PFR and a CSTR a he mean residence ime? (h) Example Load he Living Example Problem. How would your resuls change if he reacion was pseudo firs order wih k 1 = C A k =.8 min 1 2? If he reacion was hird order wih k C =.8 min 1 A? If he 12 reacion was half order wih k C =.8 min 1 A? Describe any rends. (i) Example Load he Living Example Problem. If he acivaion energies in cal/mol are E 1 = 5,, E 2 = 1,, and E 3 = 9,, how would he seleciviies and conversion of A change as he emperaure was raised or lowered around 35 K? (j) Hea Effecs. Redo Living Example Problems 13-7 and 13-8 for he case when he reacion is carried ou adiabaically wih (1) Exohermic reacion wih H T( K) T Rx Ê ˆX 3215X Ë C P (13-2.j.1) Hea effecs wih k given a 32 K and E = 1, cal/mol. (2) Endohermic reacion wih (13-2.j.2) and E = 45 kj/mol. How will your answers change? (k) you were asked o compare he resuls from Example 13-9 for he asymmeric and bimodal disribuions in Tables E and E Wha similariies and differences do you observe? Wha generalizaions can you make? (l) Repea (h) above using he RTD in Polymah program E13-8 o predic and compare conversions prediced by he segregaion model. 12 (m) he reacion in Example 13-5 was half order wih kc.8 min 1 A? How would your answers change? Hin: Modify he Living Example 13-8 program. (n) you were asked o vary he specific reacion raes k 1 and k 2 in he series k reacion A æææ 1 k B æææ 2 C given on he Solved Problems CD-ROM? Wha would you find? (o) you were asked o vary he isohermal emperaure in Example 13-9 from 3 K, a which he rae consans are given, up o a emperaure of 5 K? The acivaion energies in cal/mol are E 1 5, E 2 7, and E 3 9. How would he seleciviy change for each RTD curve? (p) he reacion in Example 13-7 were carried ou adiabaically wih he same parameers as hose in Equaion [P13-2(j).1]? How would your answers change? (q) If he reacion in Examples 13-8 and 13-5 were endohermic and carried ou adiabaically wih T(K) 32 1X and E 45 kj/mol [P13-2(j).1] how would your answers change? Wha generalizaions can you make abou he effec of emperaure on he resuls (e.g., conversion) prediced (r) T 321X from he RTD? If he reacion in Example 8-12 were carried ou in he reacor described by he RTD in Example 13-9 wih he excepion ha RTD is in seconds raher han minues (i.e., m 1.26 s), how would your answers change? 26 Pearson Educaion, Inc.

72 938 Disribuions of Residence Times for Chemical Reacors Chap. 13 P13-3 C Show ha for a firs-order reacion A æææ he exi concenraion maximum mixedness equaion dc A E ( ) kc d A ( 1F ( ) C C ) A A is he same as he exi concenraion given by he segregaion model B (P13-3.1) [Hin: Verify C A C A E()e k d (P13-3.2) is a soluion o Equaion (P13-3.1).] P13-4 C The firs-order reacion C C A () A e k E()e k d (P13-3.3) 1F ( ) A æææ wih k =.8 min 1 is carried ou in a real reacor wih he following RTD funcion B hemi (half) circle, min For 2 hen E() = 2 ( ) 2 min 1 (hemi circle) For > 2 hen E ( ) = (a) Wha is he mean residence ime? (b) Wha is he variance? (c) Wha is he conversion prediced by he segregaion model? (d) Wha is he conversion prediced by he maximum mixedness model? P13-5 B A sep racer inpu was used on a real reacor wih he following resuls: For 1 min, hen C T = For 1 3 min, hen C T = 1 g/dm 3 For 3 min, hen C T = 4 g/dm 3 The second-order reacion A Æ B wih k =.1 dm 3 /mol min is o be carried ou in he real reacor wih an enering concenraion of A of 1.25 mol/dm 3 a a volumeric flow rae of 1 dm 3 /min. Here k is given a 325 K. (a) Wha is he mean residence ime m? (b) Wha is he variance s 2? (c) Wha conversions do you expec from an ideal PFR and an ideal CSTR in a real reacor wih m? 26 Pearson Educaion, Inc.

73 Chap. 13 Quesions and Problems 939 (d) Wha is he conversion prediced by (1) he segregaion model? (2) he maximum mixedness model? (e) (f) Wha conversion is prediced by an ideal laminar flow reacor? Calculae he conversion using he segregaion model assuming T(K) = 325 5X and E/R = 5K. P13-6 B The following E() curves were obained from a racer es on wo ubular reacors in which dispersion is believed o occur..2 E() (min 1 ).2 E() (min 1 ) (a) (min) 5 (min) (b) Figure P13-6 B (a) RTD Reacor A; (b) RTD Reacor B. A second-order reacion k A æææ B wih kc A =.2 min 1 is o be carried ou in his reacor. There is no dispersion occurring eiher upsream or downsream of he reacor, bu here is dispersion inside he reacor. (a) Find he quaniies asked for in pars (a) hrough (e) in problem P13-5 B for reacor A. (b) Repea for Reacor B. P13-7 B The irreversible liquid phase reacion k 1 A æææ B is half order in A. The reacion is carried ou in a nonideal CSTR, which can be modeled using he segregaion model. RTD measuremens on he reacor gave values of 5 min and 3 min. For an enering concenraion of pure A of 1. mol/dm 3 he mean exi conversion was 1%. Esimae he specific reacion rae consan, k 1. Hin: Assume a Gaussian disribuion. P13-8 B The hird-order liquid-phase reacion wih an enering concenraion of 2M A k 3 æææ B was carried ou in a reacor ha has he following RTD E() = for 1 min E() = 1. min 1 for 1 2 min E() = for 2 min (a) For isohermal operaion, wha is he conversion prediced by 1) a CSTR, a PFR, an LFR, and he segregaion model, X seg. Hin: Find m (i.e., ) from he daa and hen use i wih E() for each of he ideal reacors. 2) he maximum mixedness model, X MM. Plo X vs. z (or ) and explain why he curve looks he way i does. (b) For isohermal operaion, a wha emperaure is he discrepancy beween X seg and X MM he greaes in he range 3 K T 35 K? 26 Pearson Educaion, Inc.

74 94 Disribuions of Residence Times for Chemical Reacors Chap. 13 (c) Suppose he reacion is carried ou adiabaically wih an enering emperaure of 35 K. Calculae X seq. Addiional Informaion k =.3 dm 6 /mol 2 /min a 3 K E/R = 2, K = 4, cal/mol C PA = C PB = 25 cal/mol/k H Rx P13-9 A Consider again he nonideal reacor characerized by he RTD daa in Example The irreversible gas-phase nonelemenary reacion A B C D is firs order in A and second order in B and is o be carried ou isohermally. Calculae he conversion for: (a) A PFR, a laminar flow reacor wih complee segregaion, and a CSTR. (b) The cases of complee segregaion and maximum mixedness. Also (c) Plo I() and () as a funcion of ime and hen deermine he mean age and he mean life expecancy. (d) How would your answers change if he reacion is carried ou adiabaically wih parameer values given by Equaion [P13-2(h).1]? Addiional informaion: C A C B.313 mol/dm 3, V 1 dm 3, v 1 dm 3 /s, k 175 dm 6 /mol 2 s a 32 K. P13-1 B An irreversible firs-order reacion akes place in a long cylindrical reacor. There is no change in volume, emperaure, or viscosiy. The use of he simplifying assumpion ha here is plug flow in he ube leads o an esimaed degree of conversion of 86.5%. Wha would be he acually aained degree of conversion if he real sae of flow is laminar, wih negligible diffusion? P13-11 A Consider a PFR, CSTR, and LFR. (a) Evaluae he firs momen abou he mean m 1 ( ) E()d for a PFR, a CSTR, and a laminar flow reacor. (b) Calculae he conversion in each of hese ideal reacors for a second-order liquid-phase reacion wih Da = 1. ( = 2 min and kc A =.5 min 1 ). P13-12 B For he caalyic reacion he rae law can be wrien as A r A æææ æææ ca C + D kc A ( 1K A C A ) 2 Which will predic he highes conversion, he maximum mixedness model or he segregaion model? Hin: Specify he differen ranges of he conversion where one model will dominae over he oher. 26 Pearson Educaion, Inc.

75 Chap. 13 Quesions and Problems 941 Addiional Informaion C A = 2 mol/dm 3 k =.1 dm 3 /mol s K A =.25 dm 3 /mol P13-13 D Too unlucky! Skip! P13-14 C The second-order liquid-phase reacion 2A k 1A æææ is carried ou in a nonideal CSTR. A 3 K he specific reacion rae is.5 dm 3 /molmin. In a racer es, he racer concenraion rose linearly up o 1 mg/dm 3 a 1. minues and hen decreased linearly o zero a exacly 2. minues. Pure A eners he reacor a a emperaure of 3 K. (a) Calculae he conversion prediced by he segregaion and maximum mixedness models. (b) Now consider ha a second reacion also akes place B (c) k 2C A B æææ C, k 2C =.12 dm 3 /mol min S BC Compare he seleciviies prediced by he segregaion and maximum mixedness models. Repea (a) for adiabaic operaion. Addiional informaion: C PA 5 J/molK, H Rx1A 75 J/mol C PB 1 J/molK, E = 1 kcal/mol C A 2 mol/dm 3 P13-15 B The reacions described in Problem P6-16 B are o be carried ou in he reacor whose RTD is described in Problem CDP13-N B. Deermine he exi seleciviies (a) Using he segregaion model. (b) Using he maximum mixedness model. (c) Compare he seleciviies in pars (a) and (b) wih hose ha would be found in an ideal PFR and ideal CSTR in which he space ime is equal o he mean residence ime. (d) Wha would your answers o pars (a) o (c) be if he reacor in Problem P13-7 B were used? P13-16 B The reacions described in Example 6-1 are o be carried ou in he reacor whose RTD is described in Example 13-7 wih C A C B.5 mol/dm 3. (a) Deermine he exi seleciviies using he segregaion model. (b) Deermine he exi seleciviies using he maximum mixedness model. (c) Compare he seleciviies in pars (a) and (b) wih hose ha would be found in an ideal PFR and ideal CSTR in which he space ime is equal o he mean residence ime. (d) Wha would your answers o pars (a) o (c) be if he RTD curve rose from zero a o a maximum of 5 mg/dm 3 afer 1 min, and hen fell linearly o zero a he end of 2 min? 26 Pearson Educaion, Inc.

76 942 Disribuions of Residence Times for Chemical Reacors Chap. 13 P13-17 B The reacions described in Problem P6-12 B are o be carried ou in he reacor whose RTD is described in Example (a) Deermine he exi seleciviies using he segregaion model. (b) Deermine he exi seleciviies using he maximum mixedness model. (c) Compare he seleciviies in pars (a) and (b) wih hose ha would be found in an ideal PFR and ideal CSTR in which he space ime is equal o he mean residence ime. P13-18 C The reacions described in Problem P6-11 B are o be carried ou in he reacor whose RTD is described in Problem CDP13-I B wih C A.8 mol/dm 3 and C B.6 mol/dm 3. (a) Deermine he exi seleciviies using he segregaion model. (b) Deermine he exi seleciviies using he maximum mixedness model. (c) Compare he seleciviies in pars (a) and (b) wih hose ha would be found in an ideal PFR and ideal CSTR in which he space ime is equal o he mean residence ime. (d) How would your answers o pars (a) and (b) change if he reacor in Problem P13-14 C were used? P13-19 B The volumeric flow rae hrough a reacor is 1 dm 3 /min. A pulse es gave he following concenraion measuremens a he oule: (min) c 1 5 (min) c (a) Plo he exernal age disribuion E() as a funcion of ime. (b) Plo he exernal age cumulaive disribuion F() as a funcion of ime. (c) Wha are he mean residence ime m and he variance, 2? (d) Wha fracion of he maerial spends beween 2 and 4 min in he reacor? (e) Wha fracion of he maerial spends longer han 6 min in he reacor? (f) Wha fracion of he maerial spends less han 3 min in he reacor? (g) Plo he normalized disribuions E() and F() as a funcion of. (h) Wha is he reacor volume? (i) Plo he inernal age disribuion I() as a funcion of ime. (j) Wha is he mean inernal age m? (k) Plo he inensiy funcion, (), as a funcion of ime. 26 Pearson Educaion, Inc.

77 Chap. 13 Quesions and Problems 943 Problem P13-19 B will be coninued in Chaper 14, P14-13 B. (l) The aciviy of a fluidized CSTR is mainained consan by feeding fresh caalys and removing spen caalys a a consan rae. Using he preceding RTD daa, wha is he mean caalyic aciviy if he caalys decays according o he rae law wih da k d D a 2 k D.1 s 1? (m) Wha conversion would be achieved in an ideal PFR for a second-order reacion wih kc A.1 min 1 and C A 1 mol/dm 3? (n) Repea (m) for a laminar flow reacor. (o) Repea (m) for an ideal CSTR. (p) Wha would be he conversion for a second-order reacion wih kc A.1 min 1 and C A 1 mol/dm 3 using he segregaion model? (q) Wha would be he conversion for a second-order reacion wih kc A.1 min 1 and C A 1 mol/dm 3 using he maximum mixedness model? Addiional Homework Problems CDP13-A C CDP13-B B CDP13-C B CDP13-D B CDP13-E B CDP13-F A CDP13-G B Afer showing ha E() for wo CSTRs in series having differen volumes is 1 E ( ) Ï ( 2m1) exp ʈ Ì Á----- exp Ó Ë m ( 1m) you are asked o make a number of calculaions. [2nd Ed. P13-11] Deermine E() from daa aken from a pulse es in which he pulse is no perfec and he inle concenraion varies wih ime. [2nd Ed. P13-15] Derive he E() curve for a Bingham plasic flowing in a cylindrical ube. [2nd Ed. P13-16] The order of a CSTR and PFR in series is invesigaed for a hird-order reacion. [2nd Ed. P13-1] Review he Murphree pilo plan daa when a second-order reacion occurs in he reacor. [1s Ed. P13-15] Calculae he mean waiing ime for gasoline a a service saion and in a parking garage. [2nd Ed. P13-3] Apply he RTD given by ÏA B ( E ( ) ) 2 Ì Ó for 2 for 2 o Examples 13-6 hrough [2nd Ed. P13-2 B ] CDP13-H B The muliple reacions in Problem 6-27 are carried ou in a reacor whose RTD is described in Example CDP13-I B Real RTD daa from an indusrial packed bed reacor operaing poorly. [3rd Ed. P13-5] CDP13-J B Real RTD daa from disribuion in a sirred ank. [3rd Ed. P13-7 B ] CDP13-K B Triangle RTD wih second-order reacion. [3rd Ed. P13-8 B ] CDP13-L B Derive E() for a urbulen flow reacor wih 1/7h power law. CDP13-M B Good problem mus use numerical echniques. [3rd Ed. P13-12 B] 26 Pearson Educaion, Inc.

78 944 Disribuions of Residence Times for Chemical Reacors Chap. 13 CDP13-N B Inernal age disribuion for a caalys. [3 rd Ed. P13-13 B] CDP13-DQEA U of M, Docoral Qualifying Exam (DQE), May, 2 CDP13-DQEB U of M, Docoral Qualifying Exam (DQE), April, 1999 CDP13-DQEC U of M, Docoral Qualifying Exam (DQE), January, 1999 CDP13-DQED U of M, Docoral Qualifying Exam (DQE), January, 1999 CDP13-DQEE U of M, Docoral Qualifying Exam (DQE), January, 1998 CDP13-DQEF U of M, Docoral Qualifying Exam (DQE), January, 1998 CDP13-ExG U of M, Graduae Class Final Exam CDP13-New New Problems will be insered from ime o ime on he web. SUPPLEMENTARY READING 1. Discussions of he measuremen and analysis of residence-ime disribuion can be found in CURL, R. L., and M. L. MCMILLIN, Accuracies in residence ime measuremens, AIChE J., 12, (1966). LEVENSPIEL, O., Chemical Reacion Engineering, 3rd ed. New York: Wiley, 1999, Chaps An excellen discussion of segregaion can be found in 3. Also see DOUGLAS, J. M., The effec of mixing on reacor design, AIChE Symp. Ser., 48, Vol. 6, p. 1 (1964). DUDUKOVIC, M., and R. FELDER, in CHEMI Modules on Chemical Reacion Engineering, Vol. 4, ed. B. Crynes and H. S. Fogler. New York: AIChE, NAUMAN, E. B., Residence ime disribuions and micromixing, Chem. Eng. Commun., 8, 53 (1981). NAUMAN, E. B., and B. A. BUFFHAM, Mixing in Coninuous Flow Sysems. New York: Wiley, ROBINSON, B. A., and J. W. TESTER, Chem. Eng. Sci., 41(3), (1986). VILLERMAUX, J., Mixing in chemical reacors, in Chemical Reacion Engineering Plenary Lecures, ACS Symposium Series 226. Washingon, D.C.: American Chemical Sociey, Pearson Educaion, Inc.