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.