Residence Time Distributions of Chemical Reactors

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2 Fogler_Ch16.fm Page 1 Tuesday, March 14, 17 6:4 PM Residence Time Disribuions of Chemical Reacors 16 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. Afer sudying his chaper he reader will be able o describe: General Consideraions. How he residence ime disribuion (RTD) can be used (Secion 16.1). Measuremen of he RTD. How o calculae he concenraion curve (i.e., he C-curve) and residence ime disribuion curve, (i.e., he E-curve (Secion 16.)). Characerisics of he RTD. How o calculae and use he cumulaive RTD funcion, F(), he mean residence ime, m, and he variance σ (Secion 16.3). The RTD in ideal reacors. How o evaluae E(), F(), m, and σ for ideal PFRs, CSTRs, and laminar flow reacors (LFRs) so ha we have a reference poin as o how much our real (i.e., nonideal) reacor deviaes form an ideal reacor (Secion 16.4). How o diagnose problems wih real reacors by comparing m, E(), and F() wih ideal reacors. This comparison will help o diagnose and roubleshoo by-passing and dead volume problems in real reacors (Secion 16.5) General Consideraions 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 16-1

3 Fogler_Ch16.fm Page Tuesday, March 14, 17 6:4 PM 16- Residence Time Disribuions of Chemical Reacors Chaper 16 We wan o analyze and characerize nonideal reacor behavior. 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 wo chapers. 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.. To predic conversion or effluen concenraions in exising/available reacors when a new chemical reacion is used in he reacor. The following wo examples illusrae reacor problems one migh find in a chemical plan. Example 1 A packed-bed reacor is shown in Figure When a reacor is packed wih caalys, he reacing fluid usually does no flow uniformly hrough he reacor. Raher, here may be secions in he packed bed ha offer lile resisance o flow (Pah 1) and, as a resul, a porion of he fluid may channel hrough his pahway. Consequenly, he molecules following his pahway do no spend much ime in he reacor. On he oher hand, if here is inernal circulaion or a high resisance o flow, he molecules could spend a long ime in he reacor (Pah ). Consequenly, we see ha here is a disribuion of imes ha molecules spend in he reacor in conac wih he caalys. Pah 1 Pah Figure 16-1 Packed-bed reacor. Example In many coninuous-sirred ank reacors, he inle and oule pipes are somewha close ogeher (Figure 16-). 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 here. Experimens were carried ou o deermine he amoun of he maerial effecively bypassed and he volume of he dead zone. See he AIChE webinar Dealing wih Difficul People : dealing-difficul-people.

4 Fogler_Ch16.fm Page 3 Tuesday, March 14, 17 6:4 PM Secion 16.1 General Consideraions 16-3 We wan o find ways of deermining he dead zone volume and he fracion of he volumeric flow rae bypassing he sysem. Dead zone Bypassing Figure 16- CSTR. The hree conceps RTD Mixing Model Chance Card: Do no pass go, proceed direcly o Chaper 17. GO 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 (RTD), 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 ideal 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 16.1 o 16.4). The nex level uses microscale (micromixing) informaion (Chaper 17) o make predicions abou he conversion in nonideal reacors. Afer compleing he firs four secions, 16.1 hrough 16.4, he reader can proceed direcly o Chaper 17 o learn how o calculae he conversion and produc disribuions exiing real reacors. Secion 16.5 closes he chaper by discussing how o use he RTD o diagnose and roubleshoo reacors. Here, we focus on wo common problems: reacors wih bypassing and dead volumes. Once he dead volumes, V D, and bypassing volumeric flow raes, b, are deermined, he sraegies in Chaper 18 o model he real reacor wih ideal reacors can be used o predic conversion 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 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 There are a number of mixing uorials on he AIChE Webinar websie and as an AIChE suden member you have free access o all hese webinars. See 1 R. B. MacMullin and M. Weber, Jr., Trans. Am. Ins. Chem. Eng., 31, 49 (1935). P. V. Danckwers, Chem. Eng. Sci.,, 1 (1953).

5 Fogler_Ch16.fm Page 4 Tuesday, March 14, 17 6:4 PM 16-4 Residence Time Disribuions of Chemical Reacors Chaper 16 Residence ime The RTD : Some molecules leave quickly, ohers oversay heir welcome. We will use he RTD o characerize nonideal reacors. 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 he BR 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 ypes of reacors in which all he aoms in he reacors have exacly 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 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 recirculaes wihin he reacor and is virually 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 in erms of conversion and produc disribuion. 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 (cf. Example 16-3), no all RTDs are unique o a paricular reacor ype; markedly differen reacors and reacor sequencing 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 16. 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 o insure ha he racer s behavior will reliably 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 In a pulse inpu, an amoun of racer N is suddenly injeced in one sho ino he feed sream enering he reacor in as shor a ime as is humanly possible. The oule concenraion is hen measured as a funcion of ime. Typical

6 Fogler_Ch16.fm Page 5 Tuesday, March 14, 17 6:4 PM Secion 16. Measuremen of he RTD 16-5 The C-curve Inerpreaion of E() d concenraion ime curves a he inle and oule of an arbirary reacor are shown in Figure 16-4 on page 77. The effluen of he racer concenraion versus ime curve is referred o as he C-curve in RTD analysis. We shall firs 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. Here, we choose an incremen of ime sufficienly small ha he concenraion of racer, C(), exiing beween ime and ime ( ) is essenially he same. The amoun of racer maerial, N, leaving he reacor beween ime and is hen N C() (16-1) where 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 N C() (16-) N N which represens he fracion of maerial ha has a residence ime in he reacor beween ime and. For pulse injecion we define C() E() N (16-3) so ha N E() (16-4) N 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. Feed Reacor Effluen Injecion Figure 16-3 Experimenal se up o deermine E(). Deecion C() The C-curve Figure 16-4 shows schemaics of he inle and oule concenraions for boh a pulse inpu and sep inpu for he experimenal se up in Figure 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 (16-1) in differenial form yields dn C() d (16-5)

7 Fogler_Ch16.fm Page 6 Tuesday, March 14, 17 6:4 PM 16-6 Residence Time Disribuions of Chemical Reacors Chaper 16 Pulse injecion Pulse response C C The C-curve Sep injecion Sep response C C Figure 16-4 RTD measuremens. Area = C()d C() and hen inegraing, we obain N C() d The volumeric flow rae is usually consan, so we can define E() as (16-6) We find he RTD funcion, E(), from he racer concenraion C() E() The E-curve E() C() C() d The E-curve is jus he C-curve divided by he area under he C-curve. (16-7) An alernaive way of inerpreing he residence ime funcion is in is inegral form: Fracion of maerial leaving he reacor ha has resided in he reacor for a ime beween 1 and 1 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 Evenually all guess mus leave E() d 1 (16-8) 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.

8 Fogler_Ch16.fm Page 7 Tuesday, March 14, 17 6:4 PM Secion 16. Measuremen of he RTD 16-7 Example 16 1 Consrucing he C() and E() Curves A sample of he racer hyane a 3 K was injeced as a pulse ino 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 a figure showing he racer concenraion C() as a funcion of ime. (b) Consruc a figure showing E() as a funcion of ime. Soluion (a) By ploing C as a funcion of ime, using he daa in Table E16-1.1, he curve shown in Figure E is obained C C() (gm/m 3 ) (min) Figure E The C-curve To conver he C() curve in Figure E o an E() curve we use he area under he C() curve. There are hree ways we can deermine he area using his daa. (1) Brue force: calculae he area by measuring he area of he squares and parial squares under he curve, and hen summing hem up. () Use he inegraion formulas given in Appendix A. (3) Fi he daa o one or more polynomials using Polymah or some oher sofware program. We will choose Polymah o fi he daa. Noe: A sep-by-sep uorial o fi he daa poins using Polymah is given on he CRE Web sie ( LEP We will use wo polynomials o fi he C -curve, one for he ascending porion, C 1 (), and one for he descending porion, C (), boh of which mee a 1. Using he Polymah polynomial fiing rouine (see uorial), he daa in Table E yields he following wo polynomials o For 4 min hen C 1 () For 4 min hen C () (E16-1.1) (E16-1.)

9 Fogler_Ch16.fm Page 8 Tuesday, March 14, 17 6:4 PM 16-8 Residence Time Disribuions of Chemical Reacors Chaper 16 Mach C() C 1 () C () 1 We hen use an if saemen in our fied curve. If ( 4 and ) hen C 1 else if ( 4 and 14) hen C else To find he area under he curve, A, we use he ODE solver. Le A represen he area he curve, hen da C () d A 14 C() d (E16-1.3) (E16-1.4) (b) Consruc E(). E () C () C () A C () d The Polymah program and resuls are shown below where we see A 51. POLYMATH Repor Ordinary Differenial Equaions Calculaed values of DEQ variables Variable Iniial value Final value 1 Area C C C Differenial equaions *1 d(area)/d() = C Explici equaions 1 C = * *^ *^ *^4 +.58*^ *1^-5*^6 C1 = * *^ *^3 3 C = If(<=4 and >=) hen C1 else if(>4 and <=14) hen C else Now ha we have he area, A (i.e., 51 g min/m 3 ), under he C-curve, we can consruc he E() curves. We now calculae E() by dividing each poin on he C() curve by 51. g min/m 3 wih he following resuls: E() C() C() g min m C() d 3 Using Table E16-1. we can consruc E() as shown in Figure E16-1. (E16-1.5) TABLE E16-1. C() AND E() (min) C() (g/m 3 ) E() (min 1 )

10 Fogler_Ch16.fm Page 9 Tuesday, March 14, 17 6:4 PM Secion 16. Measuremen of he RTD 16-9 E( ) (min 1 )..18 E (min) Figure E16-1. E()-Curve Analysis: In his example we fi he effluen concenraion daa C() from an iner racer pulse inpu o wo polynomials and hen used an If saemen o model he complee curve. We hen used he Polymah ODE solver o ge he area under he curve ha we hen used o divide he C() curve in order o obain he E(). Once we have he E() curve, we ask and easily answer such quesions as wha fracion of he modules spend beween and 4 minues in he reacor or wha is he mean residence ime m? We will address hese quesions in he following secions where we discuss characerisics of he residence ime disribuion (RTD). 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 ha 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 could be problems in fiing E() o a polynomial if he effluen concenraion ime curve were o have 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 (16-7), 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 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 relaionship beween a sep racer injecion and he corresponding concenraion in he effluen.

11 Fogler_Ch16.fm Page 1 Tuesday, March 14, 17 6:4 PM 16-1 Residence Time Disribuions of Chemical Reacors Chaper 16 C in C ou Sep Inpu The inle concenraion mos ofen akes he form of eiher a perfec pulse inpu (Dirac dela funcion), imperfec pulse injecion (see Figure 16-4), 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. The cumulaive disribuion gives he fracion of maerial F() ha has been in he reacor a ime or less. 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 ou () C, consan (16-9) 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 () C ou () C E() d F() sep (16-1) We differeniae his expression o obain he RTD funcion E(): E() df d C ou() (16-11) d d C sep Advanages and drawbacks o he sep injecion 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. 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

12 Fogler_Ch16.fm Page 11 Tuesday, March 14, 17 6:4 PM Secion 16.3 Characerisics of he RTD 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. 3 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, by inegraing E() beween ime and ime,. The cumulaive RTD funcion F() E() d F() Fracion of effluen ha has been in reacor for less han ime (16-1) Analogously, we have, by inegraing beween ime and ime E () d 1 F() Fracion of effluen ha has been in reacor for longer han ime (16-13) Because appears in he inegraion limis of hese wo expressions, Equaions (16-1) and (16-13) are boh funcions of ime. Danckwers defined Equaion (16-1) as a cumulaive disribuion funcion and called i F(). 4 We can calculae F() a various imes from he area under he curve of a plo of E() versus, i.e., he E-curve. A ypical shape of he F() curve is shown in Figure One noes from his curve ha 8% (i.e., F() =.8) of he molecules spend 8 minues or less in he reacor, and % of he molecules [1 F()] spend longer han 8 minues in he reacor. The F-curve is anoher funcion ha has been defined as he normalized response o a paricular inpu. Alernaively, Equaion (16-1) 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 indusrial example is he sudy of Wolf and Whie, who invesigaed he behavior of screw exruders in polymerizaion processes. 5 3 C. Y. Wen and L. T. Fan, Models for Flow Sysems and Chemical Reacors (New York: Marcel Dekker, 1975). 4 P. V. Danckwers, Chem. Eng. Sci.,, 1 (1953). 5 D. Wolf and D. H. Whie, AIChE J.,, 1 (1976).

13 Fogler_Ch16.fm Page 1 Tuesday, March 14, 17 6:4 PM 16-1 Residence Time Disribuions of Chemical Reacors Chaper The F-curve F() (min) Figure 16-5 Cumulaive disribuion curve, F() Mean Residence Time τ m In previous chapers reaing ideal reacors, a parameer frequenly used was he space ime or average residence ime, τ, which was defined as being equal o (V /). I will be shown ha, in he absence of dispersion, and for consan volumeric flow ( = ) no maer wha RTD exiss for a paricular reacor, ideal or nonideal, his nominal space ime, τ, 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 (16-14) We now wish o show how we can deermine he oal reacor volume using he cumulaive disribuion funcion. In he Exended Maerial for Chaper 16 on he Web, a proof is given ha for consan volumeric flow rae, he mean residence ime is equal o he space ime, i.e., m (16-15) This resul is rue only for a closed sysem (i.e., no dispersion across boundaries; see Chaper 18). The exac reacor volume is deermined from he equaion V m (16-16) 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). 6 For his purpose, hree momens are normally used. The firs is he mean residence ime, m. The 6 C. Y. Wen and L. T. Fan, Models for Flow Sysems and Chemical Reacors (New York: Decker, 1975), Chap. 11.

14 Fogler_Ch16.fm Page 13 Tuesday, March 14, 17 6:4 PM Secion 16.3 Characerisics of he RTD second momen commonly used is aken abou he mean and is called he variance, σ, 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 τ and ( m ) E() d (16-17) 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, s 3, The skewness is defined by s ( m ) 3 E() d (16-18) 3 The magniude of he hird momen measures he exen ha a disribuion is skewed in one direcion or anoher in reference o he mean. Rigorously, for a complee descripion of a disribuion, all momens mus be deermined. Pracically, hese hree are usually sufficien for a reasonable characerizaion of an RTD. Example 16 Mean Residence Time and Variance Calculaions Using he daa given in Table E16-1. in Example 16-1 (a) Consruc he F() curve. (b) Calculae he mean residence ime, m. (c) Calculae he variance abou he mean,. (d) Calculae he fracion of fluid ha spends beween 3 and 6 minues in he reacor. (e) Calculae he fracion of fluid ha spends minues or less in he reacor. (f) Calculae he fracion of he maerial ha spends 3 minues or longer in he reacor. Soluion (a) To consruc he F-curve, we simply inegrae he E-curve E () C () C () (E16-1.5) A 51 using an ODE solver such as Polymah shown in Table E16-.1 df ---- E () (16-11) d The Polymah program and resuls are shown in Table E16-1. and Figure E16-.1(b), respecively. Table E16-1. C(), AND E() (min) C() (g/m 3 ) E() (min 1 )

15 Fogler_Ch16.fm Page 14 Tuesday, March 14, 17 6:4 PM Residence Time Disribuions of Chemical Reacors Chaper 16 Calculaing he mean residence ime, m E() d (b) We also show in Table E16-.1 he Polymah program o calculae he mean residence ime, m. By differeniaing Equaion (16-14), we can easily use Polymah o find m, i.e., d m E() (E16-.1) d wih hen E and 14 hen E. Equaion (E16-.1) and he calculaed resul is also shown in Table E16-.1 where we find m 5.1 minues TABLE E16-.1 POLYMATH PROGRAM AND RESULTS TO CONTRUCT THE E- AND F- CURVES POLYMATH Repor Ordinary Differenial Equaions Calculaed values of DEQ variables Variable Iniial value Final value 1 Area C C C E 7.597E F m Differenial equaions 1 d(m)/d() = *E d(f)/d() = E Explici equaions 1 C1 = * *^ *^3 Area = 51 3 C = * *^ *^ *^4 +.58*^ *1^-5*^6 4 C = If(<=4 and >=) hen C1 else if(>4 and <=14) hen C else 5 E = C/Area Using he Polymah ploing rouines, we can consruc Figures E16-.1 (a) and (b) afer execuing he program shown in he Polymah Table E E F (a) Figure E (b) (a) E-Curve; (b) F-Curve. (c) Now ha we have found he mean residence ime m we can calculae he variance σ. Calculaing he variance ( m ) E() d We now differeniae Equaion (E16-.) wih respec o (E16-.) d ( m ) E () (E16-.3) d and hen use Polymah o inegrae beween = and = 14, which is he las poin on he E-curve.

16 Fogler_Ch16.fm Page 15 Tuesday, March 14, 17 6:4 PM Secion 16.3 Characerisics of he RTD TABLE E16-. POLYMATH PROGRAM AND RESULTS TO CALCULATE THE MEAN RESIDENCE TIME, m, AND THE VARIANCE σ POLYMATH Repor Ordinary Differenial Equaions Calculaed values of DEQ variables Variable Iniial value Final value 1 Area C C C E 7.597E Sigma mf Differenial equaions 1 d(sigma)/d() = (-mf)^ * E Explici equaions 1 C1 = * *^ *^3 Area = 51 3 C = * *^ *^ *^4 +.58*^ *1^-5*^6 4 C = If(<=4 and >=) hen C1 else if(>4 and <=14) hen C else 5 E = C/Area 6 mf = 5.1 The resuls of his inegraion are shown in Table E16-.1 where we find 6. minues, so.49 minues. (d) To find he fracion of fluid ha spends beween 3 and 6 minues, we simply inegrae he E-curve beween 3 and 6 F 3 6 The Polymah program is shown in Table E16-.3 along wih he oupu. 6 3 E () d TABLE E16-.3 POLYMATH PROGRAM TO FIND THE FRACTION OF FLUID THAT SPENDS BETWEEN 3 AND 6 MINUTES IN THE REACTOR POLYMATH Repor Ordinary Differenial Equaions Calculaed values of DEQ variables Variable Iniial value Final value 1 C C C E F Differenial equaions 1 d(f)/d() = E Explici equaions 1 C1 = * *^ *^3 C = * *^ *^ *^4 +.58*^ *1^-5*^6 3 C = If(<=4 and >=) hen C1 else if(>4 and <=14) hen C else 4 E = C/51 We see ha approximaely 5% (i.e., 49.53%) of he maerial spends beween 3 and 6 minues in he reacor. We can visualize his fracion wih he use of plo of E() versus () as shown in Figure E16-.. The shaded area in Figure E16-. represens he fracion of maerial leaving he reacor ha has resided in he reacor beween 3 and 6 min. Evaluaing his area, we find ha 5% of he maerial leaving he reacor spends beween 3 and 6 min in he reacor. (e) We shall nex 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, F(). This fracion is jus he shaded area under he curve up o minues. This area is shown in Figure E16-.3 for 3 min. Calculaing he area under he curve, we see ha approximaely % of he maerial has spen 3 min or less in he reacor.

17 Fogler_Ch16.fm Page 16 Tuesday, March 14, 17 6:4 PM Residence Time Disribuions of Chemical Reacors Chaper 16 The E curve Figure E16-. Fracion of maerial ha spends beween 3 and 6 minues in he reacor...15 E( ) (min 1 ) (min) Figure E16-.3 Fracion of maerial ha spends 3 minues or less in he reacor. (f) The fracion of fluid ha spends a ime or greaer in he reacor is Greaer han ime 1 F () 1..8 herefore 8% of he fluid spends a ime or greaer in he reacor. The square of he sandard deviaion is 6.19 min, so.49 min. Analysis: In his example we calculaed wo imporan properies of he RTD, he mean ime molecules spend in he reacors, m, and he variance abou his mean,. We will calculae hese properies from he RTD of oher nonideal reacors and hen show in Chaper 18 how o use hem o formulae models of real reacors using combinaions of ideal reacors. We will use hese models along wih reacion-rae daa o predic he conversion in he nonideal reacor we obained from he reacor sorage shed Normalized RTD Funcion, E() Frequenly, a normalized RTD is used insead of he funcion E(). If he parameer is defined as - (16-19)

18 Fogler_Ch16.fm Page 17 Tuesday, March 14, 17 6:4 PM Secion 16.3 Characerisics of he RTD E() Why we use a normalized RTD E() for a CSTR 1 1 > v 1, v The quaniy represens he number of reacor volumes of fluid, based on enrance condiions, ha have flowed hrough he reacor in ime. The dimensionless RTD funcion, E() is hen defined as E() τe() (16-) and ploed as a funcion of, as shown in he margin. 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 CSTRs differen volumes, V, and enering volumeric flow raes,. As will be shown laer in Secion 16.4., E() for a perfecly mixed CSTR and herefore 1 E() -- e (16-1) E() τe() e (16-) 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 1 and. Bu for he same value of, he value of E() is he same irrespecive of he size or volumeric flow rae of a perfecly mixed CSTR. I is a relaively easy exercise o show ha E() d 1 (16-3) and is recommended as a 93-s diverissemen. (Jofosan Universiy chemical engineers claim hey can do i in 87 s.) Inernal-Age Disribuion, I() Tombsone jail How long have you been here? I()Δ 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 now 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() is of imporance and can be of use in modeling he reacor.

19 Fogler_Ch16.fm Page 18 Tuesday, March 14, 17 6:4 PM Residence Time Disribuions of Chemical Reacors Chaper 16 The inernal-age disribuion is discussed furher on he Professional Reference Shelf (R16.) where he following relaionships beween he cumulaive inernal-age disribuion I() and he cumulaive exernal-age disribuion F() I() = (1 F())/τ (16-4) and beween E() and I() E() = d [ τi( ) ] (16-5) d are derived. For a CSTR, i is shown ha he inernal-age disribuion funcion is I(α) = 1 -- e τ (16-6) τ 16.4 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 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/ τ, or 1, as shown in Figure The E() funcion is shown in Figure 16-6(a), and F() is shown in Figure 16-6(b). In Ou E() F() 1. (a) (b) Figure 16-6 Ideal plug-flow response o a pulse racer inpu. Mahemaically, his spike is represened by he Dirac dela funcion: E() for a plugflow reacor E() ( τ) The Dirac dela funcion has he following properies: (16-7) Properies of he ( x) when x when x Dirac dela funcion ( x) d x 1 (16-8) (16-9) g( x) ( x τ) dx g ( τ) (16-3)

20 Fogler_Ch16.fm Page 19 Tuesday, March 14, 17 6:4 PM Secion 16.4 RTD in Ideal Reacors To calculae τ he mean residence ime, we se g(x) m E() d ( τ) d τ (16-31) Bu we already knew his resul, as did all chemical reacion engineering sudens a he universiy in Riça, Jofosan. To calculae he variance, we se g() = ( τ), and he variance, σ, is (τ) ( τ) d All maerial spends exacly a ime τ in he reacor, so here is no variance [ ]! The cumulaive disribuion funcion F() is F() E ()d ( τ)d From a racer balance we can deermine E(). E() and E(Θ) 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 C } = Accumulaion V dc d (16-33) Because he reacor is perfecly mixed, C in his equaion is he concenraion of he racer boh in he effluen and wihin he reacor. Separaing he variables and inegraing wih C C a yields C() C e /τ (16-34) The C-curve can be ploed from Equaion (16-34), which is he concenraion of racer in he effluen a any ime. To find E() for an ideal CSTR, we firs recall Equaion (16-7) and hen subsiue for C() using Equaion (16-34). Tha is C() C E() e τ e τ (16-35) τ C() d Evaluaing he inegral in he denominaor complees he derivaion of he RTD for an ideal CSTR and one noes hey are he same as previously given by Equaions (16-1) and (16-) E() (16-1) τ E() e C e d e τ (16-)

21 Fogler_Ch16.fm Page Tuesday, March 14, 17 6:4 PM 16- Residence Time Disribuions of Chemical Reacors Chaper 16 he cumulaive disribuion is F () E () d e τ e τ τ (16-3) Recall ha and E() = τe(). Response of an ideal CSTR E(Θ) = e Θ F(Θ) = 1 e Θ (a) Figure 16-7 (b) E(Θ) and F(Θ) for an Ideal CSTR. The cumulaive disribuion F() is F() E()d =1e (16-36) The E() and F() funcions for an ideal CSTR are shown in Figure 16-7 (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/ ), or τ. 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 τ - e /τ d τ (16-14) Thus, he nominal holding ime (space ime) τ (V/ ) is also he mean residence ime ha he maerial spends in he reacor. The second momen abou he mean is he variance and 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 /τ) For a perfecly mixed CSTR: m τ and τ. ( τ) e /τ d τ (x 1) e x dx τ (16-37) τ Then, τ. 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) 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 (i.e. cylindrical) reacor, he velociy profile is parabolic, wih he fluid in he cener of he ube spending he shores ime in he reacor. A schemaic

22 Fogler_Ch16.fm Page 1 Tuesday, March 14, 17 6:4 PM Secion 16.4 RTD in Ideal Reacors 16-1 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. Molecules near he cener spend a shorer ime in he reacor han hose close o he wall. Figure 16-8 r + dr R r dr Schemaic diagram of fluid elemens in a laminar-flow reacor. R r R U Parabolic Velociy Profile The velociy profile in a pipe of ouer radius R is r Ur () U max 1 R -- r U avg 1 R -- r R R -- (16-38) 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 L R () r L U() r 1 [ ( r R) ] [ ( r R) ] (16-39) (16-4) We are jus doing a few manipulaions o arrive a E() for an LFR The volumeric flow rae of fluid ou of he reacor beween r and (r + dr), d, is d = U(r) πrdr (16-41) The fracion of oal fluid passing ou beween r and (r + dr) is d/, i.e. d Ur ()( rdr) (16-4) The fracion of fluid beween r and (r + dr) ha has a flow rae beween and ( + d) and spends a ime beween and ( + d) in he reacor is E ()d d (16-43) We now need o relae he fluid fracion, Equaion (16-43), o he fracion of fluid spending beween ime and d in he reacor. Firs we differeniae Equaion (16-4) r dr 4 d R [ 1 ( r R) ] τ R [ 1 ( r R) r dr ] (16-44) and hen use Equaion (16-4) o subsiue for he erm in brackes o yield d r dr τ R (16-45)

23 Fogler_Ch16.fm Page Tuesday, March 14, 17 6:4 PM 16- Residence Time Disribuions of Chemical Reacors Chaper 16 Combining Equaions (16-4) and (16-45), and hen using Equaion (16-4) ha relaes for U(r) and (r), we now have he fracion of fluid spending beween ime and d in he reacor E ()d d L r dr L R τ d d 4 3 τ E () (16-46) The minimum ime he fluid may spend in he reacor is L L R V τ U max U avg R Consequenly, he complee RTD funcion for a laminar-flow reacor is A las! E() for a laminar-flow reacor E() τ τ -- τ -- (16-47) The cumulaive disribuion funcion for τ/ is F () E () d + E() d τ d d τ τ (16-48) The mean residence ime m is For LFR m τ m E() d This resul was shown previously o be rue for any reacor wihou dispersion. The mean residence ime is jus he space ime τ. The dimensionless form of he RTD funcion is τ --- τ τ τ d --- Normalized RTD funcion for a laminar-flow reacor E ( ) (16-49)

24 Fogler_Ch16.fm Page 3 Tuesday, March 14, 17 6:4 PM Secion 16.5 PFR/CSTR Series RTD 16-3 and is ploed in Figure (a) (b) Figure 16-9 (a) E(Θ) for an LFR; (b) F(Θ) for a PFR, CSTR, and LFR. The dimensionless cumulaive disribuion, F(Θ) for Θ 1/, is F( ) E( ) d d F( ) (16-5) Figure 16-9(a) shows E(Θ) for a laminar flow reacor (LFR), while Figure 9-9(b) compares F(Θ) 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. 7 Modeling he real reacor as a CSTR and a PFR in series 16.5 PFR/ CSTR Series RTD 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 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 a series arrangemen of a CSTR and a PFR. Firs consider he CSTR followed by he PFR (Figure 16-1). The mean residence ime in he CSTR will be denoed by τ s and he mean residence ime in he PFR by τ p. If a pulse of racer is injeced ino he enrance of he CSTR, 7 D. Levenspiel, Chemical Reacion Engineering, 3rd ed. (New York: Wiley, 1999), p. 34.

25 Fogler_Ch16.fm Page 4 Tuesday, March 14, 17 6:4 PM 16-4 Residence Time Disribuions of Chemical Reacors Chaper 16 CSTR PFR Figure 16-1 Real reacor modeled as a CSTR and PFR in series. he CSTR oupu concenraion as a funcion of ime will be C C e τ s This oupu will be delayed by a ime τ p a he oule of he plug-flow secion of he reacor sysem. Thus, he RTD of he reacor sysem is E() e ( p) s τ s τ p τ p (16-51) See Figure The RTD is no unique o a paricular reacor sequence. E() F() Figure RTD curves E() and F() for a CSTR and a PFR in series. Nex, consider a 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 τ p seconds laer, meaning ha he RTD of he reacor sysem will again be E() is he same no maer which reacor comes firs. E() e ( p) s τ s τ p τ p (16-51) 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 16-3.

26 Fogler_Ch16.fm Page 5 Tuesday, March 14, 17 6:4 PM Secion 16.5 PFR/CSTR Series RTD 16-5 Example 16 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 (Figure E16-3.1); in he second sysem he PFR precedes he CSTR (Figure E16-3.). To simplify he calculaions, le τ s and τ 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. For he parameers given, we noe ha in hese wo arrangemens (see Figures E and E16-3.), he RTD funcion, E(), is he same E() E() F() 1. τ p = τ s = 1 min τk = 1 m 3 /kmol and C A = 1 kmol/m 3 Soluion (a) Le s firs consider he case of early mixing when he CSTR is followed by he plug-flow secion (Figure E16-3.1). C A C Ai C A A mole balance on he CSTR secion gives Rearrranging ( C A C Ai ) kc Ai V (E16-3.1) Dividing by and rearranging, we have quadric equaion o solve for he inermediae concenraion C Ai τ s k C Ai C A Subsiuing for τ s and k Solving for C Ai gives Figure E F A r Ai Early mixing scheme. F V Ai ( C A C Ai ) C Ai kc Ai τ s k 1 min m 3 kmol m min kmol 1 4τ C s kc A Ai kmol/m τ s k 3 This concenraion will be fed ino he PFR. The PFR mole balance is df A dc A dc dv A r dv dτ A kc A p (E16-3.) (E16-3.3)

27 Fogler_Ch16.fm Page 6 Tuesday, March 14, 17 6:4 PM 16-6 Residence Time Disribuions of Chemical Reacors Chaper 16 Inegraing Equaion (16-3.3) τ p k C A C Ai (E16-3.4) CSTR PFR X.618 Subsiuing C Ai.618 kmol/m 3, τ p 1 min, k 1 m 3 /kmol/min and τ p k 1 m 3 /kmol in Equaion (E16-3.4) yields C A.38 kmol/m 3 as he concenraion of reacan in he effluen from he reacion sysem. The conversion is C X A C A C A 1.38 X (b) Now, le s consider he case of lae mixing. C Ai C A Figure E16-3. Lae mixing scheme. When he perfecly mixed secion is preceded by he plug-flow secion (Figure E16-3.), he oule of he PFR is he inle o he CSTR, C Ai. Again, solving Equaion (E16-3.3) τ p k C Ai Solving for inermediae concenraion, C Ai, given τ p k 1 m 3 /mol and a C A 1 mol/m 3 C Ai.5 kmol/m 3 C A C A PFR CSTR X.634 Early mixing X =.618 Lae mixing X.634 Nex, solve for C A exiing he CSTR. A maerial balance on he perfecly mixed secion (CSTR) gives τ s k C A C A C Ai (E16-3.5) 1 4τ C A s kc Ai kmol/m (E16-3.6) τ s k 3 as he concenraion of reacan in he effluen from he reacion sysem. The corresponding conversion is 63.4%; ha is, X 1 ( C A C A ) %. 1. Analysis: The RTD curves are idenical for boh configuraions. However, he conversion was no he same. In he firs configuraion, a conversion of 61.8% was obained; in he second configuraion, 63.4%. While he difference in he conversions is small for he parameer values chosen, he poin is ha here is a difference. Le me say ha again, he poin is here is a difference and we will explore i furher in Chapers 17 and 18.

28 Fogler_Ch16.fm Page 7 Tuesday, March 14, 17 6:4 PM Secion 16.6 Diagnosics and Troubleshooing 16-7 While E() was he same for boh reacion sysems, he conversion was no. Chance Card: Do no pass go, proceed direcly o Chaper 17. GO 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 or given reacor. However, as we jus saw, 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 in Chaper 17 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. A his poin, he reader has he necessary background o go direcly o Chaper 17 where we use he RTD o calculae he mean conversion in a real reacor using differen models of ideal chemical reacors Diagnosics and Troubleshooing General Commens As discussed in Secion 16.1, he RTD can be used o diagnose problems in exising reacors. As we will see in furher deail in Chaper 18, he RTD funcions E() and F() can be used o model he real reacor as combinaions of ideal reacors. Figure 16-1 illusraes ypical RTDs resuling from differen nonideal reacor siuaions. Figures 16-1(a) and (b) correspond o nearly ideal PFRs Ideal RTDs ha are commonly observed E() Acual E() Acual Ideal τ (a) (b) Channeling E() Channeling Dead Zone z = Dead Zones (c) z = L τ (d) Channeling Bypassing E() Long ail dead zone τ (f) Dead Zones (e) Figure 16-1 (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 in and ou.