CHEMICAL ENGINEEERING AND CHEMICAL PROCESS TECHNOLOGY Vol. III - Nonideal Flow Models in Homogeneous Chemical Reactors - A.

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1 Homogeneous hemical Reacors -. Burghard NONIDEL FLOW MODELS IN HOMOGENEOUS HEMIL RETORS. Burghard Insiue of hemical Engineering, Polish cademy of Sciences, Poland Keywords: Residence ime disribuion (RTD), momens of RTD, experimenal esimaion of RTD, RTD of ideal and nonideal flow models, axially dispersed plug flow model, anks in series model, muliparameer model, dispersion coefficien, microfluid, macrofluid, earliness and laeness of mixing, conversion in nonideal flows. onens 1. Inroducion 2. The Residence Time Disribuion (RTD) 2.1. The RTD Funcions of Ideal Flows 3. Models of Nonideal Flows 3.1 The xially Dispersed Plug Flow Model orrelaions for xial Dispersion oefficiens 3.2 The Tanks in Series Model 3.3. The Muliparameer Models 4. Influence of Mixing Phenomena on hemical Reacions 4.1 The Limiing ases of onversion. 5. Example Glossary Bibliography Biographical Skech Summary For an accurae analysis of he performance of a chemical reacor he following relaionships and funcions are necessary: reacion raes defining he chemical kineics, he RTD funcion of he exi sream from he reacor, he sae of aggregaion of he fluid (microfluid or macrofluid) and is endency o coalescence and redispersion while flowing hrough he reacor as well as earliness and laeness a which mixing occurs and is influence on he final conversion degree. These problems are he subjec of research of he discipline named chemical reacion engineering an inensively developing branch of he chemical engineering. In his chaper he funcions deermining he RTD (The densiy of RTD E() and he cumulaive RTD F() ) have been defined and he mehod of heir experimenal esimaion presened. The role of momens in comparing differen RTD curves has been discussed. The nonideal flow models like axially dispersed plug flow model, he anks in series model and he muliparameer models, which enable he esimaion of RTD funcions were developed. Special aenion has been devoed o explaining he concep of he sae of aggregaion of he fluid by inroducing wo limiing cases, i.e. he microfluid and he macrofluid and he mechanisms of inermixing ascribed o hem. Finally, he mehod of calculaing he conversion in he exi sream from he reacor has Encyclopedia of Life Suppor Sysems (EOLSS)

2 Homogeneous hemical Reacors -. Burghard been demonsraed by using he relaionships and funcions quoed above. 1. Inroducion The ideal flow paerns, i.e. he plug flow and he ideal mixed flow (discussed in Ideal Models of Reacors) are widely acceped as very useful and effecive ools in he analysis, design and operaion of chemical reacors. In he plug flow he fluid elemens which enered he reacor vessel a he same momen are moving along parallel sreamlines wih he same velociy. Thus, he residence ime of all fluid elemens in he reacor is idenical. On he oher hand in he mixed flow he mixing phenomena are assumed o be exremely inensive ensuring a complee uniformiy of he composiion of he fluid and is emperaure in he enire vessel. These wo ideal flow paerns considerably faciliae he analysis and design of chemical reacors offering sraighforward mehods of selecing he righ ype of reacor. However, in real reacors cases are ofen encounered where he flow paern deviaes from eiher of he wo limiing ideal flows, and heir applicaion in he analysis of chemical reacors can lead o subsanial errors. Generally speaking, hese deviaions can depend on he reacor ype, kind of fluid and he characerisics of flow, and are usually caused by he exising sagnan regions in he vessel, by recirculaing and bypassing he fluid as well as by mixing phenomena generaed by he flow iself (urbulen mixing). 2. The Residence Time Disribuion (RTD) From he heoreical poin of view he quaniaive descripion of he reacion processes occurring in he reacor could be deermined as a resul of he simulaneous analysis of he mass, momenum and energy balance equaions. The soluion of hese differenial equaions would lead o he esimaion of he velociy profile and he inensiy of mixing phenomena in he reacor. These quaniies would hen allow us o deermine he influence of he hydrodynamic behavior of he fluid on he performance of he chemical reacor wih a nonideal flow paern. Unforunaely such a procedure is in he majoriy of cases impossible and impracical, even wih presen-day compuers and available numerical mehods. much simpler approach sems from he fac ha he basic informaion whose knowledge is indispensable for deermining he final conversion of a fluid elemen is is residence ime in he reacor. Obviously, he residence ime of various fluid elemens which enered he vessel a he same momen = will no be he same in a nonideal flow. Therefore, in order o analyze hese problems a powerful concep he residence ime disribuion of fluid elemens in he reacor- has been inroduced. This disribuion is characerisic of he reacor ype and of he flow paern and enables he esimaion of he conversion degree of he reacing mixure leaving he reacor. The concep of a fluid elemen, inroduced by Danckwers is a volume of fluid whose size is very small in comparison o he size of he reacor vessel, bu conains a sufficien number of molecules o form a coninuous sysem. Encyclopedia of Life Suppor Sysems (EOLSS)

3 Homogeneous hemical Reacors -. Burghard The definiions of he RTD funcions have been aken from he probabiliy heory. ccording o his heory he funcion E() called he densiy of he RTD deermines he value E()d, i.e. a fracion of he fluid which enered he vessel a he momen = and leaves i beween and + d, so fracion of fluid leaving E()d = he vessel ha has a residence (1) ime (, + d) This value can also be inerpreed as he probabiliy ha a fluid elemen which enered he reacor a = will leave i a he ime, + d. Such a definiion of he funcion E() leads o he following normalizing relaionship E()d = 1 noher kind of funcion characerizing he fluid sream leaving he vessel is he cumulaive RTD F(). The value of his funcion for he ime deermines he fracion of he fluid F() which sayed in he reacor for he ime shorer han. The relaionship beween hese wo funcions can be obained from heir definiions, i.e. F() = E()d (3) Hence, he funcion F() akes he following limiing values: F() = (4) lim F() = 1 (5) Moreover, from he relaionship (3) i follows ha df() = E() (6) d (2) Figures 1 and 2 show he examples of RTD funcions E() and F() for he plug flow, ideal mixed flow and nonideal flow. I has o be noed ha he RTD funcions E() and F() are valid only for he so called closed closed vessel where plug flow has been assumed in boh he inle ube and he oule ube of he vessel. Very useful in characerizing and comparing he RTD funcions are he momens of Encyclopedia of Life Suppor Sysems (EOLSS)

4 Homogeneous hemical Reacors -. Burghard hese disribuions given by he relaion μ n, n E()d n 1,2...n (7) = = The firs wo momens are usually used for he evaluaion of he RTD funcions. The higher momens like he skewness esimaed from experimenal RTD curves comprise large errors and herefore are rarely applied in he analysis of he RTD. Figure 1. Examples of he funcion E() for various flow paerns. Figure 2. Examples of he funcion F() for various flow paerns. The firs momen equals he mean residence ime in a closed vessel, ha means: μ V = E() = = (8) 1, * V n imporan feaure of he RTD very useful in is evaluaion is he widh of he disribuion curve. This value is undoubedly a measure of he inensiy of he mixing phenomena. ccording o he probabiliy heory his widh can be bes defined by he Encyclopedia of Life Suppor Sysems (EOLSS)

5 Homogeneous hemical Reacors -. Burghard cenral second momen wih respec o he mean residence ime called variance. Thus σ = ( ) E()d = E()d (9) In comparing differen RTD curves i proved convenien o inroduce ino he foregoing equaions he dimensionless ime, defined as Θ= (1) Based on he equaliy E()d = E( Θ)dΘ (11) one obains he following dependence E and ( ) Θ = E() (12) Θ ( ) ( ) (13) F() = E()d = E Θ dθ= F Θ Using he dimensionless ime he momens (8) and (9) may be wrien as follows: ( ) μ1, Θ = Θ E Θ d Θ= 1 (14) and σ 2 2 σ 2 Θ 2 2 ( ) ( ) ( ) (15) = = Θ 1 E Θ dθ= Θ E Θ dθ 1 special echnique has been developed for deermining experimenally he RTD curves called he simulus response echnique or he racer echnique. In his echnique a fixed amoun of a racer subsance is inroduced ino he fluid sream a he inle ino he vessel by means of a sricly specified mode. Monioring he concenraion of he racer as a funcion of ime a he oule from he reacor allows us o deermine he RTD curve. The appropriaely seleced racer should no ake par in any process during he flow hrough he vessel which could cause is consumpion (like chemical reacion, absorpion or adsorpion). Moreover, is amoun should no disurb he paern of flow and change he physical properies of he fluid. Encyclopedia of Life Suppor Sysems (EOLSS)

6 Homogeneous hemical Reacors -. Burghard The wo following mehods of inroducing he racer are generally used in experimens. The firs one is he sep funcion where a = he inflowing concenraion is changed from one value, usually zero, o anoher one. The second is a pulse, where a specified amoun of racer is injeced in a shores possible ime ino he inflowing fluid. Le us assume ha in a general case he racer is inroduced in he form of an exacly defined funcion (). The funcion () is ransformed during he flow hrough he vessel according o he RTD funcion E() characerisic of he flow paern in he vessel. The moniored concenraion a he oule is hen given by () (Figure 3). Figure 3. Scheme of a general mehod of developing he RTD funcion The fluid elemen leaving he vessel a ime (recangle B) comprises he fracions deermined by he RTD funcion E() of all fluid elemens which enered he vessel over he ime period [, ]. Therefore, in order o deermine he concenraion () of he fluid elemen leaving he vessel we have o add he fracions of all fluid elemens which enered ino he vessel a ime - and sayed in he vessel for he ime (all recangles ). This sum will be expressed in he limi by means of he convoluion inegral giving finally he desired relaionship beween he concenraions () = ( ) ( ) (16) E d n equivalen form of his relaionship is (17) () = ( ) ( ) E d and applying he usual mahemaical noaion for he convoluion inegral one obains Encyclopedia of Life Suppor Sysems (EOLSS)

7 Homogeneous hemical Reacors -. Burghard = E (18) = E (19) The simples mahemaical expressions allowing us he experimenal deerminaion of he RTD funcions are obained by simulaing he inle racer injecions in he form of he Dirac δ () funcion for he pulse and he Heaviside sep funcion 1(). The firs funcion is defined as follows, for < δ ( ) =, for =, for > and he second funcion as, for < 1() = 1, for > In he firs case a fixed amoun of racer n is injeced ino he inle sream of he reacor in an infiniely shor ime, hence n ( ) = ( ) * V δ (22) Subsiuing his equaliy ino he Eq.(17) one ges: * n () = ( ) E( ) d V δ (23) (2) (21) During inegraion, he funcion δ ( ) selecs only he value of E( - ) for =, so n () = E() (24) * V n Hence, based on he moniored oule concenraion () for he simulus ( ) * V δ he E() funcion is obained as * V E() = () (25) n or, in an equivalen form Encyclopedia of Life Suppor Sysems (EOLSS)

8 Homogeneous hemical Reacors -. Burghard 1V E() = () n (26) and in a dimensionless form V E() = E( Θ ) = () (27) n n The quaniy V can be reaed as he iniial concenraion formed as a resul of an immediae uniform dispersion of he racer amoun n in he whole volume V of he vessel. Thus, E( Θ ) = () pplying he simulus in he form of he Heaviside sep funcion he inle concenraion is changed from zero o he value a he ime = ( ) ( ) = 1 (29) and from (17) one obains () = ( ) ( ) = () = () 1 E d E d F and, finally () () F (28) (3) () E d (31) = = n exac esimaion of he amoun n injeced ino he inflowing sream in a shores possible ime may be difficul. This value can hen be esimaed by monioring he concenraion () in he oule sream unil is disappearance, i.e. heoreically for an infiniely long ime * () n = V d (32) Subsiuing Eq.(32) ino Eq.(25) gives Encyclopedia of Life Suppor Sysems (EOLSS)

9 Homogeneous hemical Reacors -. Burghard () E = () () d Especially difficul is he esimaion of he E() funcion by applying an arbirary inle signal () as, in his case, E() is presen implicily in he convoluion inegral (17). In his siuaion i is mos convenien o apply he Laplace ransformaion of he convoluion simulaneously wih Borel s heorem. ccording o his heorem he Laplace ransform of he convoluion is equal o he produc of he Laplace ransforms of he original funcions forming his convoluion ( ) ( ) ( ) s = s E s (34) and hence ( ) E s = ( s) ( s) leading o he RTD funcion as a Laplace ransform E(s). The nex sep is he reverse ransformaion of E(s) o obain he original funcion E() Bibliography TO ESS LL THE 44 PGES OF THIS HPTER, Visi: hp:// (33) (35) Burghard. and Lipowska L. (1973). Mixing phenomena in a coninuous ank reacor. [Review and generalizaion of heoreical flow models. Inernaional hemical Engineering 13, (227)]. Fromen G.F. and Bischoff K.B. (199). hemical Reacion nalysis and Design. Second Ed. New York: John Wiley&Sons. [In his book a special emphasis has been placed on he descripion of nonideal flow paerns and he developmen of models accouning for hese paerns, hus leading o he esimaion of he residence ime disribuion funcion]. Levenspiel O. (1999). hemical Reacion Engineering. Third Ed. New York: John Wiley&Sons. [This book conains he chaper Basics of Nonideal Flows where he ideas of he residence ime disribuion, he sae of aggregaion of he fluid as well as he conceps of earliness and laeness of mixing of he fluid have been discussed in a deailed and insrucive manner]. Van der Laan Th. (1958). Leer o he Ediors on Noes on he diffusion ype model for he longiudinal Encyclopedia of Life Suppor Sysems (EOLSS)

10 Homogeneous hemical Reacors -. Burghard mixing in flow. hemical Engineering Science 7, (187). [In his sudy an experimenal sysem has been presened which forms he basis for analyzing a number of differen experimenal insallaions used in racer echnique. The firs momen and variance were developed for welve differen ypes of experimenal sysems]. Villermaux J. van Swaaij W.P.M. (1969). Model of he residence ime disribuion in a semi-infinie reacor wih axially dispersed pison flow and exchange of mass wih sagnan zones. pplicaion o he liquid phase for rickle flow in a column packed wih Rasching rings. hemical Engineering Science 24, (197).[ model has been developed in which he flow in he acive region is a dispersed plug flow and he sagnan region presen in he vessel is ex- changing mass of racer wih he main flow.] Wen.Y., Fan L.T. (1975). Models for Flow Sysems and hemical Reacors. New York: cademic Press. [I presens he models of nonideal flows and he residence ime disribuion funcions associaed wih hese models]. Wesererp K.R., van Swaaij P.M. and Beenackes...M. (1984). hemical Reacor Design and Operaion. Second Ed. John Wiley&Sons [ chaper in his book presens a deailed discussion of he influence of he residence ime disribuion on he performance of chemical reacors by inroducing he concep of wo kinds of fluids-microfluid and macrofluid]. Biographical Skech ndrzej Burghard was born in 1928 in Warsaw, Poland. He received his B.Sc. in 1952 and finally graduaed in hemical Engineering from he Silesian Technical Universiy, Gliwice, Poland obaining M.Sc. degree. In 1962 he received his Ph.D. while in 1965 he degree Docor in Technical Sciences from he same universiy. In 1971 he was awarded a ile of Professor in Technical Sciences by he Presiden of he Republic of Poland while some years laer, in 1983, he was eleced as a member of he Polish cademy of Sciences. In he years he was a chairman of he ommiee of hemical Engineering a he Polish cademy of Sciences. urrenly he is a member of he Presidium of he Polish cademy of Sciences and of he Scienific ouncil a he Minisry of Science and Informaics. He is also an acive member of he Working Pary on hemical Reacion Engineering of he European Federaion of hemical Engineering. In he years he was engaged a he Silesian Technical Universiy, Gliwice, Poland as an ssisan Professor lecuring in Disillaion, Mass Transfer and hemical Reacion Engineering. Then, in 1966, he moved o he Insiue of hemical Engineering of he Polish cademy of Sciences where he was designaed o he pos of direcor holding his pos unil 22. His field of research focuses mainly on he heory of mass ransfer in mulicomponen and muliphase sysems as well as on chemical reacion engineering wih special ineres in dynamics of chemical reacors. He is auhor and co-auhor of nearly 2 scienific publicaions in refereed journals and of following books and chapers in collecive ediions: hemical Reacors Engineering Vol. I and II, Examples of Reacor Design, Dynamics of Processes in a Porous aalys Pelle, Mass Transfer in Mulicomponen Sysems, ondensaion of Mulicomponen Mixures in VDI-Wärmealas (Three Ed. in German one in English). He is also a member of Ediorial Boards in several journals. Encyclopedia of Life Suppor Sysems (EOLSS)