Retrspective Theses and Dissertatins 1970 Variable-vlume peratin f a stirred tank reactr Mnty Marvin Lund Iwa State University Fllw this and additinal wrks at: http://lib.dr.iastate.edu/rtd Part f the Chemical Engineering Cmmns Recmmended Citatin Lund, Mnty Marvin, "Variable-vlume peratin f a stirred tank reactr " (1970). Retrspective Theses and Dissertatins. 4246. http://lib.dr.iastate.edu/rtd/4246 This Dissertatin is brught t yu fr free and pen access by Iwa State University Digital Repsitry. It has been accepted fr inclusin in Retrspective Theses and Dissertatins by an authrized administratr f Iwa State University Digital Repsitry. Fr mre infrmatin, please cntact digirep@iastate.edu.
70-25,803 LUND, Mnty Marvin, 1943- VARIABLE-VOLUME OPERATION OF A STIRRED TANK REACTOR. Iwa State University, Ph.D., 1970 Engineering, chemical University Micrfilms, A XEROK Cmpany, Ann Arbr, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED
VARIABLE-VOLUME OPERATION OF A STIRRED TANK REACTOR by Mnty Marvi n Lund A Dissertatin Submitted t the Graduate Faculty in Partial Fulfillment The Requirements fr the Deg-ee f DOCTOR OF PHILOSOPHY Majr Subject; Chemical Engineering Apprved: Signature was redacted fr privacy. In Charge f Majr Wrk Signature was redacted fr privacy. Head f Majr Depar'n^ent Signature was redacted fr privacy. lvia State Univers i ty Of Science and Technlgy Ames, Iwa 1970
TABLE OF CONTENTS Page NOMENCLATURE INTRODUCTION iv ' The Stirred Tank Reactr 1 Variable-Vlume Operatin 2 Purpse 3 Types f Variable-Vlume Operatin 3 Reactin Systems Studied 4 Slutins t the Mathematical Mdels k LITERATURE REVIEW 6 The Dynamics f Thermally Sensitive Reactrs 7 Reactr Mdels fr Finite Mixing 9 Yield and Selectivity fr Cmplex Chemical Reactins 12 Peridic Operatin f Chemical Reactrs 15 THE GENERAL EQUATIONS 19 CONSTANT VOLUME OPERATION 29 Isthermal 29 Adi abat i c 29 Single reactin f arbitrary rder 3' First rder reactins 3& Steady state peratin 3& Transient peratin 43 VARIABLE-VOLUME OPERATION 51 Reactr Filling and Emptying 52 Semi cntinuus Variable-Vlume Operatin 54
I I I Isthermal 70 First rder reactins 72 Effect f relative rate cnstant 81 Effect f flw average reactant cnversin 86 Effect f semibatch cycle parameters 87 The ptimal semibatch cycle 89 General characteristics f semibatch peratin 9' Reversible first rder reactins 92 Secnd rder reactins 93 Single reactins f arbitrary psitive rder 9^ Van De Vusse reactins 99 Adiabatic 108 Cntinuus Variable-Vlume Operatin 130 CONCLUSIONS 137 General 137 Isthermal 137 First rder reactins 137 Single reactins with arbitrary psitive rder 139 Van De Vusse reactins 139 Adiabatic 139 RECOMMENDATIONS 141 LITERATURE CITED 142 ACKNOWLEDGMENTS ^^5
i V NOMENCLATURE Lat i n A Heat transfer area in Equatin 3 b Cj C^^ Cj Prprtinality cnstant in Equatin 1 la Cncentratin f cmpnent j Cncentratin f cmpnent A in the feed Dimensinless cncentratin f cmpnent }, Cj/C^^ C".' Cp Cpg E Dimensinless flw average cncentratin f cmpnent j in the discharge stream Heat capacity f reactr cntents Heat capacity f clant Activatin energy in Equatin lib g(9j9') Residence time frequency functin (-AH.) (-6H?) Heat f reactin fr ith reactin Dimensinless heat f reactin fr ith reactin, (-AHj)/() K Reactin rate cnstant Reactin rate cnstant at feed temperature K L L' M N P Dimensinless reactin rate cnstant, K/K^ Dimensinless linear reactin rate parameter Dimensinless Arrhenius reactin rate parameter Dimensinless rcnctr heat transfer parameter Dimensinless culer heat transfer parameter Dimensinless re t at i ve rate cnstant 0 Discharge flw rate
Q,^ Feed flw rate Q, Feed and discharge flw rate fr the reference reactr K /V 0, Dimensinless discharge flw rate, Q.^ Q. ^ 0,^1^ R R g r r Dimensinless feed flw rate, Dimensinless discharge flw rate during k fractin f the semicntinuus cycle Dimensinless feed flw rate during k fractin f the semi cntinuus cycle Dimensinless relative thermal energy Ideal gas law cnstant Reactin rate Dimensinless reactin rate S(8, 0') Residence time distributin functin T Temperature Feed temperature T Tj. Dimensinless temperature, T/T^ Cler temperature Cler feed temperature t U V v., Vj, V V Ti me Heat transfer cefficient VIume Maximum vlume Cler vlume Minimum vlume Dimensinless vlume
vi Cnversin f cmpnent A Flw average cnversin f cmpnent A in the discharge stream Ampli tude Stichimetric cefficient fr the jth species in the ith reactin Relative yield f prduct when nly ne prduct Relative yield f prduct j Dimensinless time, t Q^/V^ Dimensînless time at end f k fractin f the semicntinuus cycle, tk 0%/%% Dimensinless time at beginning f semicntinuus cycle. Dens i ty Clant density The k fractin f the semicntinuus cycle defined by Equatin 46 Dimensinless average re si ce ice time defined by Equatin 51 Dimensinless flw average residence time defined by Equatin 55 Dimensinless time interval defined by Equatin 46 Phase lag Relative yield Frequency
vi i Subscripts i Refers t i reactin: i = i, 2, 3j j Refers t j cmpnent; j = A, B, C, D k Refers t k fractin f the semi cntinuus cycle k = F, B, E, D F - Filling B - Batch E - Emptying D - Dwn
1 INTRODUCTION The Stirred Tank Reactr The stirred tank reactr is a device used t prcess raw materials, usually in liquid frm, t desired prduct materials by means f a chemical reactin. Generally, the stirred tank reactr is a cylindrical vessel with a diameter apprximately equal t its length and an impeller t prvide stirring actin The majr distinguishing characteristic f the stirred tank reactr is the level f mixing caused by the stirring actin. At ne cnceptual extreme the reactr is "perfectly mixed" which means that the reactr cntents are unifrm in temperature and cncentratin and that feed material, upn entering the reactr is immediately dispersed thrughut the reactr, and the feed cncentratin and temperature drp instantaneusly t the cncentratin and temperature f the reactr cntents. Cnsequently, prduct material discharged frm the perfectly mixed reactr has the same cncentratin and temperature as the reactr cntents. At the ther extreme, the reactr is nt stirred and there exists a zer level f mixing. This means that there is n mixing f reactants f different levels f cncentratins in the reactr. The batch reactr is a tank reactr that has a zer level f mixing. The batch reactr is assumed t be initially charged with reactant f unifrm cncentratin and temperature. Fresh feed is nt intrduced int the batch reactr during the reactin time, and thus there is n mixing f reactants with different levels f cncentratin. The batch reactr is mathematically equivalent t a steady state perated plug flw reactr with the same residence time. The plug flw reactr is a tubular
2 reactr in which material flws thrugh the reactr as a plug with n backmixing r dispersin alng the length f the reactr. In this thesis the acrnym STR is used t dente the stirred tank reactr. In general, the STR may be perated with feed and discharge streams that are cntinuus r discntinuus peridic functins f time. The acrnym CSTR is used thrughut t dente the cntinuus stirred tank reactr fr which the feed and discharge streams are cnstant at the identical vlumetric flw rate. When used withut any mdifiers the acrnym CSTR dentes steady state peratin f the cntinuus stirred tank reactr. Variable-Vlume Operatin Variable-vlume peratin f the STR is a prcess which uses peridic feed and discharge flw rates as the driving frce t cause a peridically varying vlume and in turn, a peridic residence time. The resulting cncentratin and temperature variatins in the reactr are als peridic functins f time. Thus, the variable-vlume peratin f the STR is a special case f peridic peratin f the STR. In general, peridic peratin f the STR implies utputs frm the reactr which are peridic functins f time. The STR may be frced t generate peridic utputs by frcing the reactr with peridic inputs such as feed rate, feed cncentratin, r clant flw rate, r the STR may generate peridic utputs when the inputs are cnstant, if the parameters are such that a stable limit cycle exists abut an unstable steady state pi nt. The study f peridic peratin f the STR is a recent develpment in
3 the field f chemical reactr design. In the past, steady state peratin f the CSTR has been cnsidered t be the mst desirable. Hwever, it has been fund that sme types f peridic peratin f the STR may result in imprved perfrmance frm the standpint f increasing prfit, prductin rate, r yield» Purpse The purpse f this research is t shw that variable-vlume peratin f the STR can be used t increase the prductin rate f the STR relative t the prductin rate f the CSTR. Relative prductin rate is measured by relative yield r relative thrughput. Relative yield is defined as the rati f average yield by variable-vlume peratin f the STR t yield by the CSTR where bth reactrs have the same average thrughput f material and the same maximum vlume. Relative thrughput is defined as the rati f the average thrughput by variable-vlume peratin f the STR t the cnstant thrughput f the CSTR where bth reactrs have the same average yield and the same maximum vlume. Relative yield and relative thrughput are nt independent. They are simply tw different methds fr measuring relative prductin rate. When the relative yield and the relative thrughput are greater than unity, the variablevlume peratin is cnsidered an imprvement ver the perfrmance f the CSTR. Types f Variable-Vlume Operatin In this wrk tw general types f variable-vlume peratin f the STR are studied. These are semi cntinuus variable-vlume peratin and \
k cntinuus variable-vlume peratin. Semibatch peratin is a special case f semi cntinuus variable-vlume peratin that is studied in detai 1. Semi cntinuus variable-vlume peratin emplys feed and discharge flw rates that are peridic functins f time and are discntinuus at different pints during the semicntinuus cycle. Semibatch peratin is a simple special case f semi cntinuus variable-vlume peratin in that semibatch peratin emplys a feed flw rate nly during the filling fractin f the semibatch cycle, and a discharge flw rate nly during the emptying fractin f the semibatch cycle, whereas feed and discharge flw rates may be present thrughut the semi cntinuus cycle. The cntinuus variable-vlume peratin emplys feed and discharge flw rates that are cntinuus peridic functins f time. Reactin Systems Studied Variable-vlume peratin f the STR was studied fr varius reactin schemes. These are the single reactin f psitive rder in the isthermal STR, the first rder irreversible and exthermic reactin in the adiabatic STR, and the Van De Vusse reactins in the isthermal STR. The Van De Vusse reactins are an example f a reactin system cmpsed f cnsecutive reactins with higher rder side reactins. Mst f the analysis f the different reactin schemes is presented fr the case f semibatch peratin f the STR. Slutins t the Mathematical Mdels This thesis presents a theretical analysis f variable-vlume
5 peratin f the STR. Mathematical mdels are presented fr the varius cases f variable-vlume peratin f the STR Fr many cases where an analytical slutin is nt tractable, extensive use is made f the analg and digital cmputer t slve the systems f equatins which describe the mathematical mdels.
6 LITERATURE REVIEW In recent years interest in the peridic peratin f a chemical reactr as a means t imprve yield ver the ptimum steady state has been increasing rapidly. Peridic peratin refers t the situatin in which sme f the utput r input variables f a reactr are peridic functins f tim. The variable-vlume STR is a special case f the peridic peratin f a chemical reactr. Theretical analysis f the peridic peratin f a reactr has been applied t the CSTR and t the plug flw reactr prcessing a liquid which underges a hmgeneus liquid phase reactin. Bth the isthermal and nnisthermal cases have been cnsidered. The chemical kinetics cnsidered have been limited mainly t first r secnd rder isthermal r exthermic reactins» This research wrk is cncerned in general, with the nn:sthermal variable-vlume STR wherein a hmgeneus liquid phase exthermic reactin is ccurring» The variable-vlume STR gives results fr reactant cnversin which usually fall between the steady state CSTR cnversin and the batch rcactr r plug flw reactr cnversin with the same residence time and thus might be mdeled by a reactr with a finite mixing level. Results are btained fr varius types f reactin kinetics. Therefre, publicatins are reviewed which cncern: 1) the dynamics f thermally sensitive reactrs, 2) reactr mdels fr finite mixing, 3) yield and selectivity fr cmplex chemical reactins, and 4) peridic peratin f chemical reactrs.
7 The Dynamics f Thermally Sensitive Reactrs Van Heerden (33) prbably gave the first treatment f autthermic prcesses in the CSTR. He shws that these prcesses may be characterized by a heat versus temperature diagram n which the inlersectin f the heat generatin curve and the heat cnsumptin curve give the steady state perating pints. Van Heerden shws that fr the single exthermic reactin there are frm ne t three steady state perating pints. In the case f three steady states, the lw and high temperature steady states satisfy the necessary cnditins fr stability, while the intermediate steady state is necessarily unstable» Van Heerden called this unstable steady state the ignitin pint, since a slight increase in temperature will cause the reactin t prceed t the higher temperature steady state. In a later paper, Van Heerden (34) explains that the pssibility f mre than ne steady state can be ascribed t the backmixing f heat alng the reactin path. In rder t gain insight int the cnditins under which an exthermic prcess may have mre than ne steady state. Van Heerden perfrmed calculatins t shw the regins f three pssible steady states fr the cases f a first rder exthermic reactin in I) an adiabatic CSTR, 2) an adiabatic plug flw reactr with heat exchange between inlet and utlet, and 3) an adiabatic plug flw reactr with axial cnductin f heat. After Van Heerden's (33) first paper, Bilus and Amundsn (5) extended his wrk. They determined the shape f the heat generatin curve in the (heat, temperature) plane fr sme cmplex types f chemical reactins and shwed that smetimes mre than three steady states are pssible, sme f which are necessarily unstable. The transient equatins were
8 linearized and used t btain necessary and sufficient cnditins fr the stability f a steady state in terms f the steady state cnditins. (Cnversin, temperature) phase plane plts were made by simulating the nnlinear transient equatins n the analg cmputer. These plts shw the path f apprach t the stable states in the (cnversin-temperature) plane. Bilus and Amundsn (6) als develped an analytical methd fr predicting regins f parametric sensitivity in a tubular reactr. Parametric sensitivity refers t the fact that under the prper cnditins the thermal behavir f a tubular reactr is extremely sensitive t small changes in the perating variables» The tubular reactr des nt exhibit the type f unstable behavir fund in the CSTR. The wrk f Bilus and Amundsn (5) is extended by Aris and Amundsn (3) wh give an analysis f the stability and cntrl f the nnisthermal CSTR in which a single exthermic reactin is ccurring- The analysis is limited t lcal cntrl, since the transient equatins are linearized abut a steady state perating pint and are therefre valid fr nly small perturbatins in the inputs. Cntrl f a steady state by the three ideal mdes f cntrl and with temperature r cncentratin as the cntrl variable is discussed. It is established that an uncntrlled unstable steady state can always be made stable by the use f ideal prprtinal cntrl, if temperature is the cntrl variable. Calculatins and (cnversin, temperature) phase plane plts are presented which shw the evlutin f an uncntrlled unstable steady state t a cntrlled stable steady state by the applicatin f increasing amunts f ideal prprtinal cntrl. Kermde and Stevens (22) used the rt lcus methd t determine the amunt
9 f the ideal mdes f cntrl necessary t make an uncntrlled steady state becme stable. The rt lcus methd was applied t the linearized equatins fr the nnisthermal CSTR with a first rder exthermic reactin» The results frm the rt lcus methd agreed well with results btained by simulatin f the nnlinear equatins n the analg cmputer» Luus and Lapidus (28) develped an averaging technique which culd be used t determine the stability f secnd rder nnlinear systems» The technique can als be used t prve the existence r nnexistence f limit cycles and t shw hw fast a phase pint in the (cncentratin, temperature) phase plane appraches a singular pint r a limit cycle» The technique was successfully applied t the analysis and cntrl f a CSTR which was disturbed by sme scillating Input variable. Reactr Mdels fr Finite Mixing Levenspiel and Bischff (25) discuss the effect f backmixing n the mlecular scale n cnversin In an isthermal chemical reactr. Plts are presented which shw the rati f the residence time in the perfectly mixed r CSTR t the residence time In the unmixed r plug flw reactr versus cnversin» Except fr zer rder reactins, the CSTR always requires a larger residence time than the plug flw reactr» The effect f backmixing becmes Increasingly Imprtant fr higher rder reactins. The reactr lngitudinal dispersin number, Du/L, is presented as a means f determining cnversin at Intermediate r finite levels f mixing» Denbigh (11) shws that the decrease In cnversin caused by backmixing in an isthermal reactr Is due t a decrease In the average reactin ratec
10 Levenspiel and Bischff (2b) describe theretically the phenmenn f finite mixing r nnideal flw by such mathematical mdels as dispersin mdels, tanks in series mdels, and cmbined reactr types mdels. The residence time distributin and the internal age distributin are defined and experimental methds fr their determinatin are presented. Nauman (30) develps a thery fr residence time distributin functins in an unsteady state perfectly mixed reactr. In the unsteady state reactr there is a different residence time distributin fr each pint in time. Danckwerts (10) shws that a knwledge f the residence time distributin is nt sufficient t determine reactr cnversin in a nnlinear reacting system. In additin t the residence time f a mlecule, a knwledge f the behavir f the mlecule and its neighbrs while they reside in the reactr is required. This behavir f the mlecules may be described by their mde f dispersin. Danckwerts intrduced the cncept f segregatin t characterize the mde f dispersin. In a cmpletely segregated fluid the mlecules are assumed t pass thrugh the reactr in finite packets f small vlume cmpered t the reactr, but large enugh t cntain many mlecules. a minute batch reactr. Each little packet passes thrugh the reactr as In the nnsegregated fluid the mlecules are dispersed n a mlecular scale. Fluid in a CSTR is usually thught f as being nnsegregated unless specified therwise. Fluids whse mde f dispersin is between the cmpletely segregated and nnsegregated fluid are said t be partially segregated. Levenspiel {2k, p. 310) calls the cmpletely segregated fluid a macrfluid and the nnsegregated fluid a micrfluid. Danckwerts (10) shwed that the average reactin rate, and thus cnversin, was higher In a cmpletely segregated system if the reactin
11 rder was greater than 1, lwer if less than 1, and equal if the rder was ne» Zwietering (36) expressed mathematically the cnditin f maximum mixedness r nnsegregatin f fluid in a flw system with an arbitrary residence time distributin. Zwietering shwed that the cnversin in a chemical reactr with a knwn residence time distributin must be between the cnversins crrespnding t cmplete segregatin and maximum mixedness f the reacting fluid» Under adiabatic perating cnditins, the cnversin in a plug flw reactr is nt always greater than the cnversin in a CSTR with the same residence time» Depending n the degree f cnversin desired, a CSTR r a CSTR fllwed by a plug flw reactr will require the least residence time fr the desired cnversin. Chlette and Blanchet (9) were prbably the first t study series reactr cmbinatins under adiabatic perating cnditins. They fund that fr isthermal and endthermic reactins the plug flw reactr always gives a greater cnversin than a series cmbinatin f the CSTR and plug flw reactr. Hwever, fr exthermic reactins, a series cmbinatin f a CSTR fllwed by a plug flw reactr was fund t give a higher cnversin than the CSTR r plug flw reactr alne fr high enugh levels f cnversin. Mrever, it was established that there existed an ptimal level f mixing, that is, an ptimal prprtin f cmbined reactr vlume devted t the CSTR, which wuld maximize cnversin. N explanatin was given fr the ccurrence f the ptimum mixing level. Aris (2) gave the general analytical and graphical cnditins fr which the CSTR fllwed in series by a plug flw reactr ptimal and als an analytical expressin fr the ptimum mixing level» it was shwn that the
12 ptimum level f mixing fr the reactr cmbinatin always affrded a stable steady state peratin. Mst reactr mdels t determine cnversin at finite mixing levels have been frmulated in terms f series f CSTRs, cmbinatins f CSTRs and plug flw reactrs, and plug flw reactrs with axial dispersin. Gillespie and Carberry (20) have presented yet anther mdel whereby finite mixing levels are simulated by a plug flw reactr with a recycle stream. With n recycle, plug flw reactr cnversin is achieved. At a recycle rate f 20 r mre times the flw rate f fresh feed, CSTR cnversin is achieved. Intermediate levels f mixing are simulated by recycle rates between 0 and 20. The unique advantage f the recycle mdel is that the recycle is cntained in the bundary cnditin fr the reactr. Whenever the design equatin fr the plug flw reactr can be integrated, the effect f finite mixing can be determined. Gillespie and Carberry (20) shw the relatinship between the recycle rati in their plug flw reactr with recycle finite mixing mdel and the number f CSTRs in series which give equivalent mixing. Gillespie and Carberry (20) als apply their finite mixing mdel t the adiabatic case t shw that finite mixing levels affrd the ptimum cnversin fr sufficiently exthermic reactins. An analytical slutin is pssible whenever an analytical slutin is pssible fr the adiabatic plug flw reactr. Duglas and Eagletn (14) give an analytical slutin t the adiabatic plug flw reactr design equatin in terms f expnential integral functins. Yield and Selectivity fr Cmplex Chemical Reactiiià A cmplex reactin is cnsidered here t be a mixture f high rder
13 reactins, cnsecutive reactins, and cmpeting reactins. Fr systems f cmplex chemical reactins Carberry (7) defines yield at a pint as the rate f generatin f a desired prduct relative t the rate f cnsumptin f a key reactant and selectivity at a pint as the rate f generatin f a desired prduct relative t the rate f generatin f sme undesired prduct. Denbigh (12) defines instantaneus yield in a similar way and als defines an verall reactr yield. Denbigh (12) demnstrates the applicatin f these definitins t the determinatin f yield in tubular and mixed tank reactrs by experimental and analytical methds. Levenspiel (24, Chapter 7) discusses the effect f mixing n the yield and selectivity btainable frm systems f cmplex reactins. Plts shwing yield versus cnversin fr a mixed system f first rder cnsecutive and cmpeting reactin are given It is shwn that fr such a mixed system f reactins the maximum amunt f desired intermediate r the maximum selectivity at a given cnversin is btained when reactin materials are nt mixed at different levels f cnversin. It is als determined that backmixing r mixing at different levels f cnversin is detrimental t the yield f higher rder reactins. Van de Vusse (32) defined yield as B prduced t A fed and selectivity as B prduced t A cnsumed and cmpared yield and selectivity in the isthermal plug flw reactr and CSTR fr the fllwing cmplex reactin scheme. A! > 3 > C S A + A >D
14 Fr this scheme, degredatin f B t C reactr with a shrt residence time. is suppressed by using a plug flw Frmatin f byprduct D is kept lw if the cncentratin f A is lw; hence, a CSTR with a lng residence time is apprpriate. The chice f reactr type and residence time is gverned by the relative values f the rate cnstants. If > K^, the maximum value fr the selectivity is always in the CSTR. The maximum value fr the yield is btained in the plug flw reactr if Kg/K^ s 0.04(K^C^^/K^^ therwise the maximum yield is btained in the CSTR. Gillespie and Carberry (19) shw that an intermediate level f mixing between the CSTR and plug flw reactr may prduce a maximum yield f B fr Van de Vusse's (32) reactin scheme. They shwed that when K^C^^<: K2, all backmixing has a detrimental effect n yield and when sme Intermediate level f mixing will maximize yield. Gillespie and Carberry (20) use the plug flw reactr with recycle mdel fr finite backmixing t determine the effect f finite backmixing n selectivity and yield fr a number f linear and nnlinear yield sensitive isthermal reactin schemes. Carberry (7) extends his previus analysis t the rdiabatic reactr t shvw the effect f cmplete heat backmixing as well as cmplete mass backmixing n yield fr cmplex reactin types with different levels f exthermicity. Wei (35) states that when a reactr design equatin can be slved fr a single first rder irreversible reactin, it can be slved fr a system f cupled first rder reactins f any number f species and arbitrary ciiiplexi ty» Wei presents a matrix methd t uncuple the cupled system f linear equacins fr cmplex rt;dctiii schemes int an equivalent set f
15 independent first rder irreversible reactins. Wei slves the uncupled system f equatins t give prduce distributins fr different ideal reactr mdels. Peridic Operatin f Chemical Reactrs A chemical reactr system that is described by a nnlinear system f differential equatins r a linear system f differential equatins with variable cefficients will in general give different time average values fr peridic utput variables when frced by peridic inputs than the steady state values btained by using cnstant inputs equal t the average f the scillating inputs. Such a chemical reactr system may als act as a chemical scillatr; that is the reactr may shw stable cyclic behavir under the prper cnditins when the inputs are cnstant, in which case the time average values f the scillating variables will be different frm the crrespnding steady state values» Much f the basic thery f the peridic behavir f nnlinear systems is cntained in a bk by Minrsky (29)» Minrsky's bk is used as a surce wrk by virtually all the authrs wh have published material abut the behavir f the peridically perated chemical reactr. Duglas and Rippen (16) shw a shift in the time average value f the utput relative t the steady state utput fr an isthermal CSTR with secnd rder kinetics where the inputs are sinusidal functins f time. It was fund that a sinusidal input cncentratin wuld increase the time average cnversin slightly fr all frequencies and amplitudes. A cmbined sinusidal input flw and sinusidal input cncentratin shwed an increase r a decrease in time average cnversin, depending n the amplitude, the
16 frequency, and the phase angle between the input flw and cncentratin, A large amplitude, lw frequency, and 180 phase angle gave the largest increase in time average cnversin. Results fr the lwer limit f zer frequency were btained analytically and results at frequencies greater than zer were btained with the aid f an analg cmputer. The authrs als shw that the thermally sensitive nnisthermal CSTR which generates stable scillating utputs frm cnstant inputs will have a different time average cnversin than the crrespnding steady state value. Depending n the parameter values, the time average cnversin may be greater r less than the steady state cnversin» This result can be deduced frm the results f Aris and Amundsn (3), but these authrs are interested in the stability f the steady state. Duglas (13) gives an analytical methd fr determining the frequency respnse t sinusidal input? fr the isthermal CSTR with secnd rder kinetics. The analytical methd is t apprximate the nnlinear dynamics f the CSTR by using a perturbatin technique given by Minrsky ( 29. pp. 217-23!). This methd was fund t agree well with results btained frm the analg cmputer at lw frequencies. Duglas and Gai tnde (15) shw that a nnisthermal CSTR with a first rder chemical reactin and ne singular pint r steady state can be frced t generate stable limit cycles, the size f which are determined by the additin f ideal prprtinal cntrl n the temperature» Negative feedback tends t stabilize the reactr and decrease the size f the limit cycles and psitive feedback tends t induce instability in the reactr and increase the size f the limit cycle. A perturbatin technique
17 (Minnrsky, 1962) f nnlinear mechanics is used t evaluate the scillating CSTR relative l - ih-. ptunaj steady state peratin. Gai tnde and Duglas (18) -xtenj i.l.cii u.r'iurh'.i'iun technique t apply t any system f 2 first rder nn) i nc.- i ; r : r,.r.t i ;;i! equatins. Hwever, their wrk Is nly successful I n n I tciti ve -cnse» The directin f the shift in the Vu i LiL.if c!;i uii.i i la;l ;)f Vcir I able can be determined, but nt the fiiagnitude f- tf;;..luecar. GaI tnde, and Duglas (4) btained experi mental r..ui in re a., lr temperature scillatins In a CSTR whi ch was designed c.. g.,1^,.jlc a stable limit cycle. The reactin used fr the expcr ii.ivniû! 1 v ^ 11 th hydrlysis f acetyl chlri de. The authrs fu;sj i;.ul ihi ; e.eer 1 inencal results cmpared clsely with the iiuiikm'i cc.i In: I', I -/leni equatins. Hrii...i.i I : : shie general resul ts cncerni ng the ptimi li.. e r pel i V i:. lu..;.-',... cur a yi terns by using a variatinal apprach. Cndi li ns i\,i.. i.ich.-sr. v)f t i n,u:n steady state can be imprved by peridic iiijcruiiuii I :, i:!.. i':,,. :uùerc,: res.its irc applied t the pt i liii u ri...., r h uie ptimuci ceriipera ture cycle t prduce the iii..a I ni,1, t; _ ; i :> ii is u'c lermi ned. Chang and Bankff (8) develped à cun.i" i ta t i vn -clier.ie bailed n the methd f characteristics t ptii.iize a per i uei u i i y pe rated jacketed tubular reactr in which cnuccutl vc readier.:, ere Ue ' ng carried ut. The input is perturbed sinusidr. I t / rid t-h,. r.;i.'eer.x ui e input t maximize the time average yield,, ' i,, f ;!' i!.i':-:!:! /" ^liht increase in the time average y il. I il f i liter-...".ii ti Ce I s I..--ill /lull rclati ve t the ptimum steady state V i e 1 d.
18 Larsen (23) intrduces a special case f the semi batch peratin cycle used in this research wrk. Larsen shws that the average yield fr semibatch peratin relative t the steady state yield fr the same average residence time may be increased in the isthermal CSTR with first rder kinetics. Lund and Seagrave (27) extend this wrk t shw yield increases relative t the steady state yield fr adiabatic semi batch peratin with a first rder exthermic reactin» Fang and Engel (17) als shwed an average yield increase by semi batch peratin relative t the steady state CSTR fr isthermal first and secnd rder reactins. Fang and Engel verified their result- experimentally using the hydrlysis f acetic anhydride in water as the reactin system.
19 THE GENERAL EQUATIONS The general equatins are derived belw fr the variable-vlume peratin f a STR in which an exthermic reactin with arbitrary reactin kinetics is ccurring In the hr geneus liquid phase. The reactr is prvided with a cling jacket. The system is depicted in Figure 1. The general equatins fr the variable-vlume STR are then written in dimensinless frm in such a way that the system variables are defined with respect t a reference CSTR. Finally, tw perfrmance criteria are defined which evaluate the perfrmance f the variable-vlume STR with respect t the reference CSTR. These perfrmance criteria are called relative yield and relative thrughput. The general equatins in dimensinless frm prvide the basis fr cmputatin f relative yield and relative thrughput fr different types f variable-vlume peratin f the STR discussed later in this thesis. A material and energy balance arund the variable-vlume STR yields the fllwing independent equatins: 1- verall vlumetric balance \J = V + r (Q. - d) dt. ^ T (Eq. 1) 2 reactant j balance dc. dt 1, 2, n independent reactins
20 Q:M. (V v(0 A A; 2»-Qc(t) T,(0 Tcf "Q(t) C^(t) T(t) Figure 1. Schematic diagram f the stirred tank reactr
21 3. reactr thermal energy balance, dr n (-AH.)r V q ^ dt ^ V (T, - T) + Z f pcp pcp (Eq. 3) where q = (-^) (T-T^) 4. Cler thermal energy balance dt^ Qp Vq 'fq- 4) The assumptins used in these equatins are 1. the reactr is well stirred, that is the reactr cntents are unifrm in cncentratin and temperature, and prduct is discharged frm the reactr at the same cncentratin and temperature as the reactr cntents 2. the cler is well stirred 3. there is n vlume change due t reactin 4. (-AH.) p, Cp, Cp_, Qj,, V^, (UA/V) are cnstant. (UA/V) is cnstant i f U is cnstant and A and V are bth directly prprtinal t the liquid level in the reactr. ^ and T^. are cnstant. Fr s species and n independent reactins Equatin 2 represents a set f s reactant balances. Hwever, nly n reactant balances are independent prvided that the feed cmpsitin is cnstant and that the init cmpsitin and initial extent f each reactin in the reactr are cnipat i b1e. In this wrk cmputatins v;erc perfrmed using a reactin rate
22 cnstant that was either a linear functin f temperature r an Arrhenius functin f temperature. Only irreversible reactins fr which the rate cnstant fr the reverse reactin is zer are cnsidered fr the nnisthermal case. The rate cnstant is expressed belw as a linear functin f temperature and Arrhenius functin f temperature respectively. K = + b(t - T^) (Eq. 5a) K = exp[- I" ( I - Y")] (Eq. 5b) g f These equatins may cnveniently be put in dimensinless frm by intrducing the fllwing dimensinless variables. Cj = Cj/Caf 9 = t 0*/%^ T"' = T/T^ a"' = d/qp (Eq. 6) = K/Kf Gf = 0^/0% (-AH.) (-AH.)'= : V = V/V M ' (-AH,) The subscript R refers t the reference reactr. The reference reactr is the (STR with the same maximum vlume, feed cmpsitin, and feed temperature as the variable-vlume STR. The dimensinless cncentratin is determined with respect t the cncentratin f reactant A in the feed stream. The isthermal and adiabatic cases fr the variablevlume STR are studied in detail. The reference reactr is isthermal fr the case f the isthermal variable-vlume STR and adiabatic fr the case f the adiabatic variable-vlume STR. When the dimenbiniebb variabley are substiluted int Equatins 1
23 thrugh S, the fllwing dimensinless equatins result. r = V" + r (d* - Q")de (Eq. 7) J 9^ r dcj' Q:' n = ~ (c" - CV) + P s a,, r. (Eq. 8) d0 M i=l ' i = ], 1, n independent reactins dt Q.r n.1. = 7. (1-T") + RP Z (-AH;)" r" - M(T -T.") (Eq, 9) de M" ;=i ' ' ^ G = -il Q_'- (T _ TP + NV (T'-T ) (Eq. 10) de c cf c K" = 1 + L(t"-1 ) (Eq. 11a) K = exp[-l' ("^ -1)] (Eq. lib) ï""' The dimensinless parameters which appear in the dimensinless equatins are defined belw. Relative Rate Cnstant P = (see Table 1) Relative Thermal Energy R = ("AH^^C^^/pCpT^ Reactr Heat Transfer ^, UA \. Parameter " V ^ V^R^V Srl^etër' Linear Reactin Rate L = b T /K Parameter f f Arrhenius Reactin Rate L' = E/PL T, Parameter 9 f
24 The expressins fr reactin rate, r'.', fr the different types f reactin systems studied are given in Table 1. The general nnisthermal peratin f the variable-vlume STR is nt cnsidered in the remainder f this wrk» Only the special cases f isthermal and adiabatic peratin are cnsidered» Fr the isthermal case nly Equatins 7 and 8 are required t determine prduct cncentratins. Further, the feed temperature is arbitrarily assumed equal t the temperature f the reactr cntents, s that the reactin rate cnstant, K, is unity. Fr the adiabatic case Equatins 1, 8, 3, and 11 with M set equal t zer must be slved simultaneusly t determine cncentratin and temperature prfiles» The tw perfrmance criteria, namely, relative yield and relative thrughput, are defined belw. Fr a particular type f variable-vlume peratin either criteria may be used. The tw criteria merely represent different ways f evaluating the prductin rate f the variable-vlume STR witl'i respect t the refercncc CSTR. The tw criteria are nt independent and a methd fr determining relative thrughput when the relative yield is knwn is presented. Definitin f relative yield relative yield is defined as the rati f the flw average prduct cncentratin frm the variablei/lume STR t the cnstant prduct cncentratin frm the reference CSTR. The variable-vlume STR and the reference CSTR have the same maximum vlume, the same feed cmpsitin and temperature, and the same time average thrughput f prcess material. The relative yield f prduct j is expressed mathematically by
25 Table 1. Rate expressins fr reactin systems studied Reactin system Dîmensînless rate expressin, I vh Relative rate cnstant, P K A i-b i sthermal KfVM/Q% K A > B adi abati c K C, k a;=±b Kt isthermal ''A " ^Ae K 2a >b i sthermal -C -A- 2 a adi abat i c -.2 K"C; KfCAfVM/%R kj k, a >b >c 2A K r = ^2 CA- [[ i sthermal Kp kt
26 "i - ' flw average cncentratin f j in prduct frm variable-vlume STR cncentratin f j in reference CSTR Cj / ([])% (Eq. ]2) J ; da P - 1 where C. = g (Eq. 13) P r d9 and 0p - 0p I is the time interval fr ne variable-vlume perid fr which the cncentratin is a steady peridic functin f time. Definitin f relative thrughput relative thrughput is defined as the rati f the time average thrughput in the variable-vlume STR t the cnstant thrughput in the reference CSTR. The variablevlume STR and the reference CSTR have the same maximum vlume, the same feed cmpsitin and temperature, and the same flw average cncentratin f prduct frm the reactr. The relative thrughput is defined mathematically by V - time average thrughput f prcess material in variable-vlume STR cnstant thrughput f prcess material i p. reference CSTR.
27 a/(cl)^ (Eq 14), 8p 9p r 8p_, de where 0p - 9p_^ is the length f ne variable-vlume perid. Implicit in the abve definitins f relative yield and relative thrughput is the fact that the relative thrughput is equal t unity in the definitins f relative yield and vice versa. Cnsider the relative yield f prduct j t be knwn as a functin f the relative rate cnstant, P» The cncentratin f cmpnent j in the reference CSTR is determined by slutin f the steady state frm f the general dimsnsifiless equatins. The flw average cncentratin f cmpnent j in the reference CSTR and in the variable-vlume STR is then given respectively by ( C I = c j f + f (, %, " i j r i ' R f q - ' 5 ) Cj' = "ij (cj)^ (Eq. 16) Nw the relative thrughput, v, may be determined with respect t a secnd reference CSTR which has a cncentratin f cmpnent j given by (c:), C: (Eq. 17) J^R' J where the subscript R' dentes the secnd reference CSTR. The relative
28 thrughput with respect t the secnd reference CSTR is determined by slutin f the steady state frm f the general dimensinless equatins with P replaced by PY. Relative thrughput with respect t the secnd reference CSTR is then given by Y = ' ^ ^ (Eq. 18) -P ( Z. r.)^, 1 = 1 J After Y has been determined in Equatin 18^ the crrespnding cnversin f cmpnent A is given by \ (.%, "ia '9' 1 = 1 Equatins 15; 16; 18; and 19 determine relative thrughput as a functin f flw average cnversin f cmpnent A frm a knwledge f relative yield versus relative rate cnstant and the steady state slutins t the general dimensinless equatins.
29 CONSTANT VOLUME OPERATION Slutins t the general equatins in dimensinless frm fr transient and steady state peratin f the CSTR are given here. The steady state slutins yield the dimensinless cncentratin, temperature, and relative rate cnstant fr the reference CSTR and may be used later in the determinatin f relative yield and relative thrughput fr variablevlume peratin. The transient and steady state slutins fr the CSTR aid in understanding the cnditins fr which variable-vlume peratin f the STR may result in a relative yield r relative thrughput gr^atar than unity. The dimensinless equatins fr cnstant vlume peratin are btained frm the dimensinless frm f the general equatins by setting Q^/V equal t unity. Isthermal The steady state cncentratin in the CSTR is easily determined fr the reactin schemes cnsidered in this wrk. The transient cncentratin prfile in time can be determined analytically fr the case f first rder reactins in the CSTR. These slutins are given in Table 2. Adi abatic In the adiabatic CSTR all f the heat generated by the exthermic reactin is used t heat the incming feed t the adiabatic perating temperature. The adiabatic CSTR and the isthermal CSTR are ppsite extremes f the mre general nnisthermal CSTR.
30 Table le Slutins t isthermal cnstant vlume equatins Reactin Steady state Transient cncentratin scheme cncentratin, C prfile, C"(9) A^B C^'=l/(1 + P) C^' = ~ (l-exp[-(l+p)0]) + C^'(O) exp[-(l+p)e] K I+PC" 1 + PC" " "T+p (i-exp[-(i+p)e]) k' + (0) exp[-(l+p)e] 2a_!L,r r" - (2P+n - s/(2p+])^- 4P^ K, K, -d+p) + i (1+P)^ + 4PK,C /K K, 2A 40 Cg = PC^/CI + PK^/K^)
31 Single reacti n f arbi trary rder The reactin rate fr a single reactin in which each species cncentratin can be related t the extent f reactin may be expressed as a functin f the cncentratin f species A and temperature r* = r"(c* T*). Figure 2 shws the frm f curves f cnstant reactin rate fr the exthermic irreversible reactin in the (1 - T ) plane. At steady state the dimensinless frm f the general equatins fr the CSTR becme 1 - C^' - Pr'"'(Ç T"') = 0 (Eq 20) 1 - t" + RP r''(c^; T") = 0 (Eq, 21 ) Equatins 20 and 21 may be cmbined t eliminate r and give the energy balance line I - C^' = ^ T - i (Eq. 22) Equatin 22 als hlds fr the adiabatic plug flw reactr. Equatin 22 is pltted as line FG in Figure ^. FG gives the lci f all pssible prduct states frm the adiabatic CSTR. FG always crsses the T axis at the pint (Oj I) The slpe f FG is determined by the feed cnditins fr a particular reactin. This is apparent frm the definitin f R. R. (Eq. 23) p Cp T^ K is the dimensinless adiabatic temperature rises At cmplete cnversin K = t" - l (Eq. 24)
32. G TEMPERATURE, T" Figure 2. Lci f cnstant reactic# rate fr an exthermic irreversible reactin The curves are lines f cnstant reactin rate
33 If the values f reactin rate alng FG are pltted against cnversin, 1 - the reactin rate will g thrugh a maximum at (1 - crrespnding t pint A n FG. The slpe f this curve is called the adiabatic derivative f reactin rate by Aris (1^ Chapter 8) and is given by r"(c T*) = - + R (Eq. 25) d(l-c^) ST" At a pint A n FG d^ r /d(l-c^) = 0^ and it fllws that d (l-c^)/dt = 1/R at pint A» Frm Equatin 20 it is seen that the prductin rate, (l"c^')/pj in the adiabatic CSTR is als a maximum at pint A Aris (1, Chapter 8) als shws that if the energy balance line fr adiabatic peratin is given by line FG in Figure 2, then the least residence time is achieved under the fllwing cnditins: 1. If a cnversin less than (1-C^)^ is desired, then the least residence time is achieved in the adiabatic CSTR 2. If a cnversin greater than (1-C^)^ is desired, then the least ttal residence time is achieved by an adiabatic CSTR with cnversin fllwed in series by an adiabatic plug flw reactr t increase the cnversin t the desired value. This is called the ptimal adiabatic CSTR and plug flw reactr cmbinatin» The average reactin rate is a maximum in the ptimal reactr cmbinatin. In general, the reactr type with the maximum average reactin rate will require the least residence time t perate at a specified cnvui si Oil.
34 The stability f the steady state in an adiabatic CSTR is als an imprtant cnsideratin. A knwledge f the stability f the steady state will give insight int the stability prblems t be expected with the adiabatic variable-vlume 5TR. Cnsider Equatins 20 and 22 fr the adiabatic CSTR. The lci f Equatins 20 and 22 in the (1-C^,T ) plane are dented by ç lines and line FG respectively in Figure 3^ The intersectin f Ç lines with FG represent steady state slutins t Equatins 20 and 22. If an adiabatic CSTR is perating at steady state C, a slight increase in residence time will ignite the reactin and the steady state will shift t steady state I. Similarly a slight decrease in residence time in an adiabatic CSTR with steady state E will cause the reactin t be blwn ut, and the steady state will shift t A. If the reference adiabatic CSTR has the steady state crrespnding t pint E, the reactin may be blwn ut in the adiabatic variable-vlume STR, since the instantaneus residence time in the adiabatic variable-vlume STR may decrease belw the residence time in the reference adiabatic CSTR. Als the reactin may be reignited as the instantaneus residence time is increasing in the adiabatic variable-vlume STR. The pssibility f such a blwing ut and reigniting f the reactin is much less when the reference adiabatic CSTR has steady state J and deviatins f the instantaneus residence time in the adiabatic variable-vlume STR are sufficiently small abut the average residence time which crrespnds t the residence time in the reference adiabatic CSTR. Of curse, this analysis des nt apply accurately t the variable-vlume STR since it perates in a transient manner. Hwever, the pssibility
35 G increasing residence time t TEMPERATURE, T Figure 3. Lci f material and energy balance lines fr the adiabatic cntinuus stirred tank reactr
36 f this very undesirable type f peratin is still present. Only if the residence time in the variable-vlume STR changes very slwly s that steady state cnditins are always present, will the abve analysis be crrect, Fi rst rder reactins Steady state and transient peratin f the adiabatic CSTR are cnsidered here fr the case f an exthermic and irreversible first rder reactin. The rate cnstant, K, is assumed t be a linear functin f temperature as given by Equatin 11a. Stead, state peratin At steady state che dimensinless frm; f the general equatins becme 1 - C* - PK" C^= 0 (Eq. 26) 1 - T + RPK"C^' =0 (Eq. 2?) K" = 1 + L (T" - 1 ) (Eq. 1 la) The intersectin f the material balance 1 i tie uelermined by Equatin 26 and the energy balance line determined by Equatin 22 in the (1-C^,T ) plane gives the steady state perating pint. Since the rate cnstant, K, is a linear functin f temperature, there can be nly ne pssible steady state perating pint fr fixed values f R, L, and P. (If the rate cnstant, K, were given by the Arrhenius equatin, then as many as 3 steady states wuld be pssible.) The steady state perating pint always satisfies the necessary cnditin fr stability» There is n ignitin pint r blw ut pint fr the reactin in the adiabatic CSTR-
37 Equatins 26, 21, and Ha may be slved simultaneusly t give cn centratin as a functin f R, Lj and P. The result is The rate cnstant, K, the temperature, T, and the reactin rate, r, may als be written as a functin f Rj L, and P Figures 4 and 5 shw T ; K j and r pltted against lg^gcloo P) with R and L as parameters. The analg cmputer was used t generate these plts using lg^q (100 P) as the independent variable. The plt f reactin rate versus cnversin f reactar,t A in Figure 6 is a plt f the values f reactin rate alng the energy balance line. The slpe f the curve fr reactin rate versus cnversin in Figure 6 is the adiabatic derivative defined by Equatin 25» Fr values f cnversin fr which the adiabatic derivative is psitive, an adiabatic CSTR will achieve the greatest cnversin. Fr values f cnversin fr which the adiabatic derivative is negative, the ptimal adiabatic CSTR and plug flw reactr cmbinatin will achieve the greatest cnversin» Frm Equatin 28 it is seen that R and L always appear as the prduct RL Therefre = C^tRL, P). (Eq 29) By cnsidering Equatins 11a, 22, 26, and 28 it is seen that K' = K'(RL,P) r = r (RL, P) (Eq. 30) (Eq 31) T " = T"'(R,L, P). (Eq. 32) R and L mpppar As the prduct RL in these expressins because the
Figure h. Cncentratin and temperature versus the lgarithm f the relative rate cnstant
DI MENS IONLESS CONCENTRAT I ON, DIMENSIONLESS TEMPERATURE, T" VD / /
Figure 5. Reactin rate and rate cnstant versus the lgarithm f the relative rate cnstant
DI MENS IONLESS REACTION RATf,K 0 1 MENS IONLESS RATE CONSTATN,K" CO 5 \ë, CO
42 RL = 80 RL = 60 RL = 40 0 6 8.4 CONVERSION, (I - C Figure 6. Reactin rate vs. cnversin fr an exthermic reactin under adiabatic cnditins
43 rate cnstant, K, is a linear functin f temperature. If K were given by the Arrhenius expressin, all the variables wuld depend n the individual values f R, L, and P» The maximum prductin rate, given by (1-C^)/P, ccurs when the reactin rate is a maximum and the adiabatic derivative is zer. The maximum reactin rate and crrespnding values f the ther variables may be determined as a functin f RL The resulting expressins are given belw. (i<) = r - = (Eq. 33) P -A- max 4RL r max max P (Eq. 3 5) max (RL+1) T'V" = I + ^ - jj- (Eq. 36) max K max (Eq. 37) It'is interesting t nte that (1-C^)... appraches.5 as RL becmes 1arge Transient peratin r max Sme transient slutins f the cnstant vlume adiabatic CSTR equatins were determined. The analg cmputer was used t slve the system f equatins, since analytical slutins f these cupled and nnlinear first rder differential equatins are nt
44 knwn. Plts f the results are shwn in Figures 7^ 8^ and 9. The effect f the prduct RL is shwn in Figure 1, and the effect f the relative rate cnstant, P, is shwn in Figure 8. During the transient peratin, the cncentratin and the rate cnstant and thus the reactin velcity depend n the prduct RL, as has als been shwn fr steady state peratin. This is shwn by cmbining the transient equatins and thereby btaining ne secnd rder differential equatin in which R and L always ccur in cmbinatin Fr a given RL, the reactin rate attains larger values during the transient peratin than the steady state reactin rate nly fr values f the relative rate cnstant, P, greater than P * max Figure 9 shws the effect f the initial cncentratin f reactant, C^(0), in the reactr n the transient behavir fr fixed P and RL It is interesting t nte that the transient reactin rate ges thrugh the maximum value, r^^^ fr unrealizeable case where values f P greater than P - nly fr the max the initial dimensinless cncentratin is equal t unity.