Fundamentals of Polymer Engineering Part 5 ppsx

4 ReactionEngineeringof Step-GrowthPolymerization 4.1INTRODUCTION[1,2] Vesselsinwhichpolymerizationiscarriedoutarecalledreactors;theyare classifiedaccordingtoflowconditionsexistinginthem.Batchreactors(schema- ticallyshowninFig.4.1a)arethoseinwhichmaterialsarechargedinitiallyand polymerizationiscarriedouttothedesiredtime,whichinturn,dependsonthe propertiesofthematerialrequired.Wehavealreadyobservedthatpolymerization islimitedinbatchreactorsbyequilibriumconversion.Becausewewishtoform polymersofhighmolecularweights,wecanovercomethislimitationbyapplying highvacuumtothereactionmass.Onapplicationoflowpressures,thereaction massbeginstoboilandthecondensationproductisdrivenoutofthereactor,as showninFigure4.1b.Suchbatchreactorsarecalledsemibatchreactors. Batchandsemibatchreactorsareidealwhentheproductionrateofthe polymerneededissmall.Inlarger-capacityplants,continuousreactorsare preferred.Inthese,therawmaterialsarepumpedincontinuouslywhilethe productsareremovedattheotherend.Oneexampleoftheseisatubularreactor (showninFig.4.1c).Itislikeanordinarytubeintowhichmaterialispumpedat oneend.Polymerizationoccursinthetubularreactor,andtheproductstream consistsofthepolymeralongwiththeunreactedmonomer.Sometimes,astirred vessel(showninFig.4.1d)isemployedinsteadofatubularreactor.The advantage of such a reactor is that the concentration and temperature variations 153 Copyright © 2003 Marcel Dekker, Inc. within it are removed due to vigorous stirring, making it possible to control reactor conditions more easily. It may be mentioned that batch, semibatch, tubular, and stirred-tank reactors serve as mere idealizations of actual reactors. Consider, for example, FIGURE 4.1 Some ideal reactors. 154 Chapter 4 Copyright © 2003 Marcel Dekker, Inc. the industrial V.K. tube (Vereinfacht Kontinuierliches Rohr) reactor, which is used for nylon 6 polymerization. Its schematic diagram is given in Figure 4.2a, in which e-caprolactam monomer is mixed with water (serving as the ring opener) and introduced as feed. In the top region, the temperature is about 220–270  C and the reaction mass is vigorously boiling. The rising vapors produce intense agitation of the reaction mass and ultimately condense in the reflux exchanger. A small amoun t of e-caprolactam also evaporates in this section of the reactor and FIGURE 4.2 Schematic diagram of industrial nylon 6 reactors and reactor model. (Reprinted from Ref. 1 with the permission of Plenum Publishing Corporation.) Reaction Engineering of Step-Growth Polymerization 155 Copyright © 2003 Marcel Dekker, Inc. isrecycledtothereactor,asshown.Asthematerialmovesdownward,thereactor pressureincreasesduetogravityandtheboilingofthereactionmassstops.Inthe secondstage,mostofthee-caprolactamisreacted,and,inordertopushthe polymerizationtohighconversions,itisdesiredtoremovethecondensation product(water)fromthereactionmass.Tofacilitatethis,thereactionmassis purgedwithasuitableinertgas(say,nitrogen).Inthethirdstage,theviscosityof thereactionmassisveryhighandwatercannotberemovedbypurginganymore. Sufficientresidencetimeisprovidedsoastoachievethedesiredmolecular weightofthepolymer.Figure4.1showssimplereactorsandFigure4.2bmodels complexreactors(e.g.,V.K.tubesfornylon6)intermsofacombinationofthese. Duetointenseagitationexistinginthefirsttwostages,theentireV.K.columnhas beenviewedasatrainoftwohomogeneouscontinuous-flowstirred-tankreactors (HCSTRs)followedbyaplugflowreactor. FromtheexampleoftheV.K.tubefornylon6,weobservethatsimple reactors(Fig.4.1)arebuildingblocksofmorecomplexones.Thischapter focusesonanalyzingsimplereactorscarryingstep-growthpolymerization. Chapter3hasalreadyconsideredpolymerizationinthebatchreactor.Wefirst studytheperformanceofsemibatchreactorsandexaminetheeffectofflashingof thecondensationproductonit. 4.2ANALYSISOFSEMIBATCHREACTORS[1,3] Wehavealreadyobservedinearlierchaptersthatengineeringmaterialsshould havealargeaveragechainlength.Supposeitisdesiredtohavem n equalto100, whichwouldimplya99.9%conversionoffunctionalgroups.Step-growth polymerizationislimitedbyitsequilibriumconversion,andthereisaneedto pushthereactionintheforwarddirection.Thisisdoneinindustrybyapplying highvacuumtothereactionmass,whereuponthereactionmassbeginstoboil undertheappliedlowpressure.Weknowthatpolymerchainshaveverylow vaporpressuresand,undernormalconditionsofoperation,theydonotvaporize; however,themonomercan.Thisclearlymeansthatinthepresenceofflashing, theconcentrationofanygivenspecieschangesnotonlybypolymerizationbut alsobychangeinvolumeVofthereactionmass.Weshowtheschematicdiagram ofthesemibatchreactorinFigure4.3,andintheanalysispresentedhere,we considerthechangeinVasanexplicitvariable.Weassumethatunderthe existingreactorconditions,thecondensationproductWandthemonomerP 1 can flashoutofthereactor.Inallsemibatchreactors,themonomerinthevaporphase iscondensedinasuitableseparatorandrecycledbecauseofitshighcost.Itis assumedthatthereactorisoperatingisothermally,attotalpressureP T .The volumeoftheliquidphaseofthereactor,V,changeswithtimeasflashingofW andP 1 occurs.Weaccountforthistimedependenceasfollows.Wedefinep n as 156Chapter4 Copyright © 2003 Marcel Dekker, Inc. the total moles of species ðP n ðn ¼ 1; 2Þ and w as total moles of W in the liqu id phase. The mole balance relations of these, on the dotted control volume shown in Figure 4.3, are given by dp 1 dt ¼ k p l 0 p 1  k 0 p wðl 0  p 1 Þ V ð4:2:1aÞ dp n ¼½2k p p n l 0 þ k p P n1 r¼1 p r p nr  k 0 p wðn  1Þp n þ 2k 0 p w P 1 nþ1 p r V 1 ; n  2 ð4:2:1bÞ dw dt ¼ k p l 2 0  k 0 p wðl 1  l 0 Þ V  Q w ð4:2:1cÞ where k p and k 0 p are the forward and reverse rate constants, respectively and l 0 and l 1 the zeroth and first moments, which are defined as follows: l 0 ¼ P p n ð4:2:2aÞ l 1 ¼ P np n ð4:2:2bÞ The zeroth moment l 0 gives the total moles of polymer at any time, whereas l 1 gives the total count of repeat units, which can be shown to be time invariant. FIGURE 4.3 Schematic diagram of semibatch reactor with monomer and condensation product evaporating. Reaction Engineering of Step-Growth Polymerization 157 Copyright © 2003 Marcel Dekker, Inc. Equations (4.2.1a) and (4.2.1b) are suitably added to determine the generation relation of the zeroth moment l 0 and first moment l 1 as dl 0 dt ¼ k p l 2 0  k 0 p wðl 1  l 0 Þ V ð4:2:3aÞ dl 1 dt ¼ 0 ð4:2:3bÞ Equation (4.2.3b) implies that the first moment l 1 is time invariant and its value can be obtained from the feed conditions. In order to solve for the molecular-weight distribution (MWD) of the polymer, as given by Eqs. (4.2.1) and (4.2.2), we must know the volume, V ,of the liquid phase of the reactor and the rate of vaporization, Q w . The rate of change of volume V is given by dV dt ¼v w Q w ð4:2:4Þ where v w is the molar volume of the condensation product W. In this development, there are seven unknowns ½p 1 , p n ðn  2Þ, W , l 0 , l 1 , V , and Q w ], but we have only six ordinary differential equations [(4.2.1a)– (4.2.1c), (4.2.3a), (4.2.3b), and (4.2.4)] connecting them. Thus, one more equation is required. This is found by using the appropriate vapor–liquid equilibrium condition. Herein, to keep the mathematics simple, we assume the simplest relation given by Raoult’s law. 4.2.1 Vapor^Liquid Equilibrium Governed by Raoult’s Law We assume that all the oligomers, p n , n ¼2 are nonvolatile and that the condensation product W and the monomer P 1 can vaporize. If P 0 w and P 0 p1 are the vapor pressures and x w and x p1 are the mole fractions of W and P 1 respectively, then the partial pressures are given by Raoult’s law as follows: P w ¼ P 0 w x w ð4:2:5aÞ P p 1 ¼ P 0 p 1 x p 1 ð4:2:5bÞ where x w ¼ w l 0 þ w ð4:2:6aÞ x p 1 ¼ p 1 l 0 þ w ð4:2:6bÞ 158 Chapter 4 Copyright © 2003 Marcel Dekker, Inc. The total pressure P T is then the sum of partial pressures; that is, P T ¼ ðP 0 0 P 1 þ P 0 w wÞ l 0 þ w ð4:2:7Þ 4.2.2 Volume of Reaction Mass The previous chapter shows that the MWD of the polymer obtained from batch reactors is given by Flory’s distribution. Now, let us show that, in the presence of flashing, the MWD is still given by a similar relation. Let us assume that the feed to the semibatch reactor is pure monomer; that is, at t ¼ 0, p 1 ¼ p 10 ; ð4:2:8aÞ p n ¼ 0 for n  2 ð4:2:8bÞ then l 1 ¼ l 10 ¼ p 10 ð4:2:9aÞ l 00 ¼ p 10 ð4:2:9bÞ We propose that the MWD of the polymer is of the form p n ¼ xðtÞyðtÞ n1 ð4:2:10Þ where xðtÞ and yðtÞ are independent of the chain length n. On direct substitution of Eq. (4.2.10) into Eqs. (4.2.1a) and (4.2.1b), it is seen that the result satisfies the mole balance relation, no matter what the concentration of W. It is thus seen that the form of MWD remains unaffected by flashing. The xðtÞ and yðtÞ terms in Eq. (4.2.11), however, are now independent because of the invariance of P P n ¼ x x  y ¼ l 0 ð4:2:11aÞ P nP n ¼ xð1  yÞ 2 ¼ l 1 ð4:2:11bÞ These give yðtÞ¼1  l 0 l 10 ð4:3:12aÞ xðtÞ¼ l 2 0 l 10 ð4:2:12bÞ The addition of Eqs. (4.2.1c) and (4.2.3a) gives dðw þ l 0 Þ dt ¼Q w ð4:2:13Þ Reaction Engineering of Step-Growth Polymerization 159 Copyright © 2003 Marcel Dekker, Inc. which, on substitution into Eq. (4.2.4), yields the following on integration: V  V 0 ¼v w ½ðw 0 þ l 10 Þðw þ l 0 Þ ð4:2:14Þ Herein, w 0 is the moles of condensation product in the liquid phase having total volume V 0 at time t ¼ 0. 4.2.3 Performance of the Semibatch Reactors We rewrite the vapor–liquid equilibrium in Eq. (4.2.7) as follows: w ¼ P 0 T l 0  P 0 p1 p 1 P 0 w  P T ð4:2:15Þ However, Eq. (4.2.10) gives p 1 as xðtÞ,orl 2 0 =p 10 [see Eq. (4.2.12)], which on substituting into Eq. (4.2.15), gives W P T P 0 w  P T  l 0 !  ½P 0 p 1 =ðP 0 w  P T Þl 2 0 p 10 ¼ D a 1 l 0  a 2 l 2 0 ð4:2:16aÞ where a 1 ¼ P 0 p 1 ðP 0 w  P T Þp 10 ð4:2:16b a 2 ¼ P 0 p 1 ðP 0 w  P T Þp 10 ð4:2:16cÞ Between Eqs. (4.2.14) and (4.2.16), it is thus possible to explicitly relate V to l 0 : V ¼ b 0 þ b 1 l 0  b 2 l 2 0 ð4:2:17aÞ where b 0 ¼ V 0  v w ðw 0 þ p 10 Þð4:2:17bÞ b 1 ¼ v w ða 1 þ 1Þð4:2:17cÞ b 2 ¼ v w a 2 ð4:2:17dÞ We can now substitute Eq. (4.2.16a) for w and Eq. (4.2.17a) for V into Eq. (4.2.3a) to obtain the following: ðb 0 þ b 1 l 0  b 2 l 2 0 Þ dl 0 dt ¼ k p l 2 0 þ k 0 p ðl 00  l 0 Þða 1 l 0  a 2 l 2 0 Þð4:2:18Þ This can be integrated with the initial condition that l 0 at t ¼ 0 is the same as p 10 and the final result can be derived as A 1 ln l 0 l 00  þ A 2 ln l 0  d 1 l 00  d 1   A 3 ln l 0  d 2 l 00  d 2  ¼ t ð4:2:19Þ 160 Chapter 4 Copyright © 2003 Marcel Dekker, Inc. where d 1 ¼ g 2 þðg 2 2  4g 1 g 3 Þ 0:5 2g 1 ð4:2:20aÞ d 2 ¼ g 2 ðg 2 2  4g 1 g 3 Þ 0:5 2g 1 ð4:2:20bÞ A 1 ¼ b 0 d 1 d 2 g 1 ð4:2:20cÞ A 2 ¼ b 0 þ b 1 d 1  b 2 d 2 1 d 1 g 1 ðd 1  d 2 Þ ð4:2:20dÞ A 3 ¼ b 0 þ b 1 d 2  b 2 d 2 2 g 1 d 2 ðd 1  d 2 Þ ð4:2:20eÞ g 1 ¼ k p a 2 ð4:2:20f Þ g 2 ¼ k 0 p þ l 10 k 0 p a 2 þ k 0 p a 1 ð4:2:20gÞ g 3 ¼ k 0 pa 1 l 10 ð4:2:20hÞ When monomer P 1 has very low volatility and only water flashes, a 2 ¼ b 2 ¼ 0 ð4:2:21aÞ then w ¼ P T l 0 P 0 w  P T  a 1 l 0 ð4:2:21bÞ V 0 ¼ b 0 þ b 1 l 0 ð4:2:21cÞ Equation (4.2.18) then becomes ðb 0 þ b 1 l 0 Þ dl 0 dt ¼k p l 2 0 þ k 0 p ðl 00  l 0 Þa 1 l 0 ð4:2:22Þ which can be integrated to A 4 ln l 0 l 00   A 5 g 2  ln½ðg 2 l 0  g 3 Þ=ðg 2 l 00  g 3 Þ ¼ t ð4:2:23Þ where A 4 ¼ b 0 g 3 ð4:2:24aÞ A 5 ¼ g 2 b 0 þ b 1 g 3 g 3 ð4:2:24bÞ Let us consider that some moles of monomer (say, p 10 ) are mixed with some moles (say, w 0 ) of condensation product before the mixture is charged to the Reaction Engineering of Step-Growth Polymerization 161 Copyright © 2003 Marcel Dekker, Inc. reactor. As long as the constraint of vapor–liquid equilibrium [given in Eq. (4.2.7)] is not satisfied, there is no flashing of W and P 1 , and the system behaves like a closed reactor. During polymerization, w increases and l 0 decreases, and there is a time when the condensation product begins to evaporate. This time can be determined as follows. We observe that there is no flashing for closed reactors, Q w ¼ 0 ð4:2:25aÞ and Eqs. (4.2.13) and (4.2.14) reduce to w þ l 0 ¼ p 10 þ w 0 ð4:2:25bÞ V ¼ V 0 ð4:2:25cÞ Substituting these into Eq. (4.2.3a) gives V 0 dl 0 dt ¼k p l 2 0 þ k 0 p ðw 0 þ p 10  l 0 Þðp 10  l 0 Þð4:2:26Þ which can be easily integrated to give l 0 from q q 0 ¼ exp  dt V 0  ð4:2:27Þ where d ¼ðm 2 1 þ 4m 0 m 2 Þ 1=2 ð4:2:28aÞ q ¼ 2m 2 l 0 þ m 1  d 2m 2 l 0 þ m 1 þ d ð4:2:28bÞ q 0 ¼ 2m 2 p 10 þ m 1  d 2m 2 p 10 þ m 1 þ d ð4:2:28cÞ m 0 ¼ k 0 p ðw 0 þ p 10 Þp 10 ð4:2:28dÞ m 1 ¼ k 0 p ðw 0 þ 2p 10 Þð4:2:28eÞ m 2 ¼ðk p  k 0 p Þð4:2:28f Þ Two situations are possible, relating to whether the monomer is flashing or not. When only W is evaporating, Eq. (4.2.21) holds for thermodynamic equilibrium and the intersection point is given by l c 1 0 ¼ w 0 þ l 00 1 þ a 1 ð4:2:29Þ where superscript c 1 stands for this evaporation condition (called case 1). This is now substituted into either Eq. (4.2.27) or Eq. (4.2.24). When P 1 as well as W 162 Chapter 4 Copyright © 2003 Marcel Dekker, Inc. [...]... ADVANCED STAGE OF POLYMERIZATION [5^ 11] In several cases (e.g., in the manufacture of polyethylene terephthalate), the equilibrium constants of the reactions are such that one must remove the volatile condensation products by application of a vacuum in order to obtain a polymer Copyright © 2003 Marcel Dekker, Inc 170 Chapter 4 having long chain lengths Because the desired degree of polymerization... Þ þ rw dZ2 d dZ dy d 4 .5 CONCLUSION This chapter has discussed the analysis of reactors for step-growth polymerization assuming the equal reactivity hypothesis to be valid Polymerization involves an infinite set of elementary reactions; under the assumption of this hypothesis, the polymerization can be equivalently represented by the reaction of functional groups The analysis of a batch (or tubular)... Constants arising from averaging of profiles: a0 ¼ À3 a1 ¼ 0: 75 a2 ¼ 0:6 a3 ¼ À0: 453 6 a4 ¼ 2:4 857 a5 ¼ 0: 25 ðA1Þ Time variation of zeroth moment, l00 , at the interface: 0 aw ¼ PT =ðPw À PT Þ 0 b ¼ kp =kp ai ¼ ð1 þ baw Þl0 00 bi ¼ baw ðA2Þ ci ¼ ðai À bi Þ l00 ¼ bi l0 00 ai À ci expðÀbi yÞ Film thickness as a function of l00 : a ¼ ð1 À 2a2 þ a4 Þ þ aw bð1 À a1 À a2 þ a5 Þ b ¼ ab½l0e ða2 À a3 Þ À ð1 À... Chem Eng., 54 , 1–41, 1976 Reinisch, G., H Gajeswaki, and K Zacharias, Extension of the Reaction Diffusion Model of Melt Polycondensation, Acta Polym 31, 732–733, 1980 Amon, M., and C D Denson, A Study of the Dynamics of Foam Growth: Analysis of the Growth of Closely Spaced Spherical Bubbles, Polym Eng Sci., 24, 1026–1034, 1984 Amon, M W., and C D Denson, Simplified Analysis of the Performance of Wiped... Fundam., 19, 4 15 420, 1980 Kumar, A., S Madan, N G Shah, and S K Gupta: Solution of Final Stages of Polyethylene Terephthalate Reactors Using Orthogonal Collocation Technique, Polym Eng Sci., 24, 194–204, 1984 Khanna, A., and A Kumar, Solution of Step Growth Polymerization with Finite Mass in Films with Vapour Liquid Equilibrium at the Interface in Polymer Reaction Engineering, in Polymer Reaction Engineering, ... the solution of the MWD in the film is Pn ðy; yÞ ¼ X ðy; yÞY ðy; yÞnÀ1 Show that the MWD relations are satisfied by equations of Pb4.11 Also suggest the simplest form of X ðy; yÞ and Y ðy; yÞ Also demonstrate that this form is consistent with the equilibrium of polymerization Try the form given in Eq (3 .5. 2) first 4.13 The rate of evaporation, Nw , of condensation product at y ¼ 0 in the film of Figure 4.7... concentrations of W and polymer P have flat spatial profiles In F IGURE 4.7 Schematic diagram shown the interfacial and bulk regions within the films (Reprinted from Ref 11 with the permission of VCH Verlagsgesellschaft mbH.) Copyright © 2003 Marcel Dekker, Inc Reaction Engineering of Step-Growth Polymerization 177 general, the thickness of the interfacial region for W, dw , and that for polymer P, dl0... reactor The feed is assumed to consist of 10 mol of monomer ðw0 ¼ 0Þ At the reactor temperature, let us assume that the reactor pressure is 5 mm Hg and the vapor pressure of the condensation product is 38.483 mm Hg 1 2 3 Determine the time of flashing Determine the values of l0 and w in the reactor at equilibrium and the moles of w flashed Calculate time taken to reach 101% of the equilibrium l0 Assume 0 kp... Macromolecules, 20, 220–226, 1987 Secor, R M., The Kinetics of Condensation Polymerization, AIChE J 15( 6), 861– 8 65, 1969 Hoftyzer, P J., and D W van Krevelen, The Rate of Conversion in Polycondensation Processes as Determined by Combined Mass Transfer and Chemical Reaction, in Proceedings of Fourth European Symposium on Chemical Reaction Engineering, Brussels, 1968 Shah, Y T., and M M Sharma, Desorption... occur for all values of as because, beyond a critical value, the rate of mass transfer of W is no longer limiting, and the mn versus as curve begins to flatten out where the polymer formation is once again overall reaction controlled Copyright © 2003 Marcel Dekker, Inc Reaction Engineering of Step-Growth Polymerization F IGURE 4.6 the film 173 Average chain length mn at the end of the reactor versus . distribution (MWD) of the polymer, as given by Eqs. (4.2.1) and (4.2.2), we must know the volume, V ,of the liquid phase of the reactor and the rate of vaporization, Q w . The rate of change of volume. of a vacuum in order to obtain a polymer Reaction Engineering of Step-Growth Polymerization 169 Copyright © 2003 Marcel Dekker, Inc. having long chain lengths. Because the desired degree of polymerization. Inc. reactionmassconsistsoftwomolecules: P n :A½BA n1 B and P nx ¼A½BA n1 X DeterminetheMWDofthepolymerformedinabatchreactor. Solution:Inreactorapplications,recyclingiscommon(seeProblem4 .5) and monofunctional compounds are added to control the molecular weight of the formed polymer.