Fundamentals of Polymer Engineering Part 5 ppsx

within it are removed due to vigorous stirring, making it possible to controlreactor conditions more easily.It may be mentioned that batch, semibatch, tubular, and stirred-tankreactors s

is limited in batch reactors by equilibrium conversion Because we wish to formpolymers of high molecular weights, we can overcome this limitation by applyinghigh vacuum to the reaction mass On application of low pressures, the reactionmass begins to boil and the condensation product is driven out of the reactor, asshown in Figure 4.1b Such batch reactors are called semibatch reactors.Batch and semibatch reactors are ideal when the production rate of thepolymer needed is small In larger-capacity plants, continuous reactors arepreferred In these, the raw materials are pumped in continuously while theproducts are removed at the other end One example of these is a tubular reactor(shown inFig 4.1c) It is like an ordinary tube into which material is pumped atone end Polymerization occurs in the tubular reactor, and the product streamconsists of the polymer along with the unreacted monomer Sometimes, a stirredvessel (shown in Fig 4.1d) is employed instead of a tubular reactor Theadvantage of such a reactor is that the concentration and temperature variations

within it are removed due to vigorous stirring, making it possible to controlreactor conditions more easily.

It may be mentioned that batch, semibatch, tubular, and stirred-tankreactors serve as mere idealizations of actual reactors Consider, for example,

FIGURE4.1 Some ideal reactors

the industrial V.K tube (Vereinfacht Kontinuierliches Rohr) reactor, which is usedfor nylon 6 polymerization Its schematic diagram is given in Figure 4.2a, inwhich e-caprolactam monomer is mixed with water (serving as the ring opener)and introduced as feed In the top region, the temperature is about 220–270C andthe reaction mass is vigorously boiling The rising vapors produce intenseagitation of the reaction mass and ultimately condense in the reflux exchanger.

A small amount of e-caprolactam also evaporates in this section of the reactor and

FIGURE4.2 Schematic diagram of industrial nylon 6 reactors and reactor model.(Reprinted from Ref 1 with the permission of Plenum Publishing Corporation.)

is recycled to the reactor, as shown As the material moves downward, the reactorpressure increases due to gravity and the boiling of the reaction mass stops In thesecond stage, most of the e-caprolactam is reacted, and, in order to push thepolymerization to high conversions, it is desired to remove the condensationproduct (water) from the reaction mass To facilitate this, the reaction mass ispurged with a suitable inert gas (say, nitrogen) In the third stage, the viscosity ofthe reaction mass is very high and water cannot be removed by purging anymore.Sufficient residence time is provided so as to achieve the desired molecularweight of the polymer.Figure 4.1shows simple reactors andFigure 4.2bmodelscomplex reactors (e.g., V.K tubes for nylon 6) in terms of a combination of these.Due to intense agitation existing in the first two stages, the entire V.K column hasbeen viewed as a train of two homogeneous continuous-flow stirred-tank reactors(HCSTRs) followed by a plug flow reactor.

From the example of the V.K tube for nylon 6, we observe that simplereactors (Fig 4.1) are building blocks of more complex ones This chapterfocuses on analyzing simple reactors carrying step-growth polymerization

Chapter 3 has already considered polymerization in the batch reactor We firststudy the performance of semibatch reactors and examine the effect of flashing ofthe condensation product on it

We have already observed in earlier chapters that engineering materials shouldhave a large average chain length Suppose it is desired to have mn equal to 100,which would imply a 99.9% conversion of functional groups Step-growthpolymerization is limited by its equilibrium conversion, and there is a need topush the reaction in the forward direction This is done in industry by applyinghigh vacuum to the reaction mass, whereupon the reaction mass begins to boilunder the applied low pressure We know that polymer chains have very lowvapor pressures and, under normal conditions of operation, they do not vaporize;however, the monomer can This clearly means that in the presence of flashing,the concentration of any given species changes not only by polymerization butalso by change in volume V of the reaction mass We show the schematic diagram

of the semibatch reactor in Figure 4.3, and in the analysis presented here, weconsider the change in V as an explicit variable We assume that under theexisting reactor conditions, the condensation product W and the monomer P1canflash out of the reactor In all semibatch reactors, the monomer in the vapor phase

is condensed in a suitable separator and recycled because of its high cost It isassumed that the reactor is operating isothermally, at total pressure PT Thevolume of the liquid phase of the reactor, V , changes with time as flashing of Wand P1occurs We account for this time dependence as follows We define pnas

the total moles of species ðPnðn ¼ 1; 2Þ and w as total moles of W in the liquidphase The mole balance relations of these, on the dotted control volume shown

in Figure 4.3, are given by

prV
1; n  2 ð4:2:1bÞdw

Equations (4.2.1a) and (4.2.1b) are suitably added to determine thegeneration relation of the zeroth moment l0 and first moment l1 as

of volume V is given by

dV

wherevw is the molar volume of the condensation product W

In this development, there are seven unknowns ½ p1, pnðn  2Þ, W , l0, l1,

V , and Qw], 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 moreequation is required This is found by using the appropriate vapor–liquidequilibrium condition Herein, to keep the mathematics simple, we assume thesimplest relation given by Raoult’s law

Raoult’s Law

We assume that all the oligomers, pn, n  ¼ 2 are nonvolatile and that thecondensation product W and the monomer P1can vaporize If P0

wand P0 p1are thevapor pressures and xw and xp1are the mole fractions of W and P1respectively,then the partial pressures are given by Raoult’s law as follows:

The total pressure PT is then the sum of partial pressures; that is,

PT ¼ðP00P1þ P0

wwÞ

The previous chapter shows that the MWD of the polymer obtained from batchreactors is given by Flory’s distribution Now, let us show that, in the presence offlashing, 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,

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 themole balance relation, no matter what the concentration of W It is thus seen thatthe 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 ofP

Pn ¼ x

P

nPn ¼ xð1
yÞ2¼ l1 ð4:2:11bÞThese give

yðtÞ ¼ 1
l0

xðtÞ ¼ l

2 0

The addition of Eqs (4.2.1c) and (4.2.3a) gives

dðw þ l0Þ

which, on substitution into Eq (4.2.4), yields the following on integration:

V
V0 ¼
vw½ðw0þ l10Þ
ðw þ l0Þ ð4:2:14ÞHerein, w0 is the moles of condensation product in the liquid phase having totalvolume V0 at time t ¼ 0

We rewrite the vapor–liquid equilibrium in Eq (4.2.7) as follows:

w ¼P

0

Tl0
P0 p1p1

p10 and the final result can be derived as

ðb0þ b1l0Þdl0

dt ¼
kpl20þ k0

pðl00
l0Þa1l0 ð4:2:22Þwhich can be integrated to

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 P1, and the system behaveslike a closed reactor During polymerization, w increases and l0 decreases, andthere 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,

lc1

0 ¼w0þ l00

1 þ a1

ð4:2:29Þwhere superscript c1stands for this evaporation condition (called case 1) This isnow substituted into either Eq (4.2.27) or Eq (4.2.24) When P1 as well as W

evaporates (case 2), Eq (4.2.25) is used to eliminate w from Eq (4.2.16a), and lc2

1 Determine the time of flashing

2 Determine the values of l0and w in the reactor at equilibrium and themoles of w flashed

3 Calculate time taken to reach 101% of the equilibrium l0 Assume

kp¼ 1 and k0

pðbÞ ¼ 0:1

Solution: In normal conditions, the number of moles of monomer evaporating isusually small If the evaporation of P1 is small, a2 0at the transition iscalculated from Eq (4.2.25) Once this is known, the time when the flashing startscan be calculated from either Eq (4.2.8) or (4.2.20)

The equilibrium in the presence of flashing is reached when dl0=dt ¼ 0 or

kpl20þ k0

pwðl10
l0Þ ¼ 0where w is governed by Eq (4.2.16a) On eliminating w here, we get

kpl20eqþ k0

pa1l0eqðl10
l0Þ ¼ 0which gives l0eq as

l0eq¼ ak0pl10

kp0þ ak0 p

Let us assume that the units of kp and kp0 are liters moles per hour Calculationsreveal that flashing starts at 0.015 h and l0 at this transition point is 8.7 mol=L.The equilibrium values of l0and w are determined to be 0.1471 and 0.022 mol=Land 101% of this l0is 0.1486 mol=L In order to reach this value, the time needed

is 16.62 h and the condensation product flashed is 8.5 mol

Example 4.2: A mixture of monomer AR1B is polymerized by the step-growthmechanism with a monofunction compound AR2B in a batch reactor The

reaction mass consists of two molecules:

Pn: A½BAn
1B

and

Pnx¼ A½BAn
1X

Determine the MWD of the polymer formed in a batch reactor

Solution: In reactor applications, recycling is common (see Problem 4.5) andmonofunctional compounds are added to control the molecular weight of theformed polymer The overall polymerization is assumed to be irreversible and can

dt ¼
kp½P1xlP0d½Pn

dt ¼ kp n
1P

r¼1

½Pr½Pn
r
kp½Pnf2lP0þ lx0gd½Pnx

y ¼y0ð1
pAÞ

1
y0pASimilarly,

In the following, we assume the polymerization to be reversible The mole

balance equations for various oligomers can be derived using the followinggeneral relation:

Vd½species

dt ¼ ðspecies inÞ
ðspecies outÞ þ Vr

B¼ 0 ð4:3:1Þwhere V is the volume of the reactor and rBis the rate of formation of the species

by chemical reaction Using this equation, it is possible to derive the followingmole balance relations for all oligomers—assummg steady state and no change indensity:

be proved using Eqs (4.3.2a) and (4.3.2b) that the same is true for HCSTRs Thisfact means that

l1;feed¼ l1;product¼ l10 ðsayÞ ð4:3:3Þ

If we add Eqs (4.3.24a) and (4.3.2b) for all n, we get

1

y

P1 n¼1

½Pn
P1 n¼1

½Pr½Pn
r

¼
k0

p½W P1 n¼2

ðn
1Þ½Pn

þ 2k0

p½W P1 n¼2

P1 i¼nþ1

their numerical solution is usually slow and cumbersome These can, however, bedecoupled as follows:

Let us first assume that the condensation product, W, is not flashing fromthe reactor This means that in Eq (4.3.2c), therefore, Fw1is zero On adding thiswith Eq (4.3.4), we obtain

l00þ ½W0¼ l0þ ½W ð4:3:7ÞThis result is the same as Eq (4.2.25b) and can be directly derived from thestoichiometry of polymerization If [W] is eliminated between Eqs (4.3.4) and(4.3.7), we obtain a quadratic expression in l0:

is binary mixture consisting of polymer and condensation product Their molefractions, xp and xw, are given by

In this relation Aeqgreater than or equal to unity implies that the vapor pressure of

W is less than the applied pressure, PT This means that the condensation productwould not flash from the reactor, and l0 is given y Eq (4.3.8) Otherwise, Eq.(4.3.12) is used to eliminate [W] in Eq (4.3.4), and l0is solved The moles of thecondensation product, Fw1, can be calculated from Eq (4.3.2) as

Eq (4.3.6) by sequential computations

[5^11]

In several cases (e.g., in the manufacture of polyethylene terephthalate), theequilibrium constants of the reactions are such that one must remove the volatilecondensation products by application of a vacuum in order to obtain a polymer

having long chain lengths Because the desired degree of polymerization is about

100 for PET, the conversion of over 99% of the functional group must be attained.Under such conditions, the viscosity of the reaction mass is very high and thediffusivities of the volatile condensation product are very low Special wiped-filmreactors operating under high vacuum are then required in order to increase thesurface area and reduce the resistance to diffusion The analysis presented in thissection can be contrasted with that in Section 4.2, wherein the mass transferresistance was assumed to be negligible

One design of a wiped-film reactor is shown schematically in Figure 4.4, inwhich the reaction mass in the molten state flows downstream The reactor ispartially full and a high vacuum is applied inside Inside the reactor, there is arotating blade (not shown) that continually spreads the molten liquid as a thin film

on the reactor wall and, after a certain exposure time, another set of blades scrape

it off and mix it with the bulk of the liquid It is expected that most of thecondensation product, W, is removed from the film because it is thin and its area

is large If it is assumed that the material in the bulk is close to equilibriumconditions, the W removed from the film would perturb this equilibrium and, onmixing, the reaction in the bulk would be pushed in the forward direction Thisphysical picture of the wiped-film reactor suggests that the polymerization in thebulk is different from that occurring in the film It is necessary that appropriatebalance equations for the bulk and the film be written and solved The solution isusually obtained numerically

FIGURE4.4 Schematic diagram of a wiped-film reactor (Reprinted from Ref 1 withthe permission of Plenum Publishing Corporation.)

Depending on the relative amount of material in the film compared withthat in the bulk, two models are possible for the wiped-film reactor In one model,

it is assumed that the entire reaction mass is applied as a thin film and there is noseparate bulk phase present inFigure 4.4 This would mean that after some time

of exposure, the entire film is well mixed instantaneously and applied once againwith the help of the rotating blades In the second model, it is assumed that therelative amount of material in the film is negligible compared with that inthe bulk Because the removal of condensation product in the film occurs by themechanism of diffusion in a stationary film, the governing transport equations inthe film are partial differential equations (see Appendix 4.1) On the other hand,

in the bulk, where there is chemical reaction along with axial transport, thetransport equations are ordinary differential equations The performance of anactual wiped-film reactor lies between these two limiting models Fortunately, thetwo limiting models give results that are not significantly different; thus, only one

of them (the latter), which is more realistic, is described

A differential element of the reactor is considered as shown in Figure 4.4.Mole balance equations on the condensation product W and the polymermolecules Pn are written as follows The moles of W entering this elementper unit time are QơW and those leaving are QđơW ợ dơWỡ Meanwhile,

rwAbdx mol=sec are produced by polymerization (it is assumed that merization occurs primarily in the bulk) and nwasdx mol=sec are removed byevaporation from this differential element (through the film) Thus,