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A population balance approach to modelling of continuous emulsion polymerization
A POPULATION BALANCE1 APPROACH TO MODELLING OF CONTINUOUS EMULSION POLYMERIZATION R W THOMPSON*? and J D STEVENS Department
of Chermcal Engmeermg,
Iowa State Umverslty,
(Recerved 15 January 1976, accepted
Ames, IA 50011, U S A
18 June 1976)
&&r&-A populatton balance approach LSused to mathematically model contmuous emulsion polymenzatxm The Smtth-Ewart recurston relaaon ISmcorporated duectly as the “rate of formation” relation for polymer partxles As a result of this approach, the mechamsm of free raduxd desorption from polymer partxles 1s easdy mcluded, and fimte termmafion rates m parhcles are allowed Parkle growth ISmcorporated m the model, and parhcle sue &s&butions ate easily obtamed A bnef parameter study IS conducted M&l prechctions are compared to data from the hterature Practi&l consequencesof changes m system parametersare Qscussed BACKGROUND
Emulsion polymenzation processes are widely accepted as mdustrmlly unportant Although tis polymerrzation type has been smed senously for 30yr, the complex kmeucs mherent m the reactmn scheme has long obscured delis of the process Even today there 1s disagreement among scholars regardmg some of the mechamsms involved Most early workers m the field chose to use batch reactor eqmpment to mveswate emulsion polymemtion kmetics Small scale equpment and mmunum reagent requuements led qmte naturally to the use of batch studies Batch reactors, however, tend to be tune varmnt, 1 e transient phenomena are mtroduced Contmuous emuhuon polymenzers may be operated at a truly “steady state” con&tion, thereby allowmg easier analysis of the process In add&on, as more and more mdustnal processes become contmuoqs, a complete analysis of emulsion polymemation becomes more nnportant The conceptual picture of emulsion polymenzatlon was First set down by Harkms[l] m 1947 He discussed the nature of the components and the role played by each m the polymenzation system South and Ewart[2] then developed a mathematical descnptlon of Harkms’ conceptual picture They developed a steady state recursion relation whch descllbed the transfer of free rticals between the water phase and the polymer partxles Theu recursion relation allowed for radxals to be absorbed by polymer partxles to nutiate polymer growth, termmate wthm the pmcles, and desorb from the polymer parucles Snuth and Ewart were able to solve theu recursion relation for only three lurutmg cases The hnutmg case most widely &scussed 1s theu case II kmetics wherem the rate of ternunation of free radicals m the partxles 1s assumed mfimtely fast and the desorption mechamsm 1s assumed moperatwe Essentially, tis case assumes that only one free radical may exist m a growmg polymer parhcle, I e mutual termma on occurs mstantly at the arrival of a second radical &I e result of these assump*To whom corresoondence shbuld be addressed Kurrent address -Jkpt of Chem Engng, Worcester Polytechmc Institute, Worcester, MA 01609, U S A
uons 1s that the average number of free radlcats m a partxle, ii,, 1s 0 5 for the system That is, at any tune half of the partxles present have one radical and are m a growmg state, wtie the other half are mactrve It turns out that rS, takes on the value of 0 5 for only special cases Stockmayer[3] demonstrated the use of a generatmg function to convert the recursion relation to an ordmary Merentml equation whose solution was presented m terms of motied Bessel functions Stockmayer’s solution was correct when the desorption mechamsm was moperattve but was m error when desorphon was activated O’Toole[4] presented the correct solution for all magmtudes of the desorption mechamsm Ugelstad et al [Sl set forth the notion that free radical absorption and free radical desorption are not mdependent processes Because desorbed free radicals may mamtam the= activity and agam be absorbed by polymer particles, these mechamsms are duectly coupled Ugelstad et al mcluded tis effect and presented t$e results of a numerical solution to the steady state recursion relation Gardon[6] mamtamed the tune denvative term m hrs solution to the recursion relation, reahzmg full well that a batch reactor 1s never at “steady state ” That b results deviated mnmnally from those of Stockmayer 1s evidence that the “constant rate” tie penod may be adequately approxunated by the steady state solution Sundberga] showed that the pop&&on balance approach may be used to model batch emulsion polymemtion processes His model determined translent partxle size dlstxxbutions where the desorption mechamsm was ignored He was also able to pre&ct molecular we@ &stnbutions from hs model Contmuous &red tank reactors typically have mherent advantages for large scale commercml productions -urn manpower and mamtenance, easier operation and control, more consistent product quahty, apd higher throughputs all lend favor to the contmuous process The first s&cant Qscussion of contmuous emulsion polymerizatxon was presented by Gershberg and Lonsfield[81 Theu model assumed Srmth-Ewart case II kmetics and ignored partxle stze dtstriiutions Sato and Tamyama[9] presented results of another model of contmuous emulsion polymerrzation Their work, whtch followed duectly from an earher work concemmg batch 311
312
R W
THOMPSONand
polymenza~on[l0], treated emulsion polymemtion as a senes of bimolecular reactions between free radicals and polymer particles They ignored free radial desorption and parbcle size varuttmn Recently, Thompson and Stevens[lll extended the Sato-Tamyama concept to mclude the desorption of free ra&cals from polymer par&les Desorbed free raduzals were assumed to be active 111the water phase m keepmg with the arguments presented by Ugelstad et al [fl Parhcle size varmtions were not consldered Funderburk [ 121and Stevens and Funderburk [ 131flusbated the use of the population balance m contmuous emulsion polymtnzabon Ther model was capable of preQctmg overall particle size tistnbutions leavmg the polymerizer Free r&caI desorption was ignored The d&c&y m thev model was that the size dependent average number of free radrcals per pmcle, ii, had to be calculated before obtammg solutions to then population balance equation When usmg the Stockmayer relation for is, solutions to the population balance equation requved the evaluatton of the mtegraI of a ratio of mod&A Bessel functions of the followmg form
when workmg m parhcle volume coordmates The vanable X, used here as a dummy varmble, was theu dunensionless volume When the desorption mechamsm IS mcluded, the orders of the modified Bessel functions become x-dependent Evaluation of the mtegral then becomes practically ImpossiMe In a senes of arbcles[1&16], DeG& and Poehlem demonstrated a model they developed to predict eWuent parbcle size &stnbubons from a contmuous emulsion polymenzer They coupled the residence time dlsbnbtion m continuous welI nuxed vessels to the Stockmayer model to amve at a model sun&u 111 nature to Funderburk’s Theu pre&ctions were sinular to Funderburk’s as were the shortcommgs of thev model They were reqmred to evaluate slrmlar mtegrals and could not mclude the desorption of free radicals for the same reasons The most general and extenstve pubhshed work to date regardmg mathematical modelhng of emuislon polymeru&on processes IS the large rewew ar&le by Mm and Ray[f7] After carefully revlewmg the modelhng efforts presented thus far, they proceeded to develop a model far more comprehensive than any previously formulated These authors mcorporated more mechamsms than ever presented elsewhere, however, they &d not show any quantitative results generated from thew equations The model presented herem, although developed mdependentiy of Mm and Ray’s efforts, 1s sunilar to but less detied than thev model Consequently, we are able to present quantitative results These results are presented m the hopes of furthermg understandmg of em&ton polymemtion, as well as providmg msight mto the practmal consequences of changes m system parameters
J D STEVENS POPULATlON
BALANCE
EQIJATlONS
It 1s desrred to predrct the srxe drstrrbutron of polymer partrcles leavmg a contmuous emulsion polymertxer At the outset we define a number densrty of polymer parttcles as follows N” = iV.(U, t)
(2)
such that N, dv representd the concentration of polymer pticles between the sues v and v + dv havmg n free radicals at tune t In reabty, N,, defines a fanuly of functions, as n LS a parameter To define the system completely, one must describe tlus collection of quantities ) as functions of the internal (charactetrs(N0, N1, AL tic of the parMe dnnensrons) and external (characterrstrc of the system) vanables Note that the system has been assumed homogeneous as spatral varratrons were not considered One 1s now m a posrhon to wnte a populauon balance for the N.-type parttcle using any of the techmques &scussed m the hterature[7,12,13,17-201 The resultmg populaaon balance equation, whose soiutron descrrbes the partrcle srxe distrrbuttons of all the N,-type partrcles and the overall srxe distribubon, 1s
where R., represents the volumetnc growth rate of a polymer partrcle, dv /dt, and r,, represents the net rate of formabon of the N.-type part&e Asrde from the few assumpbons made thus far, eqn (3) 1s perfectly general applymg to both batch and contmuous reactors It remams to arnve at meamngful expressrons for th& partrcle growth rate, R.,, and the rate of partrcle formabon, rN”,to solve spectal cases of eqn (3) PARTICLE GROWWRATE The volumetrrc growth rate expresston presented by Gardon[21] wrll be used here
where
(5) Note that R., IS proportronal to the number of growmg chams present, n Several of the quantrbes makmg up K may be partrcle srxe dependent For the purpose of rllustration K wdl be assumed partrcle srxe mdependent The model 1s sufficiently general to handle erther case MT% OF PARTICLE FORMATION The Snuth-Ewart recurslon relation ISm reahty nothmg more than a rate of formabon expression for the X-type polymer partuzle We wrll here use a shghtly moddied form of therr expressron, altered to allow r&Cal capture of the polymer partscles to be proportronal to the fractron
A populataonbalanceapproachto modem of &he total surface area represented by a mven par&cle type Dlscllss10n of thus pomt appears ln the hterature[12,21] Wntten m terms of the current work, the rate of formation expresslon, rN,, Ls r,, = % {(n + 2)(n + l)N”,, - n(n - l)N-“} +Bu~0v”-,-N”I+$J(n
+l)N”+I-nN.1,
(6)
where (L = _$,
p = (4rr)1’TY3PA, and
y = (4?r)“‘(3)y3ko
of contmuousemukon polymenzatIon
A, These four constants represent the ratios of the four
mechamsms (r&Cal capture, bnlk flow, radical desorpt.lon, and r&Cal termmation) to the growth rate constant, K. We are now In a postion to solve the populatmn balance equations Attentmn ~IUbe focused on the steady state contmuous stwred tank polymerrzer It 1s assumed that thxs 1s the first tank m a smes such that there IS no polymer m the feed Under such consbunts, eqn (9) reduces to $ (a_)
= -A5X,, + $ {(n + 2Mn + 1)X., - n(n - 1)X. }
Q In eqn (6) the parameter n Is to take on non-negative integer values Megatwely subscnpted IV” functions are nonexistent and their value set equal to zero AssEMBLxLI MODEL Combnnng eqns (3). (4) and (6), one anrves at the complete model
+ 30J + 2)(n + UN”,, - n(n - f)N”} +BuU-‘(N.-I-N.)+&{(n By allowmg
+ l)N.+,-nN,)
= AAX., -Xi)
&?‘LX,-, - x.1 + 3 {(PI + 1)X.+, - nx,,) BOUNDARY
COND~ON
CONSlDERATlONS
N,,=N,+N,
(12)
1=x,+x,
(13)
when expressed m dunenslouless form ‘flus assumption unphes that at C = 1, X” = 0 for II > 1 No boundary comhtion IS required for the X&ype pticle, as the equation descnbmg It IS merely algebmc The boundary con&fion for the XI-type par&le ISfound by wr~tmg the equation generated If n takes on the value of zero m eqn (11) 04)
0=-A~Xo+W3Wi~+$tW~~
where all X” functions for n > 1 have been set equal to zero However, for the special case when mutual temnnation results unmedtately at the amval of a second radud to a pmcle, eqn (14) 1s mcorrect The corrected equation, 0 = - A,X, + A,J’/3(X, - X,)1+ $
+?{(n
(11)
The boundary conditions requued are that the number density functions, X., must all be known at a smgle value of i For tis purpose we wdl specify these functions at “zero s=e”, or the size of a nucelle It 1s assumed that parkles the sLze of a nucelle cannot contam more than one free radtcal. Then, at “zero size” the total number of pa&cles IS gwen by
(8)
entue fanuly of population balance equations ISreahzed Negatively subsctrpted functions are nonexistent and defined as zero Before proceedmg, eqn (8) wdl be made dunensionless The advantages of solving problems m dunenslonless vmable form are well recogmzed m many techmcal cucles Not only are the adjustable parameters whch must be exammed reduced m number, but one set of theoretical results may be used to mterpret data from a wide variety of expenmental comhtions By employmg the defimtions found m the Appendnc, eqn (8) becomes $$+$(nX.)
+
or as
n to take on non-negative mteger values, the
313
{X,}
m
+2)(n + 1)x.+,- n(n - 1)X”)
+ A,&-?X”-I - x.1 + 3
{(n + 1)x.+, - nx. 1
(9)
where, m add&on to the defimtions found UI the Appenduc, the followmg dunenslonless vanables were mtroduced
(10) The problem IS now reduced to a famdy of equations mvolvmg four dnnenslouless parameters, A,, AS, As and
shows that formation of X0 par&les IShmited only by the rate of amval of a second radml Now allowing 6 to take on the value of 10 gwes 0=-A,Xo+AJX,-X,)+A,{X,}
w&h,
when solved together wtth eqn (13) yields
A,+&
XO= 1 O+A,+2A,’
lO+A, x, = 1 O+A,+2A,’
(16)
R W ~OMPSONand J
& =
- Xl x0+x,
=
x,
D
STBVBNS
Iowa State Unrversrty The fourth order Runge-Kutta numerical mtegration was camed out m double prectsron along the drmensionless volume axrs, 5 The program developed accepts mformatron read m regardmg the values of the four mdependent drmensronless parameters After uutrahxauon of some quantmes the boundary values are calculated usmg eqn (17) Integratron for the present case involved eleven smmltaneous equattons and proceeded unttl a smgle mtegration step contributed less than lo-‘% to the total cumulauve partrcle number density At thrs pomt, the total partrcle concentratron, P, and the total acttve &am concentrauon, R, terms which had been cumulattvely determined as the mtegratron proceeded, were recorded A parameter study was conducted to mvesttgate the effect of varymg the parameters on the number den&y results Much of what follows are the results of thts study The amount of rnformatron generated was somewhat hmtted due to the extremely large computatron tunes
(17)
where A, and A3 are delined m the Appendrx and F&ISthe average number of radrcals per “zero srxe” partrcle Frgure 1 shows a plot of fiOas a funcuon of A, at a constant vahre of As, where AI was vaned as a parameter By recalhng that A, IS a fun&on of the desorptron rate constant and that A4 IS duectly proportronal to uutrator concentratron, the sumlantres of Frg 1 to Frg 2 of Ref [ 11J become apparent All quahtatrve drscussron of those results apply to Fig 1 here and are not repeated for the sake of brevrty Equatrons (17) form the boundary condrtrons for the current work. NuMERIcAL.aaguLTs The population balance equatrons for contmuous emulsion polymerrxatron were solved on computers at
0As=5
8-
106
A4
03x lo
-5
1o-4
Fla 1 Dependenceof rS.on A, and A, from population balance model (X. = o for n > 1)
A5=503x10 A6=OO0 A7 = 10
I
I
10
100 DIMENSIONLESS
Fw 2
Volumetnc
parhcle slzc Qstnbutions
m contmuous
I
I
1000 VOLUME,
em&Ion
10,000
100,00(
r
polymenzer,
demonstratmg
effect of A,
A population balance approach to modelbm18of contmuous emulsion polymertza~on Table 1 Values of the parameters used as the base case A, = lo-’ A,=503xlO-”
A,=000 A,= 10 The base case (shown m Table l), for example, reqmred 1125 set to mtegrate to completion The numencal mtegratron was normally camed out between the values 1-z 5 < 5 x 104,and the volume mcrement At was normally less than 10, bemg 0 10 for the base case Some rules of thumb were observed regardmg the effect of parameter vanations on the program Itself A decrease of A, by an order of magmtude reqmred that the mtegration be camed out roughly one half an order of magmtude further along the f axis An mcrease of A, by an order of magmtude usually reqmred that the volume mcrement, A& be decreased by an order of magmtude An mcrease m A6 made the program run shghtly slower as a result of the ad&tional terms mcluded Varmtion of A, seemed to have httle effect on the computation tnne The results presented herem are based on the equations generated by lettmg n take on values from 0 to 5 It was found early m the work that the results were somewhat senstive to termmation of the mlimte set of equations To mmumze these effects, the mtegration scheme mcluded eleven equations whde only the iirst su. were used for computatzonal purposes
requwed
VAltLwmNS
partxles Tlus 1s logwal If one notes that as the rate of generation of free radicals (proporhonal to I,, and therefore to A,) mcreases, the rate of rtical absorption
by parhcles mcreases Faster r&Cal arrival rate unphes faster rate of termmafion m the pticles That Is, on the average when radtcal amval rate mcreases, growmg chams are not permitted to grow as long before a rtical 8mves to cause termmation Partxles m such a situation wdl grow less than ones where termmation occurs less to often As A, mcreases there are more free rticals contnbute to termmation The quantity A, IS also mversely proportional to So, the
soap concentration As So decreases (A. mcreases) fewer mlcelles become avadable for radical capture Thus there are more radicals avdable for termmabon, wbch m turn produces more small polymer pticles Fwe 3 shows cumulative pticle size &stnbution for the same set of parameter values hsted m FW 2 The curves m Fw 3 depict the area under the curves of Fg 2 as a function of p The quantity P’, defined as
(18) represents the dunensloniess cumulative concentration of parQcles havmg size up to t When the curves become honzontal, there are essentially no more par&cles to be counted The dmensionless total concentration of parhcles IS gven by P=I-Xrdf
(19)
-.
IN &
Several computer runs were made m wluch the effects of varymg A, (ratio of r&Cal absorption rate constant to parMe growth rate constant) were tested F-e 2 shows the overall pm%cle size &stibution as a function of dunenslonless volume (Throughout tlus &cle “overall” means that the contibtion of each group of N.-type par&les 1s m&&d m the quantity bemg &scussed ) It appears from Fe 2 that an mcrease m A, (proporhonal to IO/SO) yields fewer large polymer
1
315
10
No conclusions concermng absolute numbers of par& cles can be drawn from pigs 2 and 3 unless the value of NrO(used m X,) is known for a given system Fme 4 shows the size dependent value of A and the cumulative system average value, A:, both as functions of 5 These quantities, defined as ri _ XnX zx. ’
loo DIMENSIONLESS
Fhg 3 Cumulatwe danenslonless pticle
10,ow
1.VOLUME,
wze &stnbt&ons,
r
showmg effect of A.
(20)
100,000
R W
316
--hKMPSON andJ
D STEVEE~S
J
A5=5 03x A6 = 0.00 AT=10
4-
1O-5
3-
1
10
100 DIMENSIONLESS
Fhg 4
A
(21)
gve some m&cation of how the free raduxls are dlstibuted m particles of varrous sizes When the curves of A: vs g become homntal all of the par&les have essentmlly been mcluded m the mtegration The value of a: at the hollzontal level then becomes the value of Is,,the system value of ii That IS, _ I I
nT
d5
‘,
(22) XT
10,000
100.00
, c
andfi: as functionsof parkle size.&Wrabng effectof A.
and
ii,=RIP=
1000 VOLUME
dS
1
where rtT ISthe total number of free ra&cals in a partxle and R IS the dunenslonless total number of growmg polymer chams F-e 5 shows a recapttulation of these results with A, the mdependent varmble The values of P (overall partxle concentration), R (propotional to the system rate of polymerrzation). and fi, are shown as functions of A, These results are not altogether consistent wtth some data reported m the hterature [ M-161 They report that the total concentration of pa&cles (and the rate of polymenzatlon) IS an mcreasmg function of So, vvlth whch we are in agreement However, they show that these quantutes are mvanant vvlth respect to I0 Our results would uxhcate that the total concentration of parhcles (and the rate of polymenzation) are decreasmg functions of I0 Thus Qscrepancy IS probably a result of usmg Merent values for the dunensronless constants Ad, , A, One attractive feature of tbs model IS that m soMng the parkle size dlstnbution eqwons one obtams
mformation regardmg the unportance of each number density function mcluded m the Megration Fwe 6 shows these functions for the base case Note the dommance of the X0 and XI funchons up to about I = 100 Beyond C = 1000 the contnbufions to XT from X0 and XI grow lncreasmgly nehble As g mcreases, each X,, function becomes important for a short tune, then, as Its unportance wanes, X. functions for higher II levels begm to dommate As n mcreases, the absolute maxunum of the X. functions gets smaller Physically, these results indicate that larger pticles may house a greater number of free radicals, but that there are mcreasmgly fewer of these larger partxles Results shown in Frg 6 gve an m&cation of how many X,, functions nught be reqmred m an mtegration scheme Unfortunately, thus mformation ISgathered after the fact, 1 e one learns whch X. functions are relevant only after performmg the integration DESORPTION OF FRFB RADICALS
Fwe 7 shows a sun&r composite plot where now the desorption mechamsm has been “turned on” m the equations by vn-tue of allowmg As to take on a value of 0 05 Note the Merent shape of the XT vs g curve, and that the m&vldual X, curves are much suppressed The effect of the desorption mechamsm IS to force free radicals, once mslde the par&les, back mto the water phase The desorption phenomena keeps the X. functions substantmlly lower then the same quanmes shown when desorption was inactive The effect is so dramatic that the Xs function does not even appear on the plot m Fw 7 Fme 8 dustrates the effect of the desorption mechamsm on the overall polymer parkle
size dlstrrbu-
tion In tl~s drawmg, the magmtude of the desorption parameter has been vmed When A6 = 0 00 the curve represents the base case results shown earher It IS obvious that the desorption mechamsm may have a
A populahon balance approach to modellmg of contmuws emulsmn polymenzattoo
\ A5= 5 03 x Ag=O 00 A,= 10
0%
1O-5
04-
13-
I2 1o-5
1o-4
1O-3
1c
A4 Fw
5
Overall partxle concentrahon, active polymer cham concentrahon, and average number of ralcaals per parhle as functions of A,
BASE CASE
DIMENSIONLESS Fig 6
VOLUME,
5
Par&le size dlclirlbuwm results for X funchons
base case
317
R W
~OMI'SONand J D
STEVENS
A4 = 1O-3 A5 = 5 03 x 1O-5 A6=0 05 A,= 10
DIMENSIONLESSVOLUME, 6 Fig 7 Part&
1
size distribution results for X,, functrons desorphon mechamsm activated
10
100
1000
DIMENSIONLESS
VOLUME,
10,000
100,000
r
Fa 8 Partxle s&e dlstnbuhons showmg the effect of the desorptlon mechamsm drastic effect on the shape and posttions of the final parbcle sue dlstriiubon curves The effect noted IS as expected Desorption of free ra&cals lowers the growth rate m the particles, gvmg a product havmg fewer large par&les Figure 9 shows the ri and ii: functions as a function of pax%cle size for various values of the desorption
parameter, A, The results show a marked shrft of free radicals out of the partuzles, as expected One shortcommg of the current model IS found m the rate of generation of radicals. p_, In the current work It was tacitly assumed that pA was constant However, Ugelstad et al [5] have shown that desorbed radicals may mamtam their a&v&y and agam attack polymer parbcles
A populationbalanceapproachto modelhagof contuu~ous emulsion polymewzahon
_
A4=
A5’5
319
10-3
03 x 1O-5
A, = 10
1
IO l%g 9
100 DIMENSIONLESS
TLMB, VARIATIONS
One prmclpal mdependent parameter m any physical contmuous system IS the residence tune m the reactor, T In order to mvesmte the effects of 0u.s parameter several computer runs were made where A, (mversely propotional to I) was vaned These results are shown m Fig 10 where overall parttcle size dlstnbution results are plotted Values of A5 were chosen both a factor of ten lugher and a factor of ten lower than A, used m the base case All other parameter values were held constant (The effect of 7 vanations in A, were neghgiile ) As one m&t expect, shorter residence tunes yield products havmg relatively more smaller polymer part.~l& and fewer larger parhcles That IS. the shorter growmg tunes produce a latex havmg a narrower parhcle size &stibution Surprismgly, however, for these values of the patame ters, varzations m A5 showed muumal effect on the
Cl!%Vol
32 No
3-F
C
10,Ow
100,000
Effectof dcsorptmn of radicals on fi and 6: fundons
to uutiate polymer growth or cause termmahon of a growmg cham In such a case, pA would become a fun&on of the desorptlon mechamsm and the pmcle sue dlstibution To correct the situation would unply an iterative solution In that case the current results would be merely the lirst step m the Iterative scheme The results presented here are str~tly correct when desorption of free radicals does not occur The error wdl be s&cant only when the desorpfion mechamsm contnbutes appreclabiy to the concentration of free radials m the water phase In many real systems this 1s not the case, 1e desorptlon of free ra&cals IS a somewhat mmor phenomenon These results are no more subject to error than others m the hterature[3,4] Other workers m the field dlsrmss the problem by neglectmg desorption aftogetber[l2-161 mC!E
1000 VOLUME,
results Whereas two orders of magmtude cover about the entire useful range m T, scatter m typlcal pticle size data would make dlstmction of T varmtions vutually unposslble These results m&t be satisfymg to those operatmg a productlon unit, for they mdlcate that 7 may fluctuate wlthout much consequence The reader should keep 111 mmd that for other choices of the other independent parameters these conclusions may change shghtly For example, our results deviate from those reported by DeGraff and Poehlem[14-161, probably because of the dtierences m dunenslonless constants ANALYSISOF FUNDERRURR’S RUN #4 One set of data was extracted from the work by Stevens and Funderburk[13] Data from theu run # 4 (Fig 10 m then work) was converted to a form amenable to analysis by thusmodel Shown m Fe 11 are theu data, the values of the parameters calculated from their work, and the theoretical predictions from tlus work We the agreement between experunental and theoretical results 1s not as good as desued, observe that the theoretical predictions do fall m the same general vlcmlty as the data Secondly, whereas Stevens and Funderburk mampulated two adJustable constants to a&eve a good fit to their data, the current model predictions are based on thev parameter values m unaltered form The most questionable parameter fittmg done by Stevens and Funderburk was an attempt to mathematitally back out the effect of an m&bltor m the monomer feed wtuch had not been physically removed They dlv&d the actual residence tune mto an mdumon permd and an effective residence tune, arnvmg at an effective polymemfion tune of about two-tids the actual residence tnne The theoretical prtictions presented here
R W
-hiOMPSON
and J D
th3VENS
A4 = TO-~ A&=0
00
A,=10
0
I”
Fa
s
E - ^. vui
g Z %
5
n
Effect of residence tune v-hens
VOLUME,
r
on the overall partxle size Lstnbution
Ad=5
644x
lo-”
\
A5=3
299x
1O-4
0
I
5
2 2
10
0001
!
1
10,OC
loo0
IUJ
DIMENSIONLESS
0
A6=0
00
A7 = T 1005 x lo2 DATA FROM I 10
STEVENS
AND
FUNDERBURK
(13) 1 1000
100 DIMENSIONLESS
VOLUME,
10,OWJ
Fu 11 Population balance model predxfions of data from Funderburk’s wn used theu calculated effective residence time It seems feasible to suggest that using a higher reszdence tune m&t have provided a better fit of the data, although 111 v-zew of the results of FIN 10 it LS doubtful tis change would have made s&cant Merence m the results Also m FU 11 are shown some theoretical curves where the desorpuon parameter, Ad, was activated From these results one nught suggest that the desorption mechamsm was shghtly activated m Funderburk’s expertments Because It would have reqmred extensive computer tune and because of the questionable value of T, further study of the data was not conducted
100,001
(r # 4
CONCLUSIONS
A population balance model of contmuous emulsion polymemtion processes was developed and solved A paramemc study demonstrated several mterestmg features of the model and the mechamsm of emulsion polymemtion and provtded ms@t mto some of the engmeermg ram&cations of such processes Advantages of the model mclude the fact that free rtical desorption from polymer pmcles may be easdy included Comphcated mtegrals of mod&d Bessel functions need not be evaluated Also, whde other workers[12-151 were reqmed to postulate a value of R (or Is.) for their models to
A population balance approach to modellmg of contmuous
these quantaties are found here as a consequence of obtammg partxle size dlstriiuttons from the polymerlzef The effects of several key parameters on the partxle size dlstrrbution of the emuls&n polymer product were mvestigated These parameters embody specfic physlcal constants of the system (1) The parameter A, (a measure of absorption of radxals relative to growth rate) is propotional to &/SO, two controllable system vanables As shown m Figs 2-5, an increase m A, (marufested by an mcrease m I0 or a comparable decrease m S,,) narrows the pmcle size &stnbution Decreasing A4 tends to broaden the dlstibutlon and produce more large pticles A decrease m A4 also causes a reduction 111polymer production rate (2) In tbs study, variations m the average residence tune (mcluded m AS) produced little vanation m the overall particle size dlstnbution Wrth ddferent values of some of the other parameters, residence tune vanatlons may mdeed be s&cant In general, longer residence tnnes should produce broader partxle size dlstnbutions, although the effect noted herein was muumal (3) Vmations m the parameter A, (ratio of free radical desorption rate constant to pmcle growth rate constant) produced s&cant changes m the partxcle size dlstnbubon of the emulsion polymer product we this mechaNsm is a function of the particular monomerpolymer radical system under consideration and not tiectly controllable, the model presents a means of estunatmg the magmtude of its effect An increase m the magmtude of the free radrcal desorption mechanrsm can smcantly reduce the numbers of large partxles 111the product streams (Figs 8 and 9) Such an increase would also result m a lower polymer production rate Actual par&b size &strrbufion data may be viewed from the perspective of various degrees of the desorption mechanism as shown m Fig 11 A further advantage of the current model IS that mformation IS generated dEectly regarding the pticle size Qstibution of partxles contammg it = 0, 1,2,3, active free ra&cals As many of the X. functions may be mcluded m the integration scheme as reqmred by the mvest@ator without sigmficantly affectmg the complexity of the model Figures 6 and 7 show typical results obtamed for a particular set of parameter values The value of A6 (free radical desorptlon mecharusm) was the only parameter changed between the two graphs From these results the values of A (or ii,) are determmed, as IS the complete particle size dlstnbution functton,
emulsion
the course of tlus work R
W Thompson has received financial assMance from the Umon Carbide, DuPont, and Proctor and Gamble Compames Both authors have been supported by the Engmeer&g Research Institute at Iowa State Umvers@, who also supported the computer calculations and the techmcal drawmgs presented herem Our appreciation IS here pubhcly extended to these organuations NOTATION
AI, Aa
, A,
dunenslonless pend=
constants defined II-IAp-
321
IO feed
concentration of m&&r, gmmole/l K growth rate constant, cm’/hrI # k, uutitor decompostion rate, hr-’ desorptton rate constant polymeruat8on rate constant % termmatmn rate constant N.4 Avogadro’s number Nl number density of polymer partxles m the system, #Iv2 number density of polymer N feed, I Iv” N,, NTO the number density of polymer parhcles at 2ero sue”, # Iv2 n number of radxals per parhcle, # t7 sue dependent average number of radlcals per parWle, # average number of ra&cals m “zero size” partxles, # iiS = RIP total system value of rS, # IT:= R’lP’ cumulative overall system value of fi, # P =$;“X,dc dunenslonless total partxle concentration P’ =$fXT dl cumulative dnnenslonless total partmle concentration R=f;nTdc dunensionless total active cham concentration R’ =s:?lT dC cumulative dlmenslonless total active cham concentration R., volumetic growth rate of N.-type par& cle, cm’lhr rate of formation N,-type partxle, # /cm6 hr s total parkle surface area so feed concentration of emulsdier, gmmole/l t tune, hr micellar volume, cm3 VO V parkle volume, cm3 dlmenslonless dummy vanable N eqn (1) XX dunenslonless number density X: dunenslonless overall number density for polymer product Greek symbols 7
average residence tune m reactor, hr
Pm monomer denstty, gN/cm3 PP polymer density. gm/cm’ PA 8 c
Acknowledgements-Dunng
polymenzation
W&Y hI
rate of free radical production dnnensionless tune dunenslonless partxle volume constants defined m eqn (9) monomer fraction m parkles, sionless
diien-
[l] Harhns W D , I Am Chem Sot 1947 69 1428 [2] Snuth W V and Ewart R H , I Chem Phys 1948 18 592
131 StockmayerW H , I Poiy SCI. 195724 314 [41 OToole J T , J Appl Poly Sm. 1%5 9 1291 [Sj Ugelstad J ,MiirkP C and Aasen J 0 .I Poly SCI Part A-l 1967 5 2281 [6] Garden J L , J Poly Scr Part A-l 1968 6 665
322
R W THOMPSON
STEVENS
[fl Sundberg D C , Ph D dissertation, [8]
[9] [ 101 1111
Umversrty of Delaware 1970 Gershberg D B aad Longfield J Et unpubhshed paper presented at the Synrp Poly Kwettcs Cat Systenis, 54th AIChE Meetmg, New York, l%l, preprmt No 10 Sato T and Tamyama I , Kogyo Kuguku Zasshr 1965 68 106 Sato T and Tamyama I , Kogyo KagoRu Znsshl 1964 68 67 Thompson R W and Stevens J D , Chem Engng Scr 1975 30
and J D
667 -__ [12] yunierburk
J 0
, Ph D hssertation,
Iowa State Umverslty
[13] Stevens J D and Funderburk J 0 , I&ECPmc Des &Den 1972 11360 1143 DeGraff A W , Ph D dlssertatlon, I&& Umversity 1970 rl5l_ DeGratTA W andPoehlemG W .I. Polv_ Scl Part A-Z 1971 _ 9 1955 1161 Poehlem G W, Advances m Emul Poly and Latex Tech 1973, 4th Annual Short Course. Letih Umverslty, 378 [17] Mm K W and Ray W_ H., J Mamma01 >cr -Revs Macromol Chem 1974 Cll(2) 177 [18] Behnken D W , Horowitz J and Katz S , I&EC Fund 1963 2 212 [19] Hulbert H M and Katz S , &em Engng Scr 1964 19 555 [20] Randolph A D and Larson M A, Theory of Partrculute Processes Academic Press, New York 1971 [21] Gardon J L , J Poly Scr Part A-l 1968 6 623
The seven dunenstonless defined for the convemence competm8 mechamsms
_ radrcal absorpucm rate constant partxcle growth rate constant
A, = Ndw,u - 4M) km& space velocity 3 partxle growth rate constant
= radxal desorption rate constant p&cle growth rate constant
=
radical terounation rate constant pticle growth rate constant
A,=AJA,
APPmDm
A, = AT/A,
parameters used m tis study are here of the reader They represent ratios of
A, = AJA,
of Chermcal Engmeermg,
Iowa State Umverslty,
(Recerved 15 January 1976, accepted
Ames, IA 50011, U S A
18 June 1976)
&&r&-A populatton balance approach LSused to mathematically model contmuous emulsion polymenzatxm The Smtth-Ewart recurston relaaon ISmcorporated duectly as the “rate of formation” relation for polymer partxles As a result of this approach, the mechamsm of free raduxd desorption from polymer partxles 1s easdy mcluded, and fimte termmafion rates m parhcles are allowed Parkle growth ISmcorporated m the model, and parhcle sue &s&butions ate easily obtamed A bnef parameter study IS conducted M&l prechctions are compared to data from the hterature Practi&l consequencesof changes m system parametersare Qscussed BACKGROUND
Emulsion polymenzation processes are widely accepted as mdustrmlly unportant Although tis polymerrzation type has been smed senously for 30yr, the complex kmeucs mherent m the reactmn scheme has long obscured delis of the process Even today there 1s disagreement among scholars regardmg some of the mechamsms involved Most early workers m the field chose to use batch reactor eqmpment to mveswate emulsion polymemtion kmetics Small scale equpment and mmunum reagent requuements led qmte naturally to the use of batch studies Batch reactors, however, tend to be tune varmnt, 1 e transient phenomena are mtroduced Contmuous emuhuon polymenzers may be operated at a truly “steady state” con&tion, thereby allowmg easier analysis of the process In add&on, as more and more mdustnal processes become contmuoqs, a complete analysis of emulsion polymemation becomes more nnportant The conceptual picture of emulsion polymenzatlon was First set down by Harkms[l] m 1947 He discussed the nature of the components and the role played by each m the polymenzation system South and Ewart[2] then developed a mathematical descnptlon of Harkms’ conceptual picture They developed a steady state recursion relation whch descllbed the transfer of free rticals between the water phase and the polymer partxles Theu recursion relation allowed for radxals to be absorbed by polymer partxles to nutiate polymer growth, termmate wthm the pmcles, and desorb from the polymer parucles Snuth and Ewart were able to solve theu recursion relation for only three lurutmg cases The hnutmg case most widely &scussed 1s theu case II kmetics wherem the rate of ternunation of free radicals m the partxles 1s assumed mfimtely fast and the desorption mechamsm 1s assumed moperatwe Essentially, tis case assumes that only one free radical may exist m a growmg polymer parhcle, I e mutual termma on occurs mstantly at the arrival of a second radical &I e result of these assump*To whom corresoondence shbuld be addressed Kurrent address -Jkpt of Chem Engng, Worcester Polytechmc Institute, Worcester, MA 01609, U S A
uons 1s that the average number of free radlcats m a partxle, ii,, 1s 0 5 for the system That is, at any tune half of the partxles present have one radical and are m a growmg state, wtie the other half are mactrve It turns out that rS, takes on the value of 0 5 for only special cases Stockmayer[3] demonstrated the use of a generatmg function to convert the recursion relation to an ordmary Merentml equation whose solution was presented m terms of motied Bessel functions Stockmayer’s solution was correct when the desorption mechamsm was moperattve but was m error when desorphon was activated O’Toole[4] presented the correct solution for all magmtudes of the desorption mechamsm Ugelstad et al [Sl set forth the notion that free radical absorption and free radical desorption are not mdependent processes Because desorbed free radicals may mamtam the= activity and agam be absorbed by polymer particles, these mechamsms are duectly coupled Ugelstad et al mcluded tis effect and presented t$e results of a numerical solution to the steady state recursion relation Gardon[6] mamtamed the tune denvative term m hrs solution to the recursion relation, reahzmg full well that a batch reactor 1s never at “steady state ” That b results deviated mnmnally from those of Stockmayer 1s evidence that the “constant rate” tie penod may be adequately approxunated by the steady state solution Sundberga] showed that the pop&&on balance approach may be used to model batch emulsion polymemtion processes His model determined translent partxle size dlstxxbutions where the desorption mechamsm was ignored He was also able to pre&ct molecular we@ &stnbutions from hs model Contmuous &red tank reactors typically have mherent advantages for large scale commercml productions -urn manpower and mamtenance, easier operation and control, more consistent product quahty, apd higher throughputs all lend favor to the contmuous process The first s&cant Qscussion of contmuous emulsion polymerizatxon was presented by Gershberg and Lonsfield[81 Theu model assumed Srmth-Ewart case II kmetics and ignored partxle stze dtstriiutions Sato and Tamyama[9] presented results of another model of contmuous emulsion polymerrzation Their work, whtch followed duectly from an earher work concemmg batch 311
312
R W
THOMPSONand
polymenza~on[l0], treated emulsion polymemtion as a senes of bimolecular reactions between free radicals and polymer particles They ignored free radial desorption and parbcle size varuttmn Recently, Thompson and Stevens[lll extended the Sato-Tamyama concept to mclude the desorption of free ra&cals from polymer par&les Desorbed free raduzals were assumed to be active 111the water phase m keepmg with the arguments presented by Ugelstad et al [fl Parhcle size varmtions were not consldered Funderburk [ 121and Stevens and Funderburk [ 131flusbated the use of the population balance m contmuous emulsion polymtnzabon Ther model was capable of preQctmg overall particle size tistnbutions leavmg the polymerizer Free r&caI desorption was ignored The d&c&y m thev model was that the size dependent average number of free radrcals per pmcle, ii, had to be calculated before obtammg solutions to then population balance equation When usmg the Stockmayer relation for is, solutions to the population balance equation requved the evaluatton of the mtegraI of a ratio of mod&A Bessel functions of the followmg form
when workmg m parhcle volume coordmates The vanable X, used here as a dummy varmble, was theu dunensionless volume When the desorption mechamsm IS mcluded, the orders of the modified Bessel functions become x-dependent Evaluation of the mtegral then becomes practically ImpossiMe In a senes of arbcles[1&16], DeG& and Poehlem demonstrated a model they developed to predict eWuent parbcle size &stnbubons from a contmuous emulsion polymenzer They coupled the residence time dlsbnbtion m continuous welI nuxed vessels to the Stockmayer model to amve at a model sun&u 111 nature to Funderburk’s Theu pre&ctions were sinular to Funderburk’s as were the shortcommgs of thev model They were reqmred to evaluate slrmlar mtegrals and could not mclude the desorption of free radicals for the same reasons The most general and extenstve pubhshed work to date regardmg mathematical modelhng of emuislon polymeru&on processes IS the large rewew ar&le by Mm and Ray[f7] After carefully revlewmg the modelhng efforts presented thus far, they proceeded to develop a model far more comprehensive than any previously formulated These authors mcorporated more mechamsms than ever presented elsewhere, however, they &d not show any quantitative results generated from thew equations The model presented herem, although developed mdependentiy of Mm and Ray’s efforts, 1s sunilar to but less detied than thev model Consequently, we are able to present quantitative results These results are presented m the hopes of furthermg understandmg of em&ton polymemtion, as well as providmg msight mto the practmal consequences of changes m system parameters
J D STEVENS POPULATlON
BALANCE
EQIJATlONS
It 1s desrred to predrct the srxe drstrrbutron of polymer partrcles leavmg a contmuous emulsion polymertxer At the outset we define a number densrty of polymer parttcles as follows N” = iV.(U, t)
(2)
such that N, dv representd the concentration of polymer pticles between the sues v and v + dv havmg n free radicals at tune t In reabty, N,, defines a fanuly of functions, as n LS a parameter To define the system completely, one must describe tlus collection of quantities ) as functions of the internal (charactetrs(N0, N1, AL tic of the parMe dnnensrons) and external (characterrstrc of the system) vanables Note that the system has been assumed homogeneous as spatral varratrons were not considered One 1s now m a posrhon to wnte a populauon balance for the N.-type parttcle using any of the techmques &scussed m the hterature[7,12,13,17-201 The resultmg populaaon balance equation, whose soiutron descrrbes the partrcle srxe distrrbuttons of all the N,-type partrcles and the overall srxe distribubon, 1s
where R., represents the volumetnc growth rate of a polymer partrcle, dv /dt, and r,, represents the net rate of formabon of the N.-type part&e Asrde from the few assumpbons made thus far, eqn (3) 1s perfectly general applymg to both batch and contmuous reactors It remams to arnve at meamngful expressrons for th& partrcle growth rate, R.,, and the rate of partrcle formabon, rN”,to solve spectal cases of eqn (3) PARTICLE GROWWRATE The volumetrrc growth rate expresston presented by Gardon[21] wrll be used here
where
(5) Note that R., IS proportronal to the number of growmg chams present, n Several of the quantrbes makmg up K may be partrcle srxe dependent For the purpose of rllustration K wdl be assumed partrcle srxe mdependent The model 1s sufficiently general to handle erther case MT% OF PARTICLE FORMATION The Snuth-Ewart recurslon relation ISm reahty nothmg more than a rate of formabon expression for the X-type polymer partuzle We wrll here use a shghtly moddied form of therr expressron, altered to allow r&Cal capture of the polymer partscles to be proportronal to the fractron
A populataonbalanceapproachto modem of &he total surface area represented by a mven par&cle type Dlscllss10n of thus pomt appears ln the hterature[12,21] Wntten m terms of the current work, the rate of formation expresslon, rN,, Ls r,, = % {(n + 2)(n + l)N”,, - n(n - l)N-“} +Bu~0v”-,-N”I+$J(n
+l)N”+I-nN.1,
(6)
where (L = _$,
p = (4rr)1’TY3PA, and
y = (4?r)“‘(3)y3ko
of contmuousemukon polymenzatIon
A, These four constants represent the ratios of the four
mechamsms (r&Cal capture, bnlk flow, radical desorpt.lon, and r&Cal termmation) to the growth rate constant, K. We are now In a postion to solve the populatmn balance equations Attentmn ~IUbe focused on the steady state contmuous stwred tank polymerrzer It 1s assumed that thxs 1s the first tank m a smes such that there IS no polymer m the feed Under such consbunts, eqn (9) reduces to $ (a_)
= -A5X,, + $ {(n + 2Mn + 1)X., - n(n - 1)X. }
Q In eqn (6) the parameter n Is to take on non-negative integer values Megatwely subscnpted IV” functions are nonexistent and their value set equal to zero AssEMBLxLI MODEL Combnnng eqns (3). (4) and (6), one anrves at the complete model
+ 30J + 2)(n + UN”,, - n(n - f)N”} +BuU-‘(N.-I-N.)+&{(n By allowmg
+ l)N.+,-nN,)
= AAX., -Xi)
&?‘LX,-, - x.1 + 3 {(PI + 1)X.+, - nx,,) BOUNDARY
COND~ON
CONSlDERATlONS
N,,=N,+N,
(12)
1=x,+x,
(13)
when expressed m dunenslouless form ‘flus assumption unphes that at C = 1, X” = 0 for II > 1 No boundary comhtion IS required for the X&ype pticle, as the equation descnbmg It IS merely algebmc The boundary con&fion for the XI-type par&le ISfound by wr~tmg the equation generated If n takes on the value of zero m eqn (11) 04)
0=-A~Xo+W3Wi~+$tW~~
where all X” functions for n > 1 have been set equal to zero However, for the special case when mutual temnnation results unmedtately at the amval of a second radud to a pmcle, eqn (14) 1s mcorrect The corrected equation, 0 = - A,X, + A,J’/3(X, - X,)1+ $
+?{(n
(11)
The boundary conditions requued are that the number density functions, X., must all be known at a smgle value of i For tis purpose we wdl specify these functions at “zero s=e”, or the size of a nucelle It 1s assumed that parkles the sLze of a nucelle cannot contam more than one free radtcal. Then, at “zero size” the total number of pa&cles IS gwen by
(8)
entue fanuly of population balance equations ISreahzed Negatively subsctrpted functions are nonexistent and defined as zero Before proceedmg, eqn (8) wdl be made dunensionless The advantages of solving problems m dunenslonless vmable form are well recogmzed m many techmcal cucles Not only are the adjustable parameters whch must be exammed reduced m number, but one set of theoretical results may be used to mterpret data from a wide variety of expenmental comhtions By employmg the defimtions found m the Appendnc, eqn (8) becomes $$+$(nX.)
+
or as
n to take on non-negative mteger values, the
313
{X,}
m
+2)(n + 1)x.+,- n(n - 1)X”)
+ A,&-?X”-I - x.1 + 3
{(n + 1)x.+, - nx. 1
(9)
where, m add&on to the defimtions found UI the Appenduc, the followmg dunenslonless vanables were mtroduced
(10) The problem IS now reduced to a famdy of equations mvolvmg four dnnenslouless parameters, A,, AS, As and
shows that formation of X0 par&les IShmited only by the rate of amval of a second radml Now allowing 6 to take on the value of 10 gwes 0=-A,Xo+AJX,-X,)+A,{X,}
w&h,
when solved together wtth eqn (13) yields
A,+&
XO= 1 O+A,+2A,’
lO+A, x, = 1 O+A,+2A,’
(16)
R W ~OMPSONand J
& =
- Xl x0+x,
=
x,
D
STBVBNS
Iowa State Unrversrty The fourth order Runge-Kutta numerical mtegration was camed out m double prectsron along the drmensionless volume axrs, 5 The program developed accepts mformatron read m regardmg the values of the four mdependent drmensronless parameters After uutrahxauon of some quantmes the boundary values are calculated usmg eqn (17) Integratron for the present case involved eleven smmltaneous equattons and proceeded unttl a smgle mtegration step contributed less than lo-‘% to the total cumulauve partrcle number density At thrs pomt, the total partrcle concentratron, P, and the total acttve &am concentrauon, R, terms which had been cumulattvely determined as the mtegratron proceeded, were recorded A parameter study was conducted to mvesttgate the effect of varymg the parameters on the number den&y results Much of what follows are the results of thts study The amount of rnformatron generated was somewhat hmtted due to the extremely large computatron tunes
(17)
where A, and A3 are delined m the Appendrx and F&ISthe average number of radrcals per “zero srxe” partrcle Frgure 1 shows a plot of fiOas a funcuon of A, at a constant vahre of As, where AI was vaned as a parameter By recalhng that A, IS a fun&on of the desorptron rate constant and that A4 IS duectly proportronal to uutrator concentratron, the sumlantres of Frg 1 to Frg 2 of Ref [ 11J become apparent All quahtatrve drscussron of those results apply to Fig 1 here and are not repeated for the sake of brevrty Equatrons (17) form the boundary condrtrons for the current work. NuMERIcAL.aaguLTs The population balance equatrons for contmuous emulsion polymerrxatron were solved on computers at
0As=5
8-
106
A4
03x lo
-5
1o-4
Fla 1 Dependenceof rS.on A, and A, from population balance model (X. = o for n > 1)
A5=503x10 A6=OO0 A7 = 10
I
I
10
100 DIMENSIONLESS
Fw 2
Volumetnc
parhcle slzc Qstnbutions
m contmuous
I
I
1000 VOLUME,
em&Ion
10,000
100,00(
r
polymenzer,
demonstratmg
effect of A,
A population balance approach to modelbm18of contmuous emulsion polymertza~on Table 1 Values of the parameters used as the base case A, = lo-’ A,=503xlO-”
A,=000 A,= 10 The base case (shown m Table l), for example, reqmred 1125 set to mtegrate to completion The numencal mtegratron was normally camed out between the values 1-z 5 < 5 x 104,and the volume mcrement At was normally less than 10, bemg 0 10 for the base case Some rules of thumb were observed regardmg the effect of parameter vanations on the program Itself A decrease of A, by an order of magmtude reqmred that the mtegration be camed out roughly one half an order of magmtude further along the f axis An mcrease of A, by an order of magmtude usually reqmred that the volume mcrement, A& be decreased by an order of magmtude An mcrease m A6 made the program run shghtly slower as a result of the ad&tional terms mcluded Varmtion of A, seemed to have httle effect on the computation tnne The results presented herem are based on the equations generated by lettmg n take on values from 0 to 5 It was found early m the work that the results were somewhat senstive to termmation of the mlimte set of equations To mmumze these effects, the mtegration scheme mcluded eleven equations whde only the iirst su. were used for computatzonal purposes
requwed
VAltLwmNS
partxles Tlus 1s logwal If one notes that as the rate of generation of free radicals (proporhonal to I,, and therefore to A,) mcreases, the rate of rtical absorption
by parhcles mcreases Faster r&Cal arrival rate unphes faster rate of termmafion m the pticles That Is, on the average when radtcal amval rate mcreases, growmg chams are not permitted to grow as long before a rtical 8mves to cause termmation Partxles m such a situation wdl grow less than ones where termmation occurs less to often As A, mcreases there are more free rticals contnbute to termmation The quantity A, IS also mversely proportional to So, the
soap concentration As So decreases (A. mcreases) fewer mlcelles become avadable for radical capture Thus there are more radicals avdable for termmabon, wbch m turn produces more small polymer pticles Fwe 3 shows cumulative pticle size &stnbution for the same set of parameter values hsted m FW 2 The curves m Fw 3 depict the area under the curves of Fg 2 as a function of p The quantity P’, defined as
(18) represents the dunensloniess cumulative concentration of parQcles havmg size up to t When the curves become honzontal, there are essentially no more par&cles to be counted The dmensionless total concentration of parhcles IS gven by P=I-Xrdf
(19)
-.
IN &
Several computer runs were made m wluch the effects of varymg A, (ratio of r&Cal absorption rate constant to parMe growth rate constant) were tested F-e 2 shows the overall pm%cle size &stibution as a function of dunenslonless volume (Throughout tlus &cle “overall” means that the contibtion of each group of N.-type par&les 1s m&&d m the quantity bemg &scussed ) It appears from Fe 2 that an mcrease m A, (proporhonal to IO/SO) yields fewer large polymer
1
315
10
No conclusions concermng absolute numbers of par& cles can be drawn from pigs 2 and 3 unless the value of NrO(used m X,) is known for a given system Fme 4 shows the size dependent value of A and the cumulative system average value, A:, both as functions of 5 These quantities, defined as ri _ XnX zx. ’
loo DIMENSIONLESS
Fhg 3 Cumulatwe danenslonless pticle
10,ow
1.VOLUME,
wze &stnbt&ons,
r
showmg effect of A.
(20)
100,000
R W
316
--hKMPSON andJ
D STEVEE~S
J
A5=5 03x A6 = 0.00 AT=10
4-
1O-5
3-
1
10
100 DIMENSIONLESS
Fhg 4
A
(21)
gve some m&cation of how the free raduxls are dlstibuted m particles of varrous sizes When the curves of A: vs g become homntal all of the par&les have essentmlly been mcluded m the mtegration The value of a: at the hollzontal level then becomes the value of Is,,the system value of ii That IS, _ I I
nT
d5
‘,
(22) XT
10,000
100.00
, c
andfi: as functionsof parkle size.&Wrabng effectof A.
and
ii,=RIP=
1000 VOLUME
dS
1
where rtT ISthe total number of free ra&cals in a partxle and R IS the dunenslonless total number of growmg polymer chams F-e 5 shows a recapttulation of these results with A, the mdependent varmble The values of P (overall partxle concentration), R (propotional to the system rate of polymerrzation). and fi, are shown as functions of A, These results are not altogether consistent wtth some data reported m the hterature [ M-161 They report that the total concentration of pa&cles (and the rate of polymenzatlon) IS an mcreasmg function of So, vvlth whch we are in agreement However, they show that these quantutes are mvanant vvlth respect to I0 Our results would uxhcate that the total concentration of parhcles (and the rate of polymenzation) are decreasmg functions of I0 Thus Qscrepancy IS probably a result of usmg Merent values for the dunensronless constants Ad, , A, One attractive feature of tbs model IS that m soMng the parkle size dlstnbution eqwons one obtams
mformation regardmg the unportance of each number density function mcluded m the Megration Fwe 6 shows these functions for the base case Note the dommance of the X0 and XI funchons up to about I = 100 Beyond C = 1000 the contnbufions to XT from X0 and XI grow lncreasmgly nehble As g mcreases, each X,, function becomes important for a short tune, then, as Its unportance wanes, X. functions for higher II levels begm to dommate As n mcreases, the absolute maxunum of the X. functions gets smaller Physically, these results indicate that larger pticles may house a greater number of free radicals, but that there are mcreasmgly fewer of these larger partxles Results shown in Frg 6 gve an m&cation of how many X,, functions nught be reqmred m an mtegration scheme Unfortunately, thus mformation ISgathered after the fact, 1 e one learns whch X. functions are relevant only after performmg the integration DESORPTION OF FRFB RADICALS
Fwe 7 shows a sun&r composite plot where now the desorption mechamsm has been “turned on” m the equations by vn-tue of allowmg As to take on a value of 0 05 Note the Merent shape of the XT vs g curve, and that the m&vldual X, curves are much suppressed The effect of the desorption mechamsm IS to force free radicals, once mslde the par&les, back mto the water phase The desorption phenomena keeps the X. functions substantmlly lower then the same quanmes shown when desorption was inactive The effect is so dramatic that the Xs function does not even appear on the plot m Fw 7 Fme 8 dustrates the effect of the desorption mechamsm on the overall polymer parkle
size dlstrrbu-
tion In tl~s drawmg, the magmtude of the desorption parameter has been vmed When A6 = 0 00 the curve represents the base case results shown earher It IS obvious that the desorption mechamsm may have a
A populahon balance approach to modellmg of contmuws emulsmn polymenzattoo
\ A5= 5 03 x Ag=O 00 A,= 10
0%
1O-5
04-
13-
I2 1o-5
1o-4
1O-3
1c
A4 Fw
5
Overall partxle concentrahon, active polymer cham concentrahon, and average number of ralcaals per parhle as functions of A,
BASE CASE
DIMENSIONLESS Fig 6
VOLUME,
5
Par&le size dlclirlbuwm results for X funchons
base case
317
R W
~OMI'SONand J D
STEVENS
A4 = 1O-3 A5 = 5 03 x 1O-5 A6=0 05 A,= 10
DIMENSIONLESSVOLUME, 6 Fig 7 Part&
1
size distribution results for X,, functrons desorphon mechamsm activated
10
100
1000
DIMENSIONLESS
VOLUME,
10,000
100,000
r
Fa 8 Partxle s&e dlstnbuhons showmg the effect of the desorptlon mechamsm drastic effect on the shape and posttions of the final parbcle sue dlstriiubon curves The effect noted IS as expected Desorption of free ra&cals lowers the growth rate m the particles, gvmg a product havmg fewer large par&les Figure 9 shows the ri and ii: functions as a function of pax%cle size for various values of the desorption
parameter, A, The results show a marked shrft of free radicals out of the partuzles, as expected One shortcommg of the current model IS found m the rate of generation of radicals. p_, In the current work It was tacitly assumed that pA was constant However, Ugelstad et al [5] have shown that desorbed radicals may mamtam their a&v&y and agam attack polymer parbcles
A populationbalanceapproachto modelhagof contuu~ous emulsion polymewzahon
_
A4=
A5’5
319
10-3
03 x 1O-5
A, = 10
1
IO l%g 9
100 DIMENSIONLESS
TLMB, VARIATIONS
One prmclpal mdependent parameter m any physical contmuous system IS the residence tune m the reactor, T In order to mvesmte the effects of 0u.s parameter several computer runs were made where A, (mversely propotional to I) was vaned These results are shown m Fig 10 where overall parttcle size dlstnbution results are plotted Values of A5 were chosen both a factor of ten lugher and a factor of ten lower than A, used m the base case All other parameter values were held constant (The effect of 7 vanations in A, were neghgiile ) As one m&t expect, shorter residence tunes yield products havmg relatively more smaller polymer part.~l& and fewer larger parhcles That IS. the shorter growmg tunes produce a latex havmg a narrower parhcle size &stibution Surprismgly, however, for these values of the patame ters, varzations m A5 showed muumal effect on the
Cl!%Vol
32 No
3-F
C
10,Ow
100,000
Effectof dcsorptmn of radicals on fi and 6: fundons
to uutiate polymer growth or cause termmahon of a growmg cham In such a case, pA would become a fun&on of the desorptlon mechamsm and the pmcle sue dlstibution To correct the situation would unply an iterative solution In that case the current results would be merely the lirst step m the Iterative scheme The results presented here are str~tly correct when desorption of free radicals does not occur The error wdl be s&cant only when the desorpfion mechamsm contnbutes appreclabiy to the concentration of free radials m the water phase In many real systems this 1s not the case, 1e desorptlon of free ra&cals IS a somewhat mmor phenomenon These results are no more subject to error than others m the hterature[3,4] Other workers m the field dlsrmss the problem by neglectmg desorption aftogetber[l2-161 mC!E
1000 VOLUME,
results Whereas two orders of magmtude cover about the entire useful range m T, scatter m typlcal pticle size data would make dlstmction of T varmtions vutually unposslble These results m&t be satisfymg to those operatmg a productlon unit, for they mdlcate that 7 may fluctuate wlthout much consequence The reader should keep 111 mmd that for other choices of the other independent parameters these conclusions may change shghtly For example, our results deviate from those reported by DeGraff and Poehlem[14-161, probably because of the dtierences m dunenslonless constants ANALYSISOF FUNDERRURR’S RUN #4 One set of data was extracted from the work by Stevens and Funderburk[13] Data from theu run # 4 (Fig 10 m then work) was converted to a form amenable to analysis by thusmodel Shown m Fe 11 are theu data, the values of the parameters calculated from their work, and the theoretical predictions from tlus work We the agreement between experunental and theoretical results 1s not as good as desued, observe that the theoretical predictions do fall m the same general vlcmlty as the data Secondly, whereas Stevens and Funderburk mampulated two adJustable constants to a&eve a good fit to their data, the current model predictions are based on thev parameter values m unaltered form The most questionable parameter fittmg done by Stevens and Funderburk was an attempt to mathematitally back out the effect of an m&bltor m the monomer feed wtuch had not been physically removed They dlv&d the actual residence tune mto an mdumon permd and an effective residence tune, arnvmg at an effective polymemfion tune of about two-tids the actual residence tnne The theoretical prtictions presented here
R W
-hiOMPSON
and J D
th3VENS
A4 = TO-~ A&=0
00
A,=10
0
I”
Fa
s
E - ^. vui
g Z %
5
n
Effect of residence tune v-hens
VOLUME,
r
on the overall partxle size Lstnbution
Ad=5
644x
lo-”
\
A5=3
299x
1O-4
0
I
5
2 2
10
0001
!
1
10,OC
loo0
IUJ
DIMENSIONLESS
0
A6=0
00
A7 = T 1005 x lo2 DATA FROM I 10
STEVENS
AND
FUNDERBURK
(13) 1 1000
100 DIMENSIONLESS
VOLUME,
10,OWJ
Fu 11 Population balance model predxfions of data from Funderburk’s wn used theu calculated effective residence time It seems feasible to suggest that using a higher reszdence tune m&t have provided a better fit of the data, although 111 v-zew of the results of FIN 10 it LS doubtful tis change would have made s&cant Merence m the results Also m FU 11 are shown some theoretical curves where the desorpuon parameter, Ad, was activated From these results one nught suggest that the desorption mechamsm was shghtly activated m Funderburk’s expertments Because It would have reqmred extensive computer tune and because of the questionable value of T, further study of the data was not conducted
100,001
(r # 4
CONCLUSIONS
A population balance model of contmuous emulsion polymemtion processes was developed and solved A paramemc study demonstrated several mterestmg features of the model and the mechamsm of emulsion polymemtion and provtded ms@t mto some of the engmeermg ram&cations of such processes Advantages of the model mclude the fact that free rtical desorption from polymer pmcles may be easdy included Comphcated mtegrals of mod&d Bessel functions need not be evaluated Also, whde other workers[12-151 were reqmed to postulate a value of R (or Is.) for their models to
A population balance approach to modellmg of contmuous
these quantaties are found here as a consequence of obtammg partxle size dlstriiuttons from the polymerlzef The effects of several key parameters on the partxle size dlstrrbution of the emuls&n polymer product were mvestigated These parameters embody specfic physlcal constants of the system (1) The parameter A, (a measure of absorption of radxals relative to growth rate) is propotional to &/SO, two controllable system vanables As shown m Figs 2-5, an increase m A, (marufested by an mcrease m I0 or a comparable decrease m S,,) narrows the pmcle size &stnbution Decreasing A4 tends to broaden the dlstibutlon and produce more large pticles A decrease m A4 also causes a reduction 111polymer production rate (2) In tbs study, variations m the average residence tune (mcluded m AS) produced little vanation m the overall particle size dlstnbution Wrth ddferent values of some of the other parameters, residence tune vanatlons may mdeed be s&cant In general, longer residence tnnes should produce broader partxle size dlstnbutions, although the effect noted herein was muumal (3) Vmations m the parameter A, (ratio of free radical desorption rate constant to pmcle growth rate constant) produced s&cant changes m the partxcle size dlstnbubon of the emulsion polymer product we this mechaNsm is a function of the particular monomerpolymer radical system under consideration and not tiectly controllable, the model presents a means of estunatmg the magmtude of its effect An increase m the magmtude of the free radrcal desorption mechanrsm can smcantly reduce the numbers of large partxles 111the product streams (Figs 8 and 9) Such an increase would also result m a lower polymer production rate Actual par&b size &strrbufion data may be viewed from the perspective of various degrees of the desorption mechanism as shown m Fig 11 A further advantage of the current model IS that mformation IS generated dEectly regarding the pticle size Qstibution of partxles contammg it = 0, 1,2,3, active free ra&cals As many of the X. functions may be mcluded m the integration scheme as reqmred by the mvest@ator without sigmficantly affectmg the complexity of the model Figures 6 and 7 show typical results obtamed for a particular set of parameter values The value of A6 (free radical desorptlon mecharusm) was the only parameter changed between the two graphs From these results the values of A (or ii,) are determmed, as IS the complete particle size dlstnbution functton,
emulsion
the course of tlus work R
W Thompson has received financial assMance from the Umon Carbide, DuPont, and Proctor and Gamble Compames Both authors have been supported by the Engmeer&g Research Institute at Iowa State Umvers@, who also supported the computer calculations and the techmcal drawmgs presented herem Our appreciation IS here pubhcly extended to these organuations NOTATION
AI, Aa
, A,
dunenslonless pend=
constants defined II-IAp-
321
IO feed
concentration of m&&r, gmmole/l K growth rate constant, cm’/hrI # k, uutitor decompostion rate, hr-’ desorptton rate constant polymeruat8on rate constant % termmatmn rate constant N.4 Avogadro’s number Nl number density of polymer partxles m the system, #Iv2 number density of polymer N feed, I Iv” N,, NTO the number density of polymer parhcles at 2ero sue”, # Iv2 n number of radxals per parhcle, # t7 sue dependent average number of radlcals per parWle, # average number of ra&cals m “zero size” partxles, # iiS = RIP total system value of rS, # IT:= R’lP’ cumulative overall system value of fi, # P =$;“X,dc dunenslonless total partxle concentration P’ =$fXT dl cumulative dnnenslonless total partmle concentration R=f;nTdc dunensionless total active cham concentration R’ =s:?lT dC cumulative dlmenslonless total active cham concentration R., volumetic growth rate of N.-type par& cle, cm’lhr rate of formation N,-type partxle, # /cm6 hr s total parkle surface area so feed concentration of emulsdier, gmmole/l t tune, hr micellar volume, cm3 VO V parkle volume, cm3 dlmenslonless dummy vanable N eqn (1) XX dunenslonless number density X: dunenslonless overall number density for polymer product Greek symbols 7
average residence tune m reactor, hr
Pm monomer denstty, gN/cm3 PP polymer density. gm/cm’ PA 8 c
Acknowledgements-Dunng
polymenzation
W&Y hI
rate of free radical production dnnensionless tune dunenslonless partxle volume constants defined m eqn (9) monomer fraction m parkles, sionless
diien-
[l] Harhns W D , I Am Chem Sot 1947 69 1428 [2] Snuth W V and Ewart R H , I Chem Phys 1948 18 592
131 StockmayerW H , I Poiy SCI. 195724 314 [41 OToole J T , J Appl Poly Sm. 1%5 9 1291 [Sj Ugelstad J ,MiirkP C and Aasen J 0 .I Poly SCI Part A-l 1967 5 2281 [6] Garden J L , J Poly Scr Part A-l 1968 6 665
322
R W THOMPSON
STEVENS
[fl Sundberg D C , Ph D dissertation, [8]
[9] [ 101 1111
Umversrty of Delaware 1970 Gershberg D B aad Longfield J Et unpubhshed paper presented at the Synrp Poly Kwettcs Cat Systenis, 54th AIChE Meetmg, New York, l%l, preprmt No 10 Sato T and Tamyama I , Kogyo Kuguku Zasshr 1965 68 106 Sato T and Tamyama I , Kogyo KagoRu Znsshl 1964 68 67 Thompson R W and Stevens J D , Chem Engng Scr 1975 30
and J D
667 -__ [12] yunierburk
J 0
, Ph D hssertation,
Iowa State Umverslty
[13] Stevens J D and Funderburk J 0 , I&ECPmc Des &Den 1972 11360 1143 DeGraff A W , Ph D dlssertatlon, I&& Umversity 1970 rl5l_ DeGratTA W andPoehlemG W .I. Polv_ Scl Part A-Z 1971 _ 9 1955 1161 Poehlem G W, Advances m Emul Poly and Latex Tech 1973, 4th Annual Short Course. Letih Umverslty, 378 [17] Mm K W and Ray W_ H., J Mamma01 >cr -Revs Macromol Chem 1974 Cll(2) 177 [18] Behnken D W , Horowitz J and Katz S , I&EC Fund 1963 2 212 [19] Hulbert H M and Katz S , &em Engng Scr 1964 19 555 [20] Randolph A D and Larson M A, Theory of Partrculute Processes Academic Press, New York 1971 [21] Gardon J L , J Poly Scr Part A-l 1968 6 623
The seven dunenstonless defined for the convemence competm8 mechamsms
_ radrcal absorpucm rate constant partxcle growth rate constant
A, = Ndw,u - 4M) km& space velocity 3 partxle growth rate constant
= radxal desorption rate constant p&cle growth rate constant
=
radical terounation rate constant pticle growth rate constant
A,=AJA,
APPmDm
A, = AT/A,
parameters used m tis study are here of the reader They represent ratios of
A, = AJA,