Ŕ periodica polytechnica Chemical Engeerg 56/2 (212) 55 64 doi: 1.3311/pp.ch.212-2.2 web: http:// www.pp.bme.hu/ ch c Periodica Polytechnica 212 Modellg heat exchange between drops suspension polymerization vyl chloride Ágnes Bárkányi / Béla G. Lakatos / Sándor Németh RESEARCH ARTICLE Received 212-5-15, accepted 212-1-18 Abstract A population balance model is presented for suspension polymerization vyl chloride a batch reactor to vestigate rmal properties system. Reactions are described usg a simplified reaction model focusg on heat generation highly exormic polymerization reactions. The temperature contuous phase is assumed to be homogeneous over reactor and effects some temperature rise droplets due to exormic polymerization reactions and possible heat exchange because coalescence/redispersion process are analysed. The population balance equation is solved applyg a Monte Carlo method by couplg determistic polymerization reactions side droplets with discrete event process duced by collisions droplets. The results obtaed by simulation show that rise temperature droplets over mean temperature contuous phase lead to acceleration process. The possible size distribution droplets and no smooth distribution itiator those decrease process efficiency. Keywords Suspension polymerization vyl chloride population balance model collisional heat exchange Monte Carlo method Acknowledgement This work was presented at Conference Chemical Engeerg, Veszprém, 212. This work was supported by Hungarian Scientific Research Fund under Grant K77955 which is gratefully acknowledged. The fancial support TAMOP-4.2.2/B-1/1-21- 25 project is also acknowledged Ágnes Bárkányi Department Process Engeerg, University Pannonia, H-82 Veszprém, Egyetem Street 1, Hungary e-mail: barkanyia@fmt.uni-pannon.hu Béla G. Lakatos Sándor Németh Department Process Engeerg, University Pannonia, H-82 Veszprém, Egyetem Street 1, Hungary Introduction Suspension polymerization is used for commercial manufacture many important polymers cludg polyvyl chloride (PVC), polymethyl methacrylate and polystyrene. In this process, two immiscible liquids are mixed to produce a dispersion monomer droplets a contuous aqueous phase. The itiator is dissolved monomer droplets where polymerization reactions occur. Both size and size distribution droplets agitated dispersion depend on extent droplet breakup and coalescence which, turn, depend on type and concentrations suspendg agents and on volume fraction dispersed phase [1, 2]. Droplet stability is also fluenced by coalescence. The most commonly used suspendg agents for vyl chloride suspension polymerization are water-soluble polymers such as hydroxylpropyl methylcellulose (HPMC) and partially hydrolysed polyvyl acetate, commonly called PVA [3]. The adsorption such molecules at monomer-water terface reduces terfacial tension and hence reduces energy required to form droplets. Droplet stability depends largely on nature droplet stabilizer. If no stabilizers are used to protect droplets, suspension would be unstable and fal polymer particles would reach an undesirable size. The droplet size distribution and its temporal evolution can be described by applyg population balance approach [4]. In modellg suspension polymerization, however, applyg this approach requires solvg population balance equation (PBE), simultaneously with computg polymerization reactions gog on side droplets. In addition, durg course collisions droplets coalescence and breakup as well as some terchange chemical compounds and heat, termed micromixg by coalescence/ redispersion events, may occur what complicates problem even more. As analytical solutions PBEs are available very few cases, numerical techniques are essential most practical applications. There are several numerical methods available that satisfy accuracy requirements [5] but, takg to consideration random nature breakage, coalescence and micromixg processes Monte Carlo (MC) simulation also seems to be a suitable method for solvg this Modellg heat exchange between drops suspension polymerization vyl chloride 212 56 2 55
numerical problem. Recently, Bárkányi et al. [6] and Bárkányi et al. [7, 8] have analysed suspension polymerization process vyl chloride batch reactors applyg an MC method takg to account breakage and coalescence droplets, as well as micromixg species duced by collisions. Bary breakage to two equal volumes, bary coalescence and randomly varyg mass exchange between collidg droplets were considered under isormal conditions. The aim present paper is to model and analyse a suspension polymerization process vyl chloride which beside polymerization reactions side droplets only collision duced heat exchange occurs between collidg droplets. It is assumed that stabilization monomer droplets is perfect refore no coalescence, breakage or mass exchange take place reactor studyg this way some rmal aspects process. The ketic data vyl chloride polymerization are taken from literature [9, 1]. Model development General Suspension polymerization usually is carried out batch reactors. PVC is produced by powder polymerization sce PVC is soluble vyl chloride monomer (VCM) hence it immediately precipitates out formattg a separate phase. In suspension polymerization VCM free radical polymerization reactions take place monomer droplets because itiator is dissolved those. Suspension polymerization exhibits some advantages over or polymerization processes: bulk, solution and emulsion polymerization [11]. Indeed, suspension polymerization is characterized by favourable rmal conditions but advantages are not restricted to good temperature controllg. Coagulation is largely confed to droplet teriors and coalescence polymerizg droplets can be restricted. That is why suspension polymerization is used for large scale production PVC. In that case, itial droplet diameters, and fal particle sizes, range between 1 µm and 1 µm [12]. The properties PVC are fluenced by polymerization conditions: reactor temperature, stirrg conditions, reactor size and concentration itiator. The temperature polymerization is one most important parameters. Only 1 or 2 K differences temperature can duce significant changes properties polymer products. As dispersed phase, formed by a large population monomer droplets movg stochastically contuous carrier phase tensively stirred batch reactor becomes fully developed reactor is heated to temperature at which polymerization process is started. Assumg homogeneous temperature distribution over contuous phase reactor, subsequently two relevant processes are gog on parallel reactor. The highly exormic polymerization reactions [6] side droplets, havg rates dependg on actual concentrations and temperature, form a contuous time determistic process. Simultaneously with that, random bary collisions may occur between droplets movg reactor space frequency which depends on size and number droplets a unit volume reactor. Assumg perfect stabilization, coalescence and breakage droplets becomes negligible. Besides, takg to account that rate diffusional mass transfer is smaller with some orders magnitude than that heat it is justified to assume that collisions droplets, next termed collision/redispersion events, duce only some heat exchange between collidg droplets, caused by ir possible temperature differences, with negligible species mass teractions. This sequence events forms, prciple, a stochastic discrete event process. In a model suspension polymerization reactor both processes, i.e. determistic polymerization reactions and discrete event process direct heat transfer between collidg droplets should be taken to consideration and this can be achieved by usg population balance approach. Durg course polymerization volume droplets is also changed due to density difference monomer and polymer that should be taken to account as well. In suspension polymerization vyl chloride, however, volume fraction dispersed phase is about.3 refore total volume mixture suspension polymerization can be considered constant. Polymerization reactions From a ketic pot view, polymerization VCM is considered to take place three stages [13]. Stage 1. Durg first stage, primary radicals formed by rmal fragmentation itiator rapidly react with monomer molecules to produce PVC macromolecules which are soluble monomer phase. The reaction mixture consists maly pure monomer, sce polymer concentration is less than its solubility limit (monomer (X <.1%). Stage 2. This stage extends from time appearance separate polymer phase to a fractional Xf at which separate monomer phase disappears (.1% < X < Xf ). The reaction mixture consists four phases, namely, monomer-rich phase, polymer-rich phase, aqueous phase, and gaseous phase. The reaction takes place monomer and polymer phases at different rates and is accompanied by transfer monomer from monomer phase to polymer one so that latter is kept saturated with monomer. The disappearance monomer phase is associated with a pressure drop reactor. Stage 3. Fally, at higher s (X > Xf ) only polymer-rich phase swollen with monomer exists. The monomer mass fraction polymer phase decreases as total monomer approaches a fal limitg value [13]. The reactions accountg for vyl chloride suspension polymerization are summarized Table 1, where I denotes itiator, I is active itiator radical, k d denotes itiator 56 Per. Pol. Chem. Eng. Ágnes Bárkányi / Béla G. Lakatos / Sándor Németh
decomposition rate coefficient [1/sec], k p is propagation rate coefficient [m 3 /(mol sec)], k t stands for rate coefficient termation [m 3 /(mol sec)], k tm is cha transfer rate coefficient [m 3 /(mol sec)], M denotes monomer, R i is growg polymer cha with cha length i, and P i is closed polymer cha with cha length i. When analysg properties polymer product beside trackg changes concentrations itiator and monomer it is reasonable to compute also first three leadg moments live and dead polymer chas. This formulation requires 8 variables and 8 differential equations and provides a sufficiently detailed description polymerization reactions [6 8]. However, here we focus our attention on modellg rmal effects system, and sce propagation step is most important one determg heat polymerization behaviour droplets can be described simply by mass balances itiator, monomer and growg polymer chas toger with energy balance [14]. Denotg fite total concentration fite number live polymer radicals as ν = P i (1) followg balance equations are described for each monomer droplet. Mass balance for itiator: d(υc I ) Mass balance for monomer: d(υc M ) i= = k d c I (2) = (2 f k d c I c M + k p c M ν + k tm c M ν ) (3) Mass balance for live polymer radicals: d(υν ) = 2 f k d c I k t ν 2 (4) From Eq. (3) monomer is computed as dx = d(υc M) and balance droplet volume takes form dυ = dυ P + dυ M M w m (5) Applyg Quasi-Steady-State Assumption (QSSA) for to growg polymer chas, Eq. (4) become: 2 f kd c I ν = (7) k t Energy balance for a droplet: ρc p d(υt) (6) = Q reaction + Q trans f er (8) In Eqs. (2)-(8) υ is droplet volume, c I denotes concentration itiator a droplet, c M is concentration monomer, T is temperature droplet, c p denotes heat capacity droplet, ρ denotes its density, Q reaction represents heat produced by exormic polymerization reactions and Q trans f er is heat removed to contuous phase. The mass balance equations were solved by usg rate coefficients published by De Roo et al. (25), applyg appropriate modifications accountg for viscous differences phases. Bimolecular termation reactions between cha radicals become diffusion controlled at high polymer concentrations or high leadg to an itial crease polymerization rate and molecular weight. This condition is known as gel effect or Trommsdorff effect. Typically termation rate coefficients are affected first by gel effect because y volve diffusion two bulky polymer radicals. The diffusional limitation is usually modelled by multiplyg low reaction rate coefficients (k ) by a gel effect factor (GF) that decreases with creasg [12]. Hence effective rate coefficient for a reaction is given by: k e f f = k GF (9) The most commonly used gel effect correlation is: GF = a 1 1 + a 2 X a 3 f (1) We can use this correlation simply by specifyg correlation parameters (a 1, a 2 and a 3 ). Durg course simulation, mass balances itiator and monomer, as well as zero moment balance were computed. Therefore, next sections vector concentrations c=(c 1,c 2,c 3,) denotes, turn, concentrations itiator, monomer, and zero moment live polymer chas. Population balance model Let now υ denote volume a droplet, c stand for vector concentrations K=3 relevant chemical species side droplets, and let T denote droplet temperature. Then, assumg that reactor is perfectly mixed at macro-scale and motion droplets contuous carrier phase is fully stochastic without orientation micro-scale state a droplet, general case, is given by vector (υ, c, T) R K+2 without dicatg its position and velocity space reactor [15]. Then, durg course process, this state vector is changed due to followg relevant micro-scale teractions: The concentrations chemical species are changed contuously time because polymerization reactions. This is a determistic, time contuous process thus concentrations a droplet are governed by set differential equations dc (t) = R r [c (t), T (t)] (11) which is formed by Eqs (2)-(4). Here, c = (c 1, c 2, c 3 ) T and (.) T denotes transposition operation. Modellg heat exchange between drops suspension polymerization vyl chloride 212 56 2 57
Tab. 1. Free-radical polymerization reactions vyl chloride Decomposition itiator: Cha itiation: Propagation: Cha transfer: I k d 2I I + M k d R 1 R 1 + M kp R 2 R 1 + M k tm R 1 + P 1 R i + M kp R i+1 R i + M k tm R i + P i Termation: R i k t P i R i + R j k tc P i+ j R i + R j k td P i + P j The volume droplets is also changed contuously time because polymerization reactions and significant difference between densities monomer and polymer. The volume a droplet is governed by a simple differential equation dυ (t) = f υ [R r (c (t), T (t))] (12) which consists terms given Eq.(6). Because generation heat by exormic polymerization reactions and heat transfer between contuous phase and droplets, temperature a droplet is varied time contuously. However, to this determistic, time contuous process discrete time process collision duced terchanges heat is superimposed, sce collisions droplets are gog on randomly. It is assumed that se coalescence/redispersion events may duce jump-like changes temperatures collidg droplets because ir possibly different temperatures extents which, dependg on actual collision conditions are also random characterized by a parameter ω [, 1] as it was determed case collision duced heat transfer solid particles gas-solid systems [16, 17]. [ ( )] mi c pi + m j c p j ω = 1 exp h c/r a c/r θ c/r m j c p j m i c pi (13) where c pi and c p j are heat capacities and m i and m j are masses collidg droplets. In Eq. (13) h c/r, a c/r and θ c/r denote, respectively, heat transfer coefficient, contact surface and contact time between droplets a coalescence/redispersion event. Sce se quantities appear to be random variable ω is also random takg values from closed terval [,1]. The value ω = means no heat exchange durg collisions while value ω = 1 denotes such case when total equalization temperatures collidg droplets occurs. As a consequence, temperature a droplet is governed by stochastic differential equation dt (t) = f T [R r (c (t), T (t))] ha [ ] T (t) T f + ψ c/r (ω, T)N (υ, ) (14) V where terms on right hand side describe, turn, rate change temperature due to polymerization reactions, droplet-contuous phase heat transfer and jump-like changes by coalescence/redispersion events. Here, h denotes droplet-contuous phase heat transfer coefficient, a is droplet surface, T f denotes contuous phase temperature, function ψ c/r provides temperature jump duced by a coalescence/redispersion event and characterized by random parameter ω, and N denotes a Poisson process countg se random events volume V suspension dependg on volumes droplets. The set differential equations (11)-(14) describes behaviour droplets entirely by trackg time evolution state each droplet dividually. However, complexity this mamatical model does not allow overcomg computational complexity related to system Eqs. (11)- (14). Instead followg evolution each sgle droplet it is reasonable to consider collective description those utilisg ir statistical similarities and defg an appropriate mesoscopic length-scale between micro- and macro-scales [15]. Defg population density function given as a mappg (υ,c,t,t) n(υ,c,t,t) by means which n(υ,c,t,t)dυdcdt provides number droplets beg volume (υ,υ+dυ) and temperature (T,T +dt) tervals and concentration region (c,c+dc) at time t a unit volume reactor. Then population balance equation takes form n (υ, c, T, t) t + T + υ [ dt n (υ, c, T, t) [ ] dυ n (υ, c, T, t) + ] c = M c/r [n (υ, c, T, t)] [ ] dc n (υ, c, T, t) + (15) where on left hand side are rates change population density function because determistic contuous processes while operation M c/r on right hand side represents collision-duced process which takes form [16, 17] M c/r [n (υ, c, T, t)] = S c/r n (υ, c, T, t) + 1 2S c/r Tmax p υ ωn(t) n [ υ, c, ( 2(T y) T m p υ ω + y)] n (υ, c, y, t) f ω (ω)dydω (16) where ω [, 1] is random number characterizg extent equalization temperatures with probability density function f ω, p υ = υ /(υ + υ ) and N denotes total number droplets. Function S c/r provides frequency bary collisions droplets volumes υ and υ resultg exchange 58 Per. Pol. Chem. Eng. Ágnes Bárkányi / Béla G. Lakatos / Sándor Németh
heat duced by a coalescence/redispersion event. The first and second terms on right hand side Eq.(19) describe, respectively, rates decrease and crease number droplets state (υ,c,t) caused by se events. Eqs. (15)-(16) is meso-scale model droplet population suspension polymerization reactor and, sce this process re are no mass transfer teractions between dispersed and contuous phases, and temperature over contuous phase reactor is assumed to be homogeneous, Eq. (15) provides, prciple, model whole reactor as well. Note, that macro-scale model can be formulated by means jot moments volume, concentrations and temperature variables defed as µ k,l1,l 2...l K,m (t) = T max T m c m c max υ max υ k c l 1 1 c l 2 2...c l K K T m n (υ, c, T, t) dυdcdt, k, l 1, l 2...l K, m =, 1, 2... (17) by means which mean temperature droplet population over reactor is expressed as 1 T (t) = µ,,..., (t) T max T m c m c max υ max Tn (υ, c, T, t) dυdcdt (18) while mean concentration k th species over reactor can be expressed as 1 c k (t) = µ,,..., (t) T max T m c m c max υ max c k n (υ, c, T, t) dυdcdt, k = 1, 2, 3 (19) The numerical solution multidimensional population balance equation (15) seems to be a crucial problem thus makg use randomness collision duced processes, it was solve by usg Monte Carlo method. Solution by means Monte Carlo method In this work we used a time-driven MC method. In timedriven simulations a time step is specified n simulation implements all possible events with that step [18]. The used MC algorithm is described this section. It is based on generatg random numbers from uniform probability distribution (,1). Fig. 1 represents scheme Monte Carlo method used durg simulation; steps this method are followg: Initialization: Initial droplet size distribution is generated and all state variables are given itial value. Number droplets is N. Set time equal to zero. Step 1: Next event time. The next event time will be after time. Set time to: t = t +. Step 2: There is a complex tra-particle dynamics, such as polymerization reactions, so, for all particles tegrate set tra-particle reactions from t- to t. In this step volume balance droplets, mass balance itiator and monomer, as well as zero moment live polymer chas are calculated. Step 3: The collision takes place. Two droplets are selected randomly which collide with each or, but volumes droplets are not changed collision. There is only heat transport between droplets. A random number is generated to calculate rate heat exchange between collidg droplets. The heat transfer between two collidg droplets was calculated usg followg equations T i = T i + T j = T j c p j m j c pi m i + c p j m j ω ( T j T i ), ω 1 (2) c pi m i c pi m i + c p j m j ω ( T j T i ), ω 1 (21) where T j and T i denote temperatures droplets before collision. Here, parameter ω was computed usg a uniform probability distribution (,1). Step 4: If fal time is reached, end simulation; orwise go to step 1. Results and discussion The correspondg computer program and all simulation runs were written and carried out MATLAB environment. In first step simulation program was identified. Durg simulation we calculated gel effect usg Eqs (8) and (9). The three parameters gel effect were defed and results were compared with measurement data [9]. The simulation and literature data run really good with each or [8]. Some prelimary simulations were carried out which effect total number elements was analysed. These simulation runs revealed that a thousand element droplet population proved to be sufficient to simulate process with reliable approximation [8]. Accordgly simulation runs a thousand-element droplet population was used examg effects micromixg processes on mean monomer. Durg course simulation perfect stabilization droplets and negligible component transport were assumed so that only heat transport occurred between collidg droplets. In our previous work component transport was analysed without heat transport [6]. The aim this simulation was to show effects heat generated exormic polymerization reactions. In all cases it was assumed that temperature contuous phase was constant and temperature droplets was calculated all time steps. The fluence heat exchange between droplets has been studied by observg mean value summarized over all droplets participatg suspension polymerization process VC. Short description simulations: Modellg heat exchange between drops suspension polymerization vyl chloride 212 56 2 59
transport between droplets. A random number is generated to calculate rate heat exchange between collidg droplets. so that only heat transport occurred between collidg droplets. In our previous component transport was analysed without heat transport [6]. The aim this simulation was to show effects heat generated ex polymerization reactions. In all cases it was assumed that temperature c phase was constant and temperature droplets was calculated all time s fluence heat exchange between droplets has been studied by observg value summarized over all droplets participatg su polymerization process VC. Short description simulations: - The droplets were mono-dispersed; itially all droplets contaed same a itiator; reaction conditions were isormal. - The droplets were mono-dispersed; droplets contaed different am itiator; reaction conditions were non-isormal. - The droplets were dispersed with an itial droplet size distribution; all contaed proportionally same amount itiator; reaction conditi isormal. - The droplets were dispersed with an itial droplet size distribution; all contaed proportionally same amount itiator; reaction conditi non-isormal. First we computed monomer when itially all droplets contaed amounts itiator and diameters droplets were equal. Fig.2 represents ca reactor was isormal, temperature contuous phase was homogen reactor and it was able to distract heat generated by reactions droplets. In curves (- -) and ( ) show evolutions mean monomer when th computed allowg maximum 1 K and 5 K differences between temperatures o and that contuous phase. In se cases, heat transfer coefficients were d Fig. Fig. 1. 1. The scheme Monte Carlo Monte method Carlo used method simulation that rise used temperature simulation droplets might achieve maximum 1 K (- -) and ( ) 5 K The The heat dropletstransfer were mono-dispersed; between itially two all collidg droplets contaed same amount itiator; reaction conditions deltat=1k droplets was calculated usg followg 1 isorm equations deltat=5k 8 were isormal. cp m j j Ti = Ti + ω ( Tj Ti ), ω 1 (2) The droplets c were m + mono-dispersed; c m droplets contaed 6 pi i p j j different amounts itiator; reaction conditions were non-isormal. c m ( T T ) 1 pi i j The = Tdroplets j were dispersed with ω an itial j droplet i, size distribution; all droplets cp mi contaed + cp mproportionally i j j same amount ω itiator; reaction conditions were isormal. T.5 1 1.5 2 2.5 3 x 1 4 where The Tdroplets j and were T i dispersed denote with antemperatures itial droplet size distribution; all droplets usg contaeda uniform proportionally probability itiator same amount distribution contaed same (,1). amounts itiator droplets Fig. 2. Evolution before time collision. monomer Here, if itially parameter all droplets ω was computed Step itiator; 4: If reaction fal conditions time is were reached, non-isormal. end simulation; orwise go to step 1. In Fig.3 we can see mean temperature droplets over reactor. temperature are seen In Fig. both 3 cases we canbetween see mean 1 temperature and 12 droplets seconds, over respective First we computed monomer when itially all look at Fig.2 than we can reactor. see Jumps that monomer temperature are seen reaches both cases be- critical c (~75%) at this time. The polymerization rate peaks at critical and drastically after that. This peak polymerization rate will result large temperature reactor [19]. We can say that effect temperature droplets is significa next cases maximum rise temperature was 1 K, because effect temperatu Results droplets contaed and discussion same amounts itiator and diameters correspondg droplets were equal. Fig. tween 1 and 12 seconds, respectively. If we look at The computer 2 representsprogram cases when and all Fig. simulation 2 than we canruns see thatwere monomer written and reaches carried out reactor was isormal, temperature contuous phase MATLAB environment. critical ( 75%) at this time. The polymerization rate was homogeneous over reactor and it was able to distract In first step simulation program was identified. peaks at critical Durg andsimulation decreases drastically we calculated after that. heat generated by reactions droplets. In Fig. quality 2 curves fal product (- may be significant. This peak polymerization rate will result large temperature -) gel and effect ( ) show usg evolutions Eqs (9) mean and monomer (1). The three parameters gel effect were defed and crease reactor [19]. We can say that effect temperature [9]. droplets The simulation is significant. Inand next literature cases maximum data run results when those were were compared computed allowg with maximum measurement 1 K and 5 K differences between temperatures droplets and that data really good with each or [8]. rise temperature was 1 K, because effect temperature contuous phase. In se cases, heat transfer coefficients Some prelimary simulations were carried out on quality which fal product effect may be significant. total number were defed so that rise temperature droplets might In next simulation runs all droplets contaed itiator but elements achieve maximum was analysed. 1 K (- -) and ( These ) 5 K. simulation runs revealed that a thousand element droplet not same amounts. The itiator was distributed between population proved to be sufficient to simulate process with reliable approximation [8]. Accordgly simulation runs a thousand-element droplet population was used examg 6 Per. Pol. Chem. Eng. Ágnes Bárkányi / Béla G. Lakatos / Sándor Németh effects micromixg processes on mean monomer. Durg course simulation perfect stabilization droplets and negligible component transport were assumed 4 2 (21) Fig. 2. Evolution time monomer if itially all droplets contaed same
T (K) 337 336 335 337 334 336 333 335 332 334 331 T (K) 333 33 332 329 deltat=1k deltat=5k 331 328 1 1 1 2 1 3 1 4 33 lution 329 time Fig. 3. Evolution mean temperature time mean droplets temperature droplets 328 xt simulation 1 1 runs all droplets 1 2 contaed 1 itiator 3 but not 1 4 same amounts. The s distributed droplets between small random droplets units. These small small random units units. were: These small units Fig. 5 were: (- -) presents results simulation for parameters n i /1) time mean temperature droplets ni =b(ni/1), (22) T = 1 K and when re was total heat exchange between (23) droplets which collided. It is compared to ni = b(ni/1) (22) i /1) imulation runs all droplets contaed itiator but not same amounts. The case when itially all droplets contaed same amount tributed total amount between itiator droplets system small random b is a units. uniformly These distributed small units random were: een and 1. ni = b(ni/1) (23) itiator and T = 1 K ( ). The le (-) represents case ) (22) we can see effects itial distribution itiator on mean when monomer it was ni =b(ni/1) but reactor was isormal. In se ) where ni is total amount itiator system and b is a simulation runs reactor was kept isormal. It is seen Fig. Fig.4 6 (- -) that presents (23) case for ni =b(ni/1), T = 1 K uniformly distributed random number between and 1. itial distribution itiator droplets on process and when may re be was total heat exchange between droplets which total amount In Fig. 4itiator we can see effects system and itial b is distribution a uniformly distributed random collided. It is compared to case when itially all droplets n and itiator 1. 1 on mean monomer. In se simulation contaed same amount itiator and T = 1 K ( ). can see ni=rand*(ni/1) runs effects reactor ni=rand*(ni/1) was kept itial isormal. distribution It is seen Fig. itiator 4 that on mean monomer The le (-) represents case when it was ni se simulation 8 same effects runs itial distribution reactor was itiator kept isormal. droplets onit is seen Fig.4 that =b(ni/1) but reactor was was isormal. Figs. 5 and 6 illustrate well that itial process distribution may be significant. itiator droplets on process may be 6 monomer reached its fal ( ) value earlier when 2.724 itially all droplets contaed same amounts (- -) itiator.6811 ( ). 1 4 ni=rand*(ni/1) In all previous cases, heat exchange ni=rand*(ni/1) ( ) between 1.8657 collidg droplets was total. Next, we analysed effects partial 8 same 2 heat exchange between droplets. The conditions were same, 6 namely itiator was distributed small random units and.5 1 1.5 2 2.5 3 x 1 4 all droplets contaed some itiator at start. In se cases, 4 olution time monomer when itiator was itially distributed as those randomly are shown Figs. 7 and 8 two curves run with each or so wear total or partial heat exchange occurs 2 monomer drawn by les ( ; - -) is compared to a coalescence/redispersion case when is not relevant. This is because droplets contaed same amount itiator ( ). The maximum maximum and mimum allowed rise temperature droplets durg polymerization is ni was 1 K and all droplets contaed itiator at itiator at start.5 polymerization 1 are 1.5 presented 2 Table 2.5 2. The 3 smaller x 1 he difference between c Im and c Imax droplets at begng 4 start process. temperature differences droplets have on time distribution Fig. 4. Evolution monomer itiator time are monomer negligible when if itiator when is properly was itiator itially small. was distributed proved not randomly high enough to produce significant effects by this evious cases, itially distributed we studied randomly effects droplets temperature and itial heat distribution contaed exchange. same amounts itiator ( ). monomer droplets mean drawn monomer by les ( ; separately. - -) is compared Next, we to analysed case when ffect both In Fig. parameters 4 monomer mean monomer drawn by. les ( ; Sce - - se 1 cases lets contaed same amount itiator ( ). The maximum and mimum same ) is diameter compared value to contaed case when different itially amounts all droplets contaed isorm itiator reaction rates ets or at were not start same polymerization which resulted are presented temperature differences Table 2. The droplets. smaller deltat=1k Then, is ni same amount itiator (-). The maximum and mimum 8 deltat=1k, same ifference f partial values or between total itiator equalization c Im at and start c Imax temperatures polymerization droplets at are collidg presented begng droplets durg process. /redispersion e distribution Table events 2. The itiator were smaller vestigated. are is ni negligible if ni smaller is is difference properly between and mimum c Im and values c Imax itiator droplets itially at begng process. small. 6 us cases, we studied effects temperature and itial distribution aximum droplets The on effects mean distribution monomer itiator are negligible separately. if ni Next, is we 4analysed t both properly parameters small. on mean monomer. Sce se cases 2 same diameter In previous value contaed cases, we studied different effects amounts temperature itiator reaction rates ere not and same itial which distribution resulted itiator temperature droplets differences on droplets. Then,.5 1 1.5 2 2.5 3 artial or mean total monomer equalization temperatures separately. Next, we analysed collidg droplets durg x 1 4 ispersion collective events effect were vestigated. both parameters on mean monomer. mimum Sce values se itiator cases droplets itially same diameter value time monomer =b(ni/1) Fig. 5. Evolution time monomer ni um and (ni =b(ni/1)) deltat=1k deltat=5k contaed different amounts itiator reaction rates droplets were not same which resulted temperature differences droplets. Then, effects partial or total equalization temperatures collidg droplets durg coalescence/redispersion events were vestigated. Tab. 2. The maximum and mimum values itiator itially Case c Im (mol/m 3 ) c Imax (mol/m 3 ) (-) 2.724 2.724 (- -).6811 5.8294 ( ) 1.8657 3.4846 Case c Im (mol/m 3 ) c Fig.5 (- -) presents results simulation for pa when re was total heat exchange between droplet case when itially all droplets contaed same am le ( ) represents case when it was ni =b(ni/1 Fig.6 (- -) presents case for ni =b(ni/1), exchange between droplets which collided. It is co droplets contaed same amount itiator and case when it was ni =b(ni/1) but reactor wa well that monomer reached its fal 1 8 6 4 2 Fig. 6. (ni =b In all previous 212 56 2 cases, heat exchange between th analysed effects partial heat exchange between namely itiator was distributed small random Modellg heat exchange between drops suspension polymerization vyl chloride 61
ed its fal e reactor was value was.5isormal. earlier 1 when 1.5 Figs itially 5 2and Fally, 6 2.5 illustrate droplets effects 3 itial droplet size.5 1 1 distribution were analysed by ge 1.5 2 2.5 3 ). itial size distribution x 1 4 usg normally distributed pseudorandom numbers. x 1 4 A typ ed its fal value earlier when itially all droplets size distribution is seen Fig.9. 5 ). Fig. Fally, 15. Evolution effects time monomer itial droplet size Fig. distribution 3 6. Evolution were time analysed monomer by generatg an isorm itial (ni =b(ni/1)) size deltat=1k distribution usg normally distributed (nipseudorandom =b(ni/1)) numbers. A typical droplet 1 25 85 9 95 1 8 size deltat=1k, same In distribution isorm all previous is seen cases, Fig.9. deltat=1k heat exchange between collidg droplets was total. Next, we 8 deltat=1k, same 2 analysed 6 3 effects partial heat exchange between droplets. The conditions were same, Fig. 9. Initial droplet distribution 15 namely 6 itiator was distributed small random units and all droplets contaed some 4 25 itiator at start. In se cases, as those are shown 1 First we Figs analysed 7 and 8 effect two curves run itial with dropl 4 each 2 or so wear 2 total or partial heat exchange were occurs a relatively a coalescence/redispersion tight size range is effect not 5 relevant. 2 This is because maximum allowed although rise we can temperature see Fig.1 droplets that even durg this tigh 3.5 1 151.5 2 2.5 3 x 1 polymerization was 1 4 t K (s) and all droplets contaed mean itiator at start process x 1 4 85 monomer 9 95. 1 The 15 mean 11 monomer 115 3 diameter (micrometer).5 1 1.5 2 2.5 3 temperature differences droplets have proved not high enough to produce significant x 1 Fig. 6. ersion 4 1 when itial droplet size distribution was as sh Evolution time time monomer monomer ni =b(ni/1) x 1 4 Fig. 9. Initial droplet distribution effects by this heat exchange. Fig. 9. Initial droplet it distribution was mono-dispersed. ersion Fig. (ni 1 =b(ni/1)) 6. Evolution time 5 monomer 1 total heat exchange First we analysed 1 1 effect total heat exchange itial droplet size distribution. Sce d itial drop size distribution (ni =b(ni/1)) partial heat exchange were a relatively tight size partial range heat exchange effect this parameter has proved not 8 8 same drop size 85 although 9 we 95can see 1 8 Fig.1 15 that even 11 this tight 115 size range caused some differ 6 mean monomer drop diameter. (micrometer) 6 The mean monomer reached fal v 6 when itial droplet size distribution was as shown Fig.9 compared with Fig. 4 9. Initial droplet distribution it was mono-dispersed. 4 e between collidg droplets was total. Next, we nge e between droplets. collidg The droplets conditions was total. were Next, same, we nge mall between random droplets. units and The all conditions droplets contaed were same, some se mall are random shown units Figs and 7 and all 8 droplets two contaed curves run some with 4 se exchange are shown occurs Figs a 7 coalescence/redispersion and 8 two curves 1 run is with not 1 itial drop size distribution isorm 2 m allowed First rise we analysed temperature effect droplets itial durg droplet 2size distribution. Sce same drop size partial heat droplet exchange sizes exchange occurs a coalescence/redispersion is not 2 8 8 total heat exchange ts contaed were itiator a relatively tight start size range process effect this parameter has proved not significant allowed rise temperature droplets durg.5 1 1.5 2 2.5 3.5 1 1.5 2 2.5 3 e proved although not we high can enough see s contaed itiator at start t to Fig.1 (s) produce that 6 even process significant this tight 6 x 1 4 size range.5 caused 1 1.5 some difference 2 2.5 3 t (sec) x 1 mean monomer. The 4 4 mean monomer reached 4 fal value slower x 1 4 e proved not high enough to produce Fig. 7. Evolution time monomer ni Fig. 1 7. Evolution time monomer significant =b(ni/1) when itial droplet size distribution was as shown Fig. 1. 8. Influence Evolution Fig.9 itial compared time droplet itial size monomer total heat exchange 2 2droplet distribution with size evolution case distribution when time (ni =b(ni/1)) partial heat exchange on (ni evolution monomer it was mono-dispersed. =b(ni/1)) time monomer 1 8 total heat exchange 1 partial heat exchange 8 itial drop size distribution 6 same drop size 8 6 4 6 4 2 drop number 4 Fig.11 shows evolutions 4 monomer when partial and total hea 2 occurred between each collidg run. droplets Aga, compared it is seen with curves case when partial temper hea 3.5 1 1.5 2 2.5 3 x 1 4 2 reactor was totally homogeneous. run x 1 4 2toger. The same itial droplet size distribution was 3.5 1 1.5 each 2 run. 2.5Aga, 3it is seen curves partial heat exchange (- -) and total heat ex x 1 4 x 1 4 ersion Fig. 8. Evolution time monomer run toger..5 1 1.5 2 2.5 3 Conclusions.5 1 1.5 2 2.5 3 Fig. 8. Evolution time monomer t (sec) ni ersion Fig. (ni =b(ni/1)) =b(ni/1) x 1 4 x 1 8. Evolution time monomer A population balance model was presented 4 for sus Conclusions Fig. 11. Fig. (ni 1. =b(ni/1)) Influence itial droplet size distribution Fally, effects itial droplet size A population distribution balance a were model batch Effects heat was polymerization exchange heat between presented for suspension reactor. exchange collidg polymerization The between droplets evolution monomer compared with case homogeneous reactor vyl stabilization on evolution time monomer analysed by generatg an itial size distribution a batch usg polymerization re collidg normally reactor. was droplets not The any stabilization component on evolution droplets transport was monomer temperature assumed between to th be distributed pseudorandom numbers. A typical re droplet was not size any distribution is seen Fig. 9. assumed partial and total heat be exchange homogeneous occurred between over collidg reactor a component transport compared between with contuous case homogeneous between droplets durg ir collisions. phase and droplet The between droplets reactor durg temperature ir collisions. The temperature contuous Fig.11 assumed to be homogeneous over reactor effects temperat First we analysed shows volutions effect itial droplet monomer size distribution. Scebetween droplet sizes were collidg a relatively droplets tight sizecompared range actorwith was totally homogeneous. case when The same temperature itial droplet size droplets due to droplets compared when exormic due with polymerization to partial casexormic and whentotal temperature heat reactions and polymerization exchange re- possible hear occurred reactor effectwas thistotally parameter homogeneous. has proved not significant The although same itial distribution droplet was assumed size distribution each run. Aga, was it assumed is seen we can see Fig. 1 that even this tight size range caused curves partial heat exchange (- -) and total heat exchange each run. Aga, it is seen curves partial heat exchange (- -) and total heat exchange ( ) some difference mean monomer. The mean ( ) run toger. run toger. monomer reached fal value slower when itial droplet size distribution was as shown Fig. 9 compared with case when it was mono-dispersed. Fig. 11 shows evolutions monomer when drop number.5 1 1 1.5 2 2.5 3 t (sec) isorm x 1 4 partial heat exchange 8 total heat exchange Fig. 1. Influence itial droplet size distribution on evolution time monomer dr drop diameter (m.5 1 1.5 2 Fig col con rea Fig. 11. Effects heat exchange collidg droplets on evolution compared with case h reactor temperature Fig.11 shows evolutions monomer conve occurred 6 between collidg droplets compared reactor was totally homogeneous. The same itia Conclusions A population balance model was presented for suspension polymerization polymerization vyl chloride a batch vyl polymerization chloride re- a batch polymerization reactor. The stabilization droplets was assumed to be perfect so re 62 was not any component Per. transport Pol. Chem. Eng. between contuous Ágnes phase Bárkányi and / Béla droplets G. Lakatos / Sándor as well Németh as between droplets durg ir collisions. The temperature contuous phase was assumed to be homogeneous over reactor and effects temperature rise Conclusions A population balance model was presented for suspension
actor. The stabilization droplets was assumed to be perfect so re was not any component transport between contuous phase and droplets as well as between droplets durg ir collisions. The temperature contuous phase was assumed to be homogeneous over reactor and effects temperature rise droplets due to exormic polymerization reactions and possible heat exchange because coalescence/redispersion were analysed.. The resulted population balance equation was solved by usg a Monte Carlo method by couplg determistic polymerization reactions with discrete event process duced by collisions droplets. The results obtaed by simulation showed that allowg maximum 5 K rise temperature droplets over mean temperature contuous phase led to significant acceleration process. The size distribution droplets and no smooth distribution itiator those decreased efficiency process. When temperature rise droplets over mean temperature contuous phase was small, i.e. maximum 1 K n fluence heat exchange between collidg droplets on process proved to be negligible. Notation a droplet surface a 1,2,3 parameters a c/r contact surface between droplets a coalescence/redispersion event c vector concentration variables a droplet c I concentration itiator a droplet [mol/ m 3 ] c M concentration monomer a droplet [mol/ m 3 ] c p heat capacity droplet [J/mol K] f factor f ω probability density function I concentration itiator [mol/ m 3 ] k low reaction rate coefficients [m 3 /(mol s)] k d itiator decomposition rate coefficient [1/s] k e f f effective rate coefficient [m 3 /(mol s)] k p propagation rate coefficient [m 3 /(mol s)] k t rate coefficient termation [m 3 /(mol s)] k tc rate coefficient termation by combation [m 3 /(mol s)] k td rate coefficient termation by disproportion [m 3 /(mol s)] k tm cha transfer to monomer rate coefficient [m 3 /(mol s)] GF gel effect factor h droplet-contuous phase heat transfer coefficient h c/r heat transfer coefficient between droplets a coalescence/redispersion event I itiator I active itiator radical M monomer M c/r collision-duced process m mass droplet [g] ni total amount itiator system n(v,χ, t) population density function droplet population N a Poisson process countg collision events N total number droplets P i closed polymer cha with cha length i P n active, growg polymer cha Q reaction generated heat exormic polymerization reactions Q trans f er removal heat to contuous phase R i growg polymer cha with cha length i S c/r frequency bary collisions droplets t time [s] T temperature [K] T f contuous phase temperature υ droplet volume [m 3 ] υ M volume monomer a droplet [m 3 ] υ P volume polymer a droplet [m 3 ] V volume suspension X monomer (%) X f critical monomer (%) Greek symbols θ c/r contact time between droplets a coalescence/redispersion event µ k kth moment dead polymer chas [mol/ m 3 ] ν k kth moment polymer radicals [mol/ m 3 ] ψ c/r temperature jump duced by a coalescence/redispersion event ω parameter (random number between -1) ρ density drop References 1 Zerfa M, Brooks B W, Vyl chloride dispersion with relation to suspension polymerization, Chemical Engeerg Science 14 (1996), 3591 3611, DOI 1.116/9-259(96)2-4. 2 Hashim S, Brooks B W, Mixg stabilised droplets suspension polymerisation, Chemical Engeerg Science 59 (24), 2321 2331, DOI 1.116/j.ces.24.3.1. 3 Saeki Y, Emura T, Technical progresses for PVC production, Progress Polymer Science 27 ( 22), 255 2131, DOI 1.116/S79-67(2)39-4. 4 Kotoulas C, Kiparissides C, A generalized balance model for prediction particle size distribution suspension polymerization reactors, Chemical Engeerg Science 612 (26), 332 346, DOI 1.116/j.ces.25.7.13. 5 Bove S, Solberg T, Hjertager B H, A novel algorithm for solvg population balance equations: parallel parent and doughter classes. Derivation, analysis and testg, Chemical Engeerg Science 6 (25), 1449 1464, DOI 1.116/j.ces.24.1.21. 6 Bárkányi Á, Németh S, Lakatos BG, Effects mixg on formation polymer particles suspension polymerization, Chemical Engeerg Transactions 24 (211), 613 618. 7 Bárkányi Á, Lakatos B G, Németh S, Modellg and simulation suspension polymerization vyl chloride via population balance model, Modellg heat exchange between drops suspension polymerization vyl chloride 212 56 2 63
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