HUNGARIAN JOURNAL OF INDUSTRY AND CHEMISTRY Vol. 46(2) pp. 79–84 (2018) hjic.mk.uni-pannon.hu DOI: 10.1515/hjic-2018-0023 COMPARISON OF PARTICLE SIZE DISTRIBUTION MODELS FOR POLYMER SWELLING ÁDÁM WIRNHARDT *1 AND TAMÁS VARGA1 1Department of Process Engineering, University of Pannonia, Egyetem u. 10, Veszprém H-8200, HUNGARY In polymer technologies, various particle shapes and size distributions can be found. One of these are heterodisperse polymer beads. The capabilities of polymer swelling can be used in industries, e.g in the production of ion-exchange resins, to intensify specific technological steps such as sulphonation in the manufacturing process of ion-exchange resins. According to the literature different approaches can be used to create models for describing the behavior of disperse systems, of which the simplest models are the particle size distribution models for a given state of the solid phase. The aim of our examination was to compare and evaluate these simple models in terms of modeling polymer swelling. Hence, most of these models examine how each of the investigated models can be applied to approximately describe growth in a heterodisperse polymer system and how the identified model parameters in each time step could be interpreted. All the models were fitted to generate particle size distributions based on a swelling rate constant. The swelling of a styrene divinylbenzene-based copolymer was chosen as the basis of our examination. A model is proposed that is capable of describing the changes in the size of beads over time in this system. Keywords: polymer swelling, particle size distribution, heterodisperse polymer beads, modeling, ion-exchange resin 1. Introduction Polymer beads are used as raw materials in a wide range of technologies, e.g. in the production of ion-exchange resins. Before the chemical modification of polymer beads, they are often swollen with different types of swelling agents such as dichloroethane, dichloromethane, toluene, etc. Monodisperse and heterodisperse types of beads are known in polymer technologies. The process of the swelling of monodisperse beads is easily measurable and easily predictable. The production of monodisperse poly- mers is a more expensive process than the production of heterodisperse beads, which makes heterodisperse poly- mer beads a more popular form. Heterodisperse polymer beads exhibit a closely nor- mal distribution in terms of particle size. The prediction of the swelling of these particle systems is more difficult because the different beads can swell at significantly dif- ferent rates due to the change in the specific area of each bead. The swelling of the polymer network system has al- ready been a subject of interest. Painter and Shenoy[1] considered the chemical properties of polymers. Schott [2] described the kinetics of polymer swelling. First or- der and second order kinetics were founded by him. A *Correspondence: wirnhardt.adam@fmt.uni-pannon.hu swelling model was formulated by Sweijen et al. [3] ac- cording to the diffusion properties of components in the polymer matrix. These models are capable of describing the swelling of polymer networks, but because of its com- plexity they are hard to apply in any kind of optimiza- tion process. Hence, the simplest particle size distribution (PSD) model, which can describe the swelling of poly- mers over time using the least number of parameters, can be of interest from a process intensification point of view. Bayat et al. collected PSD models from the last sev- enty years [4, 5]. Thirty-five models were listed. These models describe the cumulative mass fraction of poly- mers as a function of the diameter of polymer beads. The models contain one, two, three or four unknown parame- ters, which can be identified with a specific polymer frac- tion. A hyperbolic tangent distribution [6, 7] PSD model composed of four parameters was added to this list by us. This study is the first step in the process of devel- oping this model which focuses on the investigation of the swelling phenomena of the styrene divinylbenzene copolymer system. Therefore, our focus is on identifying a simple PSD model which is capable of describing the swelling of heterodisperse polymer beads. Hence, all the previously mentioned PSD models are investigated and compared. Our aim was not only to identify a PSD model which is capable of describing the distribution of the in- vestigated polymer system but to find a PSD model which mailto:wirnhardt.adam@fmt.uni-pannon.hu 80 WIRNHARDT AND VARGA exhibits a correlation between changes made to parame- ters and particle size. 2. Experimental For the modelling of heterodisperse polymer systems physical experiments with regard to swelling should be performed to obtain the data necessary to validate the model. In our case, with the lack of experiments, the data were generated from a model implemented and solved in MATLAB. A code was made in MATLAB for the gen- eration of these distributions. In the developed model, the swelling of polymer beads with a given theoretical rate of growth was calculated. The volume of each bead increased at the same rate. Hence, the diameter of the beads does not influence the rate of growth and the ten- sion caused by the swelling process of the polymer beads is neglected in this investigation. The following simplifi- cations were implemented in the model: 1. The shape of the polymer beads is a perfect sphere. 2. The particles swell until they reach a steady state. 3. The swelling rate of particles is constant until a steady state is achieved. 4. The number of particles is constant. 5. The initial PSD of the beads is close to normal. To generate the distributions after different durations of swelling an initial unswollen distribution is required. The initial distribution was calculated by MATLAB from a picture of heterodispersed particles. The Varion KS pre- form styrene divinylbenzene copolymer was used for the zero-time distribution. The polymer beads were identified by a circle detection algorithm and the size distribution of the detected particles was calculated using a reference particle. From the initial state, the size of the polymer particles starts to increase by applying a swelling agent to the sys- tem. The size of the particles increases until a steady state is achieved. The steady state in this case means that the size of the particles grows until a state when the amount of infiltration of the swelling agent is equal to the out- come amount. The size of the particles can increase until a maximum is reached because of the internal tension of the polymer. The data are generated using the parameters of the steady state. These parameters are the swelling rate (p [-]) and time required (t [sec]). PSDs were generated at different times during the swelling process. The next step was to examine the PSD models to determine if they were able to describe the dis- tribution in every instant. 2.1 PSD models In this study only PSD models that are capable of de- scribing cumulative mass fraction distribution were in- vestigated. Altogether five models with one parameter, twelve with two, two with three and four with four were examined. They are collected in Table 1. The cumulative mass fraction of particles is denoted by P(d), the maxima and minima of the particle size range are represented by dmax and dmin, respectively, and the particle diameter [mm] is denoted by d. The mod- els were fitted to all the distributions collected over time using extreme value problem solver algorithms in MAT- LAB. 2.2 Theoretical methodologies Two types of extreme value problem-solving methods were applied to fit the PSD models. One is a local ex- treme value problem solver known as the Nelder-Mead simplex algorithm and its function “fminsearch” to im- plement it in MATLAB. The other one is a global ex- treme value problem solver called “NOMAD” [8]. They are both components of the MATLAB toolkit. MATLAB 2011b was applied in all modelling steps. The minimum difference was sought between the gener- ated distribution data and calculated distributions based on each model. The parameters of PSD models were the results of this search. In every case, the goodness of fit was measured. For each model, every sample time was considered and the difference examined using the mean absolute difference. The average of the mean absolute difference of the percentage difference was calculated for every function. The extreme value problem solver at- tempted to find the minimum of the following equation Et = ∣∣P (d)′t − P(d)t∣∣ nd (1) where the mean absolute difference between the gener- ated and calculated distribution is Et at instant t. The calculated distribution is denoted by P (d)′t and the gen- erated distribution by P(d)t at instant t. The number of items of data is represented by nd. 2.3 Model selection A selection could be made according to the average per- centage differences. The selection was carried out with a criterion. This criterion was an average percentage difference of five percent because under this value the difference is not considerable but over it an unaccept- able fit is shown. Three different models, namely the Rosin-Rammler, the Exponential-power_Pasikatan and the Logarithm-Zhuang models fitted to the generated data are shown in Fig. 1. The goodness of fit for these three models is 1 %, 5 % and 11 %, respectively. As can be seen the differences of 1 % and 5 % are negligible, and are only noticeable at diameters in excess of 0.7 mm. How- ever, a considerable difference can be observed between the models with fits of 5 % and 11 %. It can be seen that a goodness of fit of under 5 % is appropriate. Hungarian Journal of Industry and Chemistry COMPARISON OF PARTICLE SIZE DISTRIBUTION MODELS FOR POLYMER SWELLING 81 Table 1: The investigated PSD models Name Model 1 parameter (k1) 1 Gaudy-Meloy P(d) = 1− (1−d/dmax)k1 2 Nesbitt-Breytenbach P(d) = 10[(1/(k1 + 1)](d/2)k1+1 +[0.1−1/(k1 + 1)](d/2)1/[1/(k1+1)−0.1] 3 Rosin-Rammler P(d) = 1−exp(−k1d) 4 Jaky P(d) = exp { − ( 1/k21 ) [ln(d/dmax)] 2 } 5 Schumann P(d) = (d/dmax)k1 2 parameters (k1, k2) 6 Power low- Paskikatan P(d) = [k1/(1−k2)]d1−k2 7 van Genuchten P(d) = [ 1 + (k1/d) k2 ]1/k2−1 8 Rosin-Rammler P(d) = 1−exp ( −k1dk2 ) 9 Fractal P(d) = exp{ln(k2))+[( 3k21 −13k1 + 14 ) / ( k21 −5k1 + 4 ) + 1 ] log(d) } 10 Power low - Gimenez P(d) = k1dk2 11 BEST P(d) = [ 1 + (k1/d) k2 ]2/k2−1 12 Bennet P(d) = k1k2dk1−1 exp ( −k2dk1 ) 13 Exponential power- Pasikatan P(d) = exp ( −k1dk2 ) 14 Logarithm-Zhuang P(d) = k1 ln(d) + k2 15 Log-exp-Kolev P(d) = k1 exp [k2 log(d)] 16 Weibull-2par P(d) = 1−exp [ −(d/k1)k2 ] 17 Lognormal- Zobeek P(d) = 1/ { k1(2π) 1/2 exp [ −(log(d)−k2)2/(2k21) ]} 3 parameters (k1, k2, k3) 18 S-Curve: Vipulanandan Ozgurel P(d) = 100 exp { −k1 [ k2 ln(d/0.001) d/k3 ]} 19 Weibull-3par P(d) = k1 −exp [ −(d/k2)k3 ] 4 parameters (k1, k2, k3, k4) 20 Gompertz P(d) = k1 + k2 exp{−exp [−k3(d−k4)]} 21 Weibull-4par P(d) = k3 + (1−k3) [ 1−exp ( −k1kk24 )] where k4 = (d−dmin)/(dmax −dmin) 22 Fredlund P(d) = { 1/ [ ln ( exp(1) + (k1/k2) k2 )]k3}{ 1− [ln(1 + k4/d)/ ln(1 + k4/0.001)]7 } 23 Tanh if 0