Modeling Particle Size Distribution in Heterogeneous Polymerization Systems Using Multimodal Lognormal Function

Abstract

-This work evaluates the usage of the multimodal lognormal function to describe Particle Size Distributions (PSD) of emulsion and suspension polymerization processes, including continuous reactions with particle re-nucleation leading to complex multimodal PSDs. A global optimization algorithm, namely Particle Swarm Optimization (PSO), was used for parameter estimation of the proposed model, minimizing the objective function defined by the mean squared errors. Statistical evaluation of the results indicated that the multimodal lognormal function could describe distinctive features of different types of PSDs with accuracy and consistency.

Physically Motivated Models for Heterogeneous Random Phenomena

“…The two extensions of the material presented in Random Phenomena presented in sections and are well-known in the statistics literature and, as noted, have been used in a variety of engineering applications. ,,,,,,− The material presented in these sections is therefore primarily a survey of known results that should be useful to, and easily understood by, those who have worked through the material in Random Phenomena . The third extension of this material, the generalized CLT proposed in section , is believed to be new.…”

“…In common with all other introductions to statistics, Random Phenomena primarily restricts consideration to homogeneous phenomena, where a single probability distribution or regression model is developed to characterize the entire data set. The primary focus of this paper has been on concepts and methods appropriate to heterogeneous phenomena like the multimodal particle size distributions discussed by Ferrari et al, 19 or the bimodal droplet size distribution example shown in the paper by Prileszky,Ogunnaike,and Furst [ref 40,Figure 4]. The class of discrete mixture distributions discussed in section 2 represents a simple and convenient way of characterizing heterogeneous behavior in examples like these.…”

“…In cases of substantial heterogeneity, as evidenced by multimodal distributions, a simple extension of the ideas presented in Random Phenomena is the class of finite mixture distributions , described in section . These models have been applied in a number of engineering applications, including process monitoring, ,,− describing multimodal particle size distributions in heterogeneous polymerization systems, and characterizing rare defects in manufacturing processes . One reason for the popularity of discrete mixture distributions is their extreme mathematical flexibility, both as approximations of other more complex distributions (see, for example, the enormous variety of the 15 Gaussian mixture distributions used by Marron and Wand in their evaluation of density estimation procedures and discussed in the book by McLachlan and Peel [ref , section 1.5.2]), and as descriptions of data generated by several distinct mechanisms, as in the multimodal particle size distributions discussed by Ferrari et al Another reason for the popularity of these models is that software for fitting them to data is widely available, as discussed in section .…”

“…Mixture models are almost always fit by the method of maximum likelihood, which can be approached using standard, general-purpose optimization procedures like Newton–Raphson or more recent alternatives like Particle Swarm Optimization, but the Expectation–Maximization (EM) algorithm is the most popular approach. , This algorithm treats the random selection variable z i in eq as missing data and develops a two-step iteration for maximizing the likelihood. A practical advantage of this approach is that the likelihood increases at each iteration and the algorithm generally converges, although convergence can be very slow in some circumstances [ref , section 2.3], a point illustrated in section .…”

Experimental characterization of polydisperse particle‐loaded flow for linear resin transfer molding injections

“…• Retained PSD f th retained is assumed to follow a lognormal distribution [46,47] whereas inlet, outlet, and blocked particles are described by their experimental distributions. The lognormal distribution [42][43][44] is given by Equation 4, where d k represents the k th particle size μ and σ, the mean and standard deviation of the distribution, respectively.…”