Numerical inversion techniques in the recovery of molecular weight distribution expressed in different transformed domains. Experimental validation

Abstract

In this work we analyze the feasibility of using numerical inversion techniques for recovering MWD of actual polymers from a transformed domain, specifically the one defined by probability generating functions. We start from known experimental MWDs, transform them, and then apply two different numerical techniques to recover the MWD. We analyze the influence of noise in the calculated probability generating functions on the quality of the recovered molecular weight distributions. We also study how the range of molecular weight selected for the inversion procedure affects the results. We compare the recovered distributions obtained by both methods and suggest a criterion for establishing the reliability of a given solution. We find that this general strategy is appropriate for the recovery of MWDs whether they are monomodal, multimodal, wide or narrow. This provides a tool for the treatment of actual polymerization systems for which there is no analytical solution for the mass balance equations.

Mathematical Modeling of Bivariate Polymer Property Distributions Using 2D Probability Generating Functions, 1 – Numerical Inversion Methods

“…Hence, there is a compromise value of the parameter that yields the best possible result, that is, the smallest error with respect to the true function. In our previous works on univariate pgf inversion,14, 23 it was shown that curves obtained with two successive values of N became closer to each other as N increased towards its best value. After that point, they started to separate due to the influence of the increasing round‐off error.…”

“…From our previous works with univariate pgfs, we have developed a variety of well‐established methods of this type. In this work two of them, the adaptations to pgf inversion of the methods proposed by Stehfest23 and Papoulis14 for Laplace transforms, were evaluated for the proposed inversion scheme of 2D pgfs. The Stehfest inversion method for univariate pgf consists of the following formula:23 where $\widehat {{\bf f}}({\bf z}) = [\widehat {f}(z1),\widehat {f}(z2), \ldots ,\widehat {f}(z_{N} )]^{{\rm T}} $ is a vector of univariate pgf transforms evaluated at $z_{n} = {\rm e}^{ - n\ln (2)/t} $ , N is an even integer, which is a parameter of the method, and k n ( n = 1,…, N ) are coefficients defined by …”

High‐Pressure Polymerization of Ethylene in Tubular Reactors: A Rigorous Dynamic Model Able to Predict the Full Molecular Weight Distribution

“…These pgfs represent the transforms of the MWD expressed as number fraction versus molecular weight ( n i ) when a = 0, weight fraction versus molecular weight ( w i ) when a = 1 and the product the product of weight fraction and molecular weight versus molecular weight [used to calculate the differential log distribution, (d W /dlog 10 $\overline M _{\rm w} $ ) i ] when a = 2. Inversion of each type of pgf allows the recovery of the three kinds of distribution independently, thereby attenuating numerical noise propagation 25. Tedious algebraic operations needed for the transformation of the mass balances can be avoided by using of the pgf transform table we previously developed 23.…”

“…The resulting terms for the reaction rates r j in Equation (20) in Table 2, with j = λ a · ϕ a,l and µ a · φ a,l , are shown in Table 3. The pgf inversion algorithm employed in this model is the adaptation of the Stehfest algorithm for pgf inversion 25. The equations corresponding to this inversion algorithm [represented by function f (.)…”

Polycondensation

“…A recent surge on this approach has led to exploitation of other more complex but hopefully more efficient methods [265], mainly developed for Laplace transform inversion, and so more adapted to high average molecular weights. (119) can be neglected for large enough N or small enough jsj.…”