Numerical study on the turbulent reacting flow in the vicinity of the injector of an LDPE tubular reactor

Three-dimensional (3-D), transient numerical simulations of the turbulent reacting flow in the vicinity of the initiator injection point of a low-density polyethylene (LDPE) tubular reactor using a large eddy simulation (LES) approach combined with a filtered density function (FDF) technique are presented. The numerical approach allows for detailed predictions of the turbulent flow field and the associated (passive and reactive) scalar mixing. The aim is to study the influence of the injector geometry and initiator injection temperature on the LDPE process in terms of product quality (average polymer chain length, and polydispersity) and process efficiency (such as initiator consumption).     

Comparison of Monte Carlo and quasi-Monte Carlo technique in structure and relaxing dynamics of polymer in dilute solution

Structure and dynamics of polymer in solvent solution is an important area of research since the functional properties of polymer are largely dependent on the morphology of the polymers in solution. This structure related properties are especially important in case of surface science where the phase-separated morphology in the micro/nano scale dictates the properties of the product. Modeling polymers in solution is an efficient way to determine the morphology and thus the properties of the products. It saves time as well as helps to design novel materials with desired properties. Polymers in solution systems are generally modeled with bead spring model and Monte Carlo or importance sampling Monte Carlo simulations is used to find the optimal configuration where the energy of the system is minimized. Often in these simulations, random numbers are used in the Monte Carlo steps. Normally random numbers try to form clusters and do not cover the entire dimension of the system. Thus the minimum energy structures obtained from simulations with random numbers are not optimal configuration of the system. In the present work a lattice-based model is used for polymer solution system and importance sampling Monte Carlo is used for simulation. Quasi-random numbers generated from Hammersley sequence sampling (HSS) are used in the simulation steps for stochastic selection polymers and its movements. Quasi-random numbers obtained from HSS are random in nature and they have n-dimensional uniformity. They do not form clusters and the structural configuration obtained using quasi-random numbers are optimal in nature. The optimal configurations of the polymers as obtained from random number and quasi-random number are compared. The result shows that simulation with HSS attains a lower energy state after initial quench. At the late stage of spinodal decomposition, the structure factor decrease-showing Ostwald ripening which is not observed from simulation with random numbers.     

Monte Carlo simulation of two interpenetrating polymer networks: Structure, swelling, and mechanical properties

The swelling and mechanical properties of various interpenetrating polymer networks (IPNs) were studied. Six networks made from permutations of a moderately crosslinked polyelectrolyte network (ref), a moderately crosslinked neutral polymer network (net1), and a highly crosslinked polyelectrolyte network (net2) were first swollen in water and structural properties such as end-to-end chain lengths and radial distribution functions were compared with the component networks' equilibrium properties. The swelling of composite IPNs was discussed in terms of a balance between the osmotic pressure due to mobile counterions and the restoring force of the network chains, which act in parallel to counteract the osmotic swelling. For the ref-net2 system, the strong stretching of net2 chains increases the network restoring force and the further swelling due to the counterions is suppressed. The swollen networks were then uniaxially stretched, and equilibrium stress-strain plots were obtained up to high extension ratios. The equilibrium volume decreased upon uniaxial extension, and the elastic moduli of IPNs of the A-A type were slightly greater than that of their respective single networks.     

Molecular weight distribution in low-density polyethylene polymerization; impact of scission mechanisms in the case of a tubular reactor

It has been shown that scission kinetics strongly affects the molecular weight distribution (MWD) of low-density polyethylene (ldPE) in a continuous stirred tank reactor (CSTR). The present paper focuses on the effects of different chemical scission mechanisms, linear and topological scission, as well as mechanical scission on MWD in batch and tubular reactors. In contrast to the CSTR, a batch reactor MWD does not show bimodalities or long tails. The tubular reactor was modeled as an industrially representative system with four initiator injection points and a proper ‘cocktail’ of different initiators. Calculated MWD was compared to one experimentally determined with SEC-MALLS for a commercial tubular product and fair agreement was found. Typically, these MWDs are broad, but not bimodal. Sensitivity studies were performed as to scission kinetics and the effect of chain transfer agent (CTA). Both batch and tubular reactor were observed to be less sensitive to scission kinetics than a CSTR. In addition, alternative CTA injection strategies (downstream positions) were tested. These showed interesting behavior leading to very broad and bimodal MWD. The model allows following the MWD broadening along the tube. We conclude that batch and tubular ldPE reactors lead to completely different MWD behavior than a CSTR and that it is possible to manipulate it to a great extent.     

Conditional Monte Carlo sampling to find branching architectures of polymers from radical polymerizations with transfer to polymer and recombination termination

A model is developed that predicts branching architectures of polymers from radical polymerization with transfer to polymer and termination by disproportionation and recombination, in a continuously stirred tank reactor (CSTR). It is a so-called conditional Monte Carlo (MC) method generating architectures of molecules of specified dimensions. The relevant dimensions in the present case are the number of branch points, n_p, and the number of combined parts a molecule consists of, n_c. These branch points and combination points together are decisive for the connectivity inside molecules. The modeling strategy is based on backtracking of the molecular growth history in terms of the chemical events determining connectivity, transfer to polymer and recombination termination. The recombination termination mechanism requires the model to develop parts of the architecture following several paths back to the initial primary polymers that form the starting points for the molecules. The algorithm requires the construction of probability density functions being evaluated using a fast Galerkin-FEM method. The architectures generated by the conditional Monte Carlo method are compared to those from a full MC method using several qualifiers. One of these is the number of initial primary polymers in a molecule as well as their lengths, another is the radius of gyration contraction factor. Perfect agreement is found between the architectures found by the conditional and full MC methods.     

Combination of a Monte Carlo approach with the contact time distribution concept for the steady-state modeling of an isothermal heterogeneous coordinated anionic ring opening polymerization reactor

In this paper, the possibility of combining a molecular level Monte Carlo simulation with a chemical engineering model of an isothermal fixed-bed reactor is demonstrated. This approach is applied to the isothermal heterogeneous coordinated anionic ring opening polymerization of ε-caprolactone. The contact time distribution (CTD) concept as defined by Orcutt et al. (Chem. Eng. Prog. Symp. Ser. 38 (58) (1962) 1), Nauman and Collinge (Chem. Eng. Sci. 23 (1968a) 1309) and Shinnar et al. (Chem. Eng. Sci. 27 (1972) 1627) is used. The contact time is the time spent by a reactant within the pores of the pellets constituting the fixed-bed. The classical monodisperse model represents the hydrodynamic and mass transfer in the reactor. The CTD is calculated from this model according to Shinnar et al. theory. From that point, the reactor outlet monomer conversion can be calculated analytically, the ε-caprolactone consumption being a first-order process. Furthermore, the molecular size distribution (MSD) of the oligomers at the reactor outlet is computed on the basis of Monte Carlo simulations. Comparisons with experimental data are given.     

Influence of molecular architecture and melt rheological characteristic on the optical properties of LDPE blown films

A systematic investigation on the origin of the haze of LDPE blown films was conducted, aiming to correlate the film haze with the molecular architecture and melt rheological properties. First of all, the haze measurement indicated that the surface haze, rather than the bulk haze, is the dominating factor for the total haze of the investigated films. No spherulitic crystals or other superstructures were observed for the LDPE blown films, implying that the crystallites formed in the film-blowing process are too small to be responsible for the optical haze. Rheological study revealed that the surface roughness was originated from the irregular flow of LDPE melt during the extrusion process. NMR, GPC and parallel-plate rheology were applied to study the molecular architecture of the LDPE resins. It was found that the LDPE sample with higher haze value exhibits distinctly larger portion of higher molecular weight component, broader molar mass distribution, significantly higher side chain branch density.     

A graph theory assisted Monte Carlo algorithm for simulating the topology structure of LDPE and connection with molecular dynamics simulation

Precise simulation of the topology structure of low-density polyethylene (LDPE) chain is significant for product properties research. However, the chain structure symbolization methods in literature all rely on specially designed methods or temporary tricks, which are hard to understand and may induce some deviations in branch distribution prediction. To overcome this problem, a graph theory assisted Monte Carlo algorithm is developed for chain topology structure simulation of LDPE. The new symbolization method provides better accuracy and is much easy for code implementation and integration in other software. In detail, the average long chain branching predicted by this method is always smaller than that predicted by traditional methods, and much closer to the experimental results. Furthermore, the predicted topology structure information of LDPE chains can be provided as input data for molecular dynamics simulation to study the crystal process, and the predicted crystallinity and density also show good agreement with experiments.     

Monte Carlo simulations of phase separation in thin polymer blend films: scaling properties of morphological measures

Surface directed phase separation in thin polymer blend film has been studied with Monte Carlo simulations using a simple model based on reptation method. Time evolution of phase structure was characterized quantitatively by morphological measures (the Minkowski functionals) in addition to the inspection of concentration versus depth profiles. It was shown that the dynamical scaling hypothesis holds for the Minkowski functionals describing morphologies at the early stages of phase separation in thin films. Two time regimes with different scaling exponents (0.25, 0.33) were found for the growth of the characteristic length scale in the system corresponding to various transport mechanisms (diffusion along- and normally to the interface). Fast decrease in the morphological measures observed at the end of phase separation was attributed to the confinement of the thin film.     

Investigating the effect of shape on particle segregation using a Monte Carlo simulation

A Monte Carlo simulation has been used to investigate the segregation potential of a range of particulate systems under conditions in which the particles undergo high amplitude low frequency shaking. These systems involve a wide range of binary powder mixtures in which complex particle shapes have been investigated, including plates and rods which represent the real world materials encountered in pharmaceutical systems such as those which include crystalline components. Previous simulations on the segregation propensity of systems with different shapes were limited to spheres and spherocylinders, with relatively low vibrational amplitude drops. A commercial computer application for particle packing—called MacroPac—has been successfully employed here, as it has been able to model systems that are more complex where the shape variation is much wider. These simulations apply to the case of macroscopic particles, in the absence of air resistance and inter-particle forces. For non-spherical shapes, an ‘effective size’ which relates to the radius of gyration of the particles is determined. Our studies indicate that with high amplitude low frequency shaking, in a mixture of particles with different shapes but with equal volumes, the particles with the larger ‘effective size’, which tend to have a lower packing fraction, segregate to the top.     

Analysis of four Monte Carlo methods for the solution of population balances in dispersed systems

Monte Carlo (MC) constitutes an important class of methods for the numerical solution of the general dynamic equation (GDE) in particulate systems. We compare four such methods in a series of seven test cases that cover typical particulate mechanisms. The four MC methods studied are: time-driven direct simulation Monte Carlo (DSMC), stepwise constant-volume Monte Carlo, constant number Monte Carlo, and multi-Monte Carlo (MMC) method. These MC's are introduced briefly and applied numerically to simulate pure coagulation, breakage, condensation/evaporation (surface growth/dissolution), nucleation, and settling (deposition). We find that when run with comparable number of particles, all methods compute the size distribution within comparable levels of error. Because each method uses different approaches for advancing time, a wider margin of error is observed in the time evolution of the number and mass concentration, with event-driven methods generally providing better accuracy than time-driven methods. The computational cost depends on algorithmic details but generally, event-driven methods perform faster than time-driven methods. Overall, very good accuracy can be achieved using reasonably small numbers of simulation particles, O(10³), requiring computational times of the order 10²−10³ s on a typical desktop computer.     

Accuracy and optimal sampling in Monte Carlo solution of population balance equations

Implementation of a Monte Carlo simulation for the solution of population balance equations (PBEs) requires choice of initial sample number (N₀), number of replicates (M), and number of bins for probability distribution reconstruction (n). It is found that Squared Hellinger Distance, H², is a useful measurement of the accuracy of Monte Carlo (MC) simulation, and can be related directly to N₀, M, and n. Asymptotic approximations of H² are deduced and tested for both one‐dimensional (1‐D) and 2‐D PBEs with coalescence. The central processing unit (CPU) cost, C, is found in a power‐law relationship, , with the CPU cost index, b, indicating the weighting of N₀ in the total CPU cost. n must be chosen to balance accuracy and resolution. For fixed n, M × N₀ determines the accuracy of MC prediction; if b > 1, then the optimal solution strategy uses multiple replications and small sample size. Conversely, if 0 < b < 1, one replicate and a large initial sample size is preferred. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2394–2402, 2015     

Synthesis and characterization of branched polymers by a free radical approach using a novel ‘macroiniferter’

Free radical bulk polymerization of styrene and methyl methacrylate (MMA) was carried out using a novel ‘macroiniferter’ which resulted in branched polymers with relatively narrow molecular weight distribution. This approach involving the novel macroiniferter; poly[3-(t-butylperoxy)propyl disulfide] (PBPPDS) that has side chain peroxide groups and main chain disulfide linkages was developed to prepare soluble branched polymers as well as to control the extent of branching in vinyl polymers synthesized via a free radical route. The synthesis, characterization and thermal degradation studies of PBPPDS are reported here for the first time. The resulting polystyrene (PS) and poly(methyl methacrylate) (PMMA) polymers were characterized using gel permeation chromatography (GPC), intrinsic viscosity [η] measurements and the degree of branching was studied by the determination of g′ factor.     

Monte Carlo investigation of hysteresis properties in ferroelectric thin-films under the effect of uniaxial stresses

The uniaxial stress dependence of the hysteresis behavior of ferroelectric films was studied. The DIFFOUR model was modified to include the uniaxial stress effect. Both the uniaxial stress and the external electric field were applied on the out-of-plane direction of the films. The polarization was measured with varying the magnitude of the applied stress and the electric field frequency via the dynamics of the polarization reversal in terms of hysteresis. The study was taken by means of Monte Carlo simulations using the spin-flip Metropolis algorithm. From the results, the district dependence of hysteresis behavior on frequency between low frequency and high frequency was prominent. On the other hand, the remanent and the coercivity significantly decreased with increasing applied stresses. Moreover, the areas under the hysteresis loops also decreased indicating smaller magnitude of energy dissipation. The results agree well with related experiments where applicable.     

Degradation of LDPE/VGO mixtures to fuels using a FCC equilibrium catalyst in a sand fluidized bed reactor

Plastic degradation for recovery of useful products or raw materials is a very interesting alternative for reducing the plastic accumulation. This paper explores the possibility of using refinery facilities to carry out the plastic cracking as well as to take the most of the products obtained. In the present work, LDPE/VGO blends with different percentages of polymer are degraded in presence of a FCC equilibrium catalyst. The reactor used in this study is a laboratory scale sand fluidized bed reactor at 500 °C, and a 7:1 catalyst:LDPE/VGO blend ratio in order to simulate the operating conditions in a large scale industrial reactor. Polyethylene blends evaluated show relative proportions of LDPE of 0, 6, 25, 75 and 100% (w/w). Gas and liquid compounds were collected and quantified. The results obtained are compared with those generated in a thermal cracking.

In all cases, the FCC equilibrium catalyst showed a high selectivity to the production of isobutane and isopentane in the volatile compounds as well as to aromatics in the liquid products.

Results shown in this paper evidences the viability of introducing plastics into FCC unit, producing potential valuable products from low value materials.     

Monte Carlo simulation of molecular weight distribution and copolymer composition in transamidation reaction of polyamides

Melt mixing of polyamides results in exchange reaction and generation of copolymers. In this work, Monte Carlo method is used to simulate the time evolution of molecular weight distribution (MWD) and copolymer composition during the exchange reaction process between polyamides with AA and BC structure. The influences of initial composition and molecular weight have been investigated. Decrease in the difference between the average molecular weight of two kinds of polyamides results in faster approach of the MWD toward Flory's distribution and higher probability of producing copolymers. When the ratio between the numbers of initial molecules of two homopolymers is increased, the number of generated copolymers is reduced and the wider MWD is obtained. © 2012 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

Influence of polymerization procedure on polymer topology and other structural properties in highly branched polymers obtained by A2 + B3 approach

Computational studies were carried out to investigate the influence of polymerization procedure on the topology and various macromolecular characteristics of the highly branched polymers formed by the reaction of A₂ and B₃ type monomers through step-growth polymerization reactions. The influence of three different polymerization procedures on the properties of the polymers formed was investigated, namely, (i) slow addition of A₂ over B₃, (ii) slow addition of B₃ over A₂, and (iii) mixed A₂ + B₃. Topology, degree of branching, number and weight average molecular weights, and polydispersity index of the polymers were determined using Monte Carlo simulations, assuming different levels of cyclization ratios during the reactions. Interestingly model polymers obtained by the slow addition of B₃ over A₂ produced much higher degree of branching or truly hyperbranched polymers, when compared with the other two methods, which mainly resulted in linear growth with slight branching.     

Prediction of Viscoelastic Properties of Enzymatically Crosslinkable Tyramine–Modified Hyaluronic Acid Solutions Using a Dynamic Monte Carlo Kinetic Approach

The present study deals with the mathematical modeling of crosslinking kinetics of polymer–phenol conjugates mediated by the Horseradish Peroxidase (HRP)-hydrogen peroxide (H₂O₂) initiation system. More specifically, a dynamic Monte Carlo (MC) kinetic model is developed to quantify the effects of crosslinking conditions (i.e., polymer concentration, degree of phenol substitution and HRP and H₂O₂ concentrations) on the gelation onset time; evolution of molecular weight distribution and number and weight average molecular weights of the crosslinkable polymer chains and gel fraction. It is shown that the MC kinetic model can faithfully describe the crosslinking kinetics of a finite sample of crosslinkable polymer chains with time, providing detailed molecular information for the crosslinkable system before and after the gelation point. The MC model is validated using experimental measurements on the crosslinking of a tyramine modified Hyaluronic Acid (HA-Tyr) polymer solution reported in the literature. Based on the rubber elasticity theory and the MC results, the dynamic evolution of hydrogel viscoelastic and molecular properties (i.e., number average molecular weight between crosslinks, M_c, and hydrogel mesh size, ξ) are calculated.     

An exploration of corrosion inhibition of mild steel in sulphuric acid solution through experimental study and Monte Carlo simulations

Abstract

4-[4-(1H-imidazol-1-yl)methyl)-1H-1,2,3-triazol-1-yl]methylbenzophenone (ITBP) and 4[4-(1H-1,2,4-triazol-1-yl)methyl)-1H-1,2,3-triazol-1-yl]methylbenzophenone (TTBP) are synthesized as new heterocyclic compounds of the triazole derivative family and tested successfully as potential inhibitors for MS in 1?M H₂SO₄ corrosive medium by using gravimetric analysis, electrochemical impedance spectroscopy, potentiodynamic polarization, and energy dispersive X-ray spectroscopy (EDX). Polarization curves show that the tested inhibitors are mixed-type inhibitors. Scanning electron microscopy (SEM) affirmed the existence of an adsorbed film on the steel surface. Monte Carlo simulations were in excellent agreement with the experimental tests.

Abbreviation: PDP: Potentiodynamic Polarization; EIS: Electrochemical impedance spectroscopy; DFT: Density functional theory; MC: Monte Carlo     

TiO2 absorption and scattering coefficients using Monte Carlo method and macroscopic balances in a photo-CREC unit

The radiation field inside photocatalytic reactors can be predicted by solving the Radiative Transfer Equation (RTE). From the solution of the RTE, the Local Volumetric Rate of Energy Absorption (LVREA) can also be obtained. This LVREA is an important parameter in photocatalytic reactor design, energy efficiency assessments and kinetic studies of photocatalytic reactions. However, when solving the RTE, two optical parameters are needed: (1) the absorption and scattering coefficients and (2) the phase function. In the present study, the Monte Carlo (MC) method along with an optimization technique is shown to be effective in predicting the wavelength-averaged absorption and scattering coefficients for three different TiO₂ powders. To accomplish this, the LVREA and the transmitted radiation (P_t) in a Photo-CREC annular photoreactor have to be determined using a macroscopic balance. The optimized coefficients are calculated ensuring that they comply with a number of physical constrains, falling in between bounds established via independent criteria. The optimization technique is demonstrated by finding the absorption and scattering coefficients for three different semiconductors that best fit the experimental values from the macroscopic balance minimizing the least-squared error of objective functions for the LVREA and P_t. The proposed approach is a general and promising one, not being restricted to reactors of concentric geometry, specific semiconductors and/or particular photocatalytic reactor unit scale.