Population Balance Modeling with Coupled Agglomeration and Disintegration Processes for TiO2 Nanoparticles Formation and Experimental Validation

Particle size distribution of nanoparticles plays an important role in modelling many scientific and engineering problems. In this article, we proposed a Finite Volume Method (FVM) to model TiO₂ nanoparticles formation using population balance equations (PBEs) by incorporating the simultaneous agglomeration and disintegration processes. The superposition of the PBEs for agglomeration and disintegration with different kernels leads to a system of partial-integro differential equations, which are numerically solved by using FVM. The precipitation of TiO₂ nanoparticles in the batch reactor is studied experimentally as well as by numerical simulations based on Austin and Diemer disintegration kernels and Shear agglomeration kernel. Finally, the capability of the precipitation model is evaluated and the experimental results on particle sizes are compared with the numerical results.

     

Using Unfold‐PCA for batch‐to‐batch start‐up process understanding and steady‐state identification in a sequencing batch reactor

In chemical and biochemical processes, steady‐state models are widely used for process assessment, control and optimisation. In these models, parameter adjustment requires data collected under nearly steady‐state conditions. Several approaches have been developed for steady‐state identification (SSID) in continuous processes, but no attempt has been made to adapt them to the singularities of batch processes. The main aim of this paper is to propose an automated method based on batch‐wise unfolding of the three‐way batch process data followed by a principal component analysis (Unfold‐PCA) in combination with the methodology of Brown and Rhinehart 2 for SSID. A second goal of this paper is to illustrate how by using Unfold‐PCA, process understanding can be gained from the batch‐to‐batch start‐ups and transitions data analysis. The potential of the proposed methodology is illustrated using historical data from a laboratory‐scale sequencing batch reactor (SBR) operated for enhanced biological phosphorus removal (EBPR). The results demonstrate that the proposed approach can be efficiently used to detect when the batches reach the steady‐state condition, to interpret the overall batch‐to‐batch process evolution and also to isolate the causes of changes between batches using contribution plots. Copyright © 2007 John Wiley & Sons, Ltd.     

A method to determine quasi-steady state in constant voltage mode isotachophoresis

Identification of the steady state is very challenging in isotachophoresis (ITP); especially in complex microgeometries, such as dog-leg channels or cross-channel junctions. In this work, an elastic matching method is applied to determine the quasi-steady state in microscale ITP. In the elastic matching method, the similarity between two profiles is calculated by comparing intensity distribution of two images or profiles. To demonstrate this similarity-based analysis technique for ITP, a constant voltage mode ITP model is developed and applied to a five-component ITP system. Hydrochloric acid and caproic acid are used as the leader and terminator, respectively, while histidine is used as the counter-ion. Two sample components, acetic acid and benzoic acid, are separated under the action of an applied electric field in both straight and dog-leg microchannels. This analysis shows that conductivity profiles provide a better measure to determine the quasi-steady state in an ITP process. For a straight microchannel, the quasi-steady state is achieved in less than a minute with a total potential drop of 100?V in a 2?cm long channel. In a straight channel, a true steady state can be achieved for ITP with appropriate countercurrent flow where stationary zones are formed, but the time it takes to reach the steady state is much longer than the without counter flow case. The numerical results indicate that a steady state cannot be reached in a dog-leg microchannel because of sample dispersion and refocusing at and near the intersections and at the branch channels. However, the elastic matching method can be used to determine the quasi-steady state in a dog-leg microchannel.     

Evaporation rate of nucleating clusters

The Becker-D?ring kinetic scheme is the most frequently used approach to vapor liquid nucleation. In the present study it has been extended so that master equations for all cluster configurations are included into consideration. In the Becker-D?ring kinetic scheme the nucleation rate is calculated through comparison of the balanced steady state and unbalanced steady state solutions of the set of kinetic equations. It is usually assumed that the balanced steady state produces equilibrium cluster distribution, and the evaporation rates are identical in the balanced and unbalanced steady state cases. In the present study we have shown that the evaporation rates are not identical in the equilibrium and unbalanced steady state cases. The evaporation rate depends on the number of clusters at the limit of the cluster definition. We have shown that the ratio of the number of n-clusters at the limit of the cluster definition to the total number of n-clusters is different in equilibrium and unbalanced steady state cases. This causes difference in evaporation rates for these cases and results in a correction factor to the nucleation rate. According to rough estimation it is 10(-1) by the order of magnitude and can be lower if carrier gas effectively equilibrates the clusters. The developed approach allows one to refine the correction factor with Monte Carlo and molecular dynamic simulations.     

Simulation of micro-behaviors including nucleation,growth, and aggregation in particle system

A new method for the solution of population balance equations (PBE) describing the micro-processes such as nucleation, growth, aggregation of particle swarms in a multiphase system is proposed. The method is based on the fixed pivot moment and allows arbitrary number of moments to be tracked simultaneously. By expressing PBEs for both batch and continuous operations in a general form, and using weighted residual method to derive the moment equations, different moments can be tracked directly. The numerical density function is assumed to be a summation of several weighted Dirac Delta functions, and the integral and derivative terms in PBEs are transformed to a summation in order to reduce computational cost. Simulations of a batch nucleation-growth process and a continuous aggregation-growth process have demonstrated good agreement with the corresponding analytical solutions, with relative errors less than 10⁻⁸%. Simulation of a combined nucleation-growth-aggregation process, which does not have an analytical solution, is also included, so as to reproduce the micro-behaviors of such a complex system, demonstrating the feasibility and reliability of this method. Supported by the National Basic Research Program of China (Grant No. 2004CB720208), National Natural Science Foundation of China (Grant Nos. 40675011 and 10872159) and Key Laboratory of Mechanics on Disaster and Environment in Western China (Grant No. 200707)     

A STUDY OF THE PHOTOCHEMICAL PROPERTIES OF SOME CINNAMATE SUNSCREENS BY STEADY STATE AND LASER FLASH PHOTOLYSIS

The E ⇄ Z photoisomerization of 4'-methoxycinnamates, used as sunscreens in cosmetics, has been studied by steady state and laser flash photolysis, in aqueous and organic solutions. Photoisomerization quantum yields are found to be fairly high (˜0.5-1), although no intermediate is detected upon laser flash photolysis. Cinnamates are not photodynamic sensitizers but are able to quench the 8-methoxy-psoralen and 5-methoxypsoralen triplets which produces E → Z isomerization.     

Systems with electrocatalytic surface reactions: Effect of the coverage dependence of the adsorption and desorption rates

Stability of steady states and possibilities for the formation of multiplicity and oscillations under potentiostatic conditions are studied for various deviations from the Langmuir adsorption kinetics in systems containing one type of substance reacting electrochemically only via its adsorptive state. If the adsorbate-adsorbate interaction between molecules of substance X along the electrode surface is repulsive or does not occur, a stable steady state exists. The multiplicity of steady states and the periodical solutions emerge only if this interaction is attractive.     

Positron–electron correlation-polarization potential model for positron binding in polyatomic molecules

Positron binding energies (PBEs) of 41 polyatomic molecules were calculated using the positron–electron correlation-polarization potential (CPP) approach and compared with experimentally measured values. In this approach, the short-range positron–electron potential is modeled using the density-functional expression, whereas the long-range potential is approximated by the attractive polarization potential. The positron–electron CPP model based on local-density approximation yields larger PBEs than experimental values; however, the calculated values can be substantially improved by introducing generalized gradient approximation. We also investigated the conformational dependence of PBEs for representative molecules.     

Network models of concentrated polymer solutions derived from the yamamoto network theory

Several choices of the functions describing the creation and destruction processes of entanglement junctions in the Yamamoto network theory of concentrated polymer solutions have been examined. These choices are simple functions of the extension of the network segments bridging the entanglement points and it is demonstrated that the moments of the distribution function describing the network conformation can be solved for analytically. This has been done for a wide range of two-dimensional flows, both for the steady state and transient start-up and relaxation problems. The macroscopic stress tensor and flow birefringence are calculated and a variety of nonlinear effects are predicted and discussed.     

Coupling of identical chemical oscillators

Identical BZ oscillators, in a CSTR, modeled by the Field-Körös-Noyes (FKN) mechanism, are coupled in a diffusion-like manner. In addition to the obvious symmetric solutions, i.e. solutions in which both CSTRs are oscillating in unison, or are in the same stable steady state, unsymmetric, broken symmetry, solutions may coexist under the same set of constraints. Thus, depending on constraints and initial conditions, the combined system can be in the following states: a) stable symmetric steady state. b) symmetric oscillations, when both cells oscillate in phase. c) coexistence of symmetric and unsymmetric steady states. d) coexistence of symmetric oscillations and unsymmetric stable steady state (broken symmetry). e) coexistence of symmetric and unsymmetric oscillations. The latter differ from the former in phase, in amplitude and in period. On the other hand, no unsymmetric oscillations were found to coexist with the symmetric steady state. All the initial conditions tried ended in either of the two, possible, stable states. The change of periods and amplitude of both types of oscillations are examined as a function of the system constraints namely, concentrations and coupling rate.     

In Situ Detection of Particle Aggregation on Electrode Surfaces

Partially blocked electrodes (PBEs) are important; many applications use non‐conductive nanoparticles (NPs) to introduce new electrode functionalities. As aggregation is a problem in NP immobilization, developing an in situ method to detect aggregation is vital to characterise such modified electrodes. We present chronoamperometry as a method for detection of NP surface aggregation and semi‐quantitative sizing of the formed aggregates, based on the diffusion limited current measured at PBEs as compared with the values calculated numerically for different blocking feature sizes. In contrast to voltammetry, no approximations on electrode kinetics are needed, making chronoamperometry a more general and reliable method. Sizing is shown for two modification methods. Upon drop casting, significant aggregation is observed, while it is minimized in electrophoretic NP deposition. The aggregate sizes determined are in semi‐quantitative agreement with ex situ microscopic analysis of the PBEs.     

Mass Transport at Microband Electrodes: Transient,Quasi‐Steady‐State,and Convective Regimes

Mass transport at microband electrodes is investigated theoretically and experimentally in unstirred solutions by chronoamperometry and cyclic voltammetry. Because natural convection limits the convection‐free domain up to which diffusion layers may only expand, several regimes of mass transport are identified through simulation by means of a previous model. A zone diagram is established which allows all relative contributions to mass transport to be delineated according to the electrode dimension, timescale of experiment, and amplitude of natural convection. In opposition to the quasi‐steady‐state regime usually expected at microband electrodes under diffusion control, a steady‐state regime always occurs at long enough times. By comparison to microdisk electrodes, a greater influence of natural convection is predicted. These results are validated experimentally by monitoring current responses and mapping steady‐state concentration profiles at microband electrodes.     

Deconvolution of Molecular Weight Distributions Using Dynamic Flory‐Schulz Distributions

Summary: The deconvolution of molecular weight distributions (MWDs) may be useful for obtaining information about the polymerization kinetics and properties of catalytic systems. However, deconvolution techniques are normally based on steady‐state assumptions and very little has been reported about the use of non‐stationary approaches for the deconvolution of MWDs. In spite of this, polymerization reactions are often performed in batch or semi‐batch modes. For this reason, dynamic solutions are proposed here for simple kinetic models and are then used for deconvolution of actual MWD data. Deconvolution results obtained with dynamic models are compared to deconvolution results obtained with the standard stationary Flory‐Schulz distributions. For coordination polymerizations, results show that dynamic MWD models are able to describe experimental data with fewer catalytic sites, which indicates that the proper interpretation of the reaction dynamics may be of fundamental importance for kinetic characterization. On the other hand, reaction dynamics induced by modification of chain transfer agent concentration seem to play a minor role in the shape of the MWD in free‐radical polymerizations.

This Figure illustrates that MWDs obtained at unsteady conditions should not be deconvoluted with standard steady‐state Flory‐Schulz distributions.     

Starting electroosmotic flow in an annulus and in a rectangular channel

The initial electroosmotic flow through a small pore or microchannel with annular or rectangular cross section is studied under the Debye-Hückel approximation. Analytical series solutions and their asymptotic behavior for small and large non-dimensional electrokinetic widths are found for these two basic cases. The explicit and accurate solutions are particularly useful for examining various geometric/physical effects on the transient time scales and the flow rates for the transient states. The steady flow rate of the smaller channel may be disproportionately smaller than a reference channel if the electric double layer is thick, but will be in close proportion to the area ratio if the electric double layer is thin. A smaller channel compared to a reference channel has a shorter transient time scale, and the transient flow has characters very different from the steady state if the electric double layer is thin. The total transient flow rate of several smaller pores or channels may exceed largely that of a single large pore or channel with the same total cross section on the transient time scale of the smaller channels. The results have important implications on liquid transport in micropores or channels by pulse voltages or more general time-varying voltages.     

Theory of time-resolved level anticrossing experiment

The purpose of the present paper is to present a general treatment of steady state and time-resolved level-anticrossing (LAC) experiments using the generalized master equation approach in which both radiate and nonradiative decays and dephasing processes have been considered. Several models are treated to demonstrate the theoretical method. The theory is applicable to both atoms and molecules. It is shown that under appropriate conditions, as derived in this paper, quantum beats can be observed in the time-resolved LAC experiment. Numerical calculations have been performed to show the time-dependent behavior (build-up and decay) of the time-resolved LAC experiment. These quantum beats are then a direct measurement of the microscopic coupling parameters in intersystem crossing, etc. It will be shown that combining the steady state and the time-resolved LAC experiments one can determine not only the microscopic coupling parameters but also the relaxation and/or the dephasing rate constants. Hence the particular virtues of time dependent LAC experiments are seen in these model calculations.     

An efficient method for computing steady state solutions with Gillespie's direct method

Gillespie's direct method is a stochastic simulation algorithm that may be used to calculate the steady state solution of a chemically reacting system. Recently the all possible states method was introduced as a way of accelerating the convergence of the simulations. We demonstrate that while the all possible states (APS) method does reduce the number of required trajectories, it is actually much slower than the original algorithm for most problems. We introduce the elapsed time method, which reformulates the process of recording the species populations. The resulting algorithm yields the same results as the original method, but is more efficient, particularly for large models. In implementing the elapsed time method, we present robust methods for recording statistics and empirical probability distributions. We demonstrate how to use the histogram distance to estimate the error in steady state solutions.     

Studies on the interfacial charge transfer processes of nanocrystalline CdSe thin film electrodes by intensity modulated photocurrent spectroscopy

Interfacial charge transfer kinetics of the nanocrystalline CdSe thin film electrodes have been studied in sodium polysulfide solutions by intensity modulated photocurrent spectroscopy (IMPS). The interfacial direct and indirect charge transfer and recombination processes were analyzed in terms of the parameters: normalized steady state photocurrents and surface state lifetimes obtained by measuring the IMPS responses under different applied potentials and different solution concentrations. IMPS responses of polycrystalline CdSe thin film electrodes were also presented for comparison.     

Fluid flow in an idealized spiral wound membrane module

Analytical and numerical solutions for steady laminar incompressible flow in an idealized spiral wound membrane (annular) duct are developed and computed. Axial and radial velocity profiles and axial pressure distributions for injection and suction are presented. The analytical method uses a perturbation technique in wall Reynolds number (Re_w) to solve the Navier--Stokes problem. The usual assumptions of uniform injection or suction and similar velocity profiles with axial distance are invoked. For most membrane processes where Re_w < 1, the analytical solution is sufficient and the assumptions are valid after the developing entrance region. For Re_w > 1, numerical calculations reveal that similarity in the velocity profiles (assumed for the analytical development) disappear and that the radial velocity profile exhibits inflections. These results could be used for estimating pumping requirements and for effecting design changes in spiral wound modules.