Steady-state Size Distribution Research Articles

A new model for prediction of dispersoid precipitation in aluminium alloys containing zirconium and scandium

A model has been developed to predict precipitation of ternary Al 3(Sc, Zr) dispersoids in aluminium alloys containing zirconium and scandium. The model is based on the classical numerical method of Kampmann and Wagner, extended to predict precipitation of a ternary phase. The model has been applied to the precipitation of dispersoids in scandium containing AA7050. The dispersoid precipitation kinetics and number density are predicted to be sensitive to the scandium concentration, whilst the dispersoid radius is not. The dispersoids are predicted to enrich in zirconium during precipitation. Coarsening has been investigated in detail and it has been predicted that a steady-state size distribution is only reached once coarsening is well advanced. The addition of scandium is predicted to eliminate the dispersoid free zones observed in scandium free 7050, greatly increasing recrystallization resistance.

Aerosol Growth in a Steady-State, Continuous Flow Chamber: Application to Studies of Secondary Aerosol Formation

An analytical solution for the steady-state aerosol size distribution achieved in a steady-state, continuous flow chamber is derived, where particle growth is occurring by gas-to-particle conversion and particle loss is occurring by deposition to the walls of the chamber. The solution is presented in the case of two condensing species. By fitting the predicted steady-state aerosol size distribution to that measured, one may infer information about the nature of the condensing species from the calculated values of the species's molecular weights. The analytical solution is applied to three sets of experiments on secondary organic aerosol formation carried out in the U.S. Environmental Protection Agency irradiated continuous flow reactor, with parent hydrocarbons: toluene, f -pinene, and a mixture of toluene and f -pinene. Fits to the observed size distributions are illustrated by assuming two condensing products for each parent hydrocarbon; this is a highly simplified picture of secondary organic aerosol formation, which is known to involve considerably more than two condensing products. While not based on a molecular-level model of the gas-to-particle conversion process, the model does allow one to evaluate the extent to which the observed size distribution agrees with that based on a simple, two-component picture of condensation, and to study the sensitivity of those size distributions to variation of the essential properties of the condensing compounds, such as molecular weight. An inherent limitation of the steady-state experiment is that it is not possible to calculate the vapor pressures of the condensing species.

Simulation and analysis of industrial crystallization processes through multidimensional population balance equations. Part 1: a resolution algorithm based on the method of classes

In order to obtain constant solid properties with particles exhibiting a low order of symmetry, it is necessary to monitor and to control several distributed parameters characterising the crystal shape and size. A bi-dimensional population balance model was developed to simulate the time variations of two characteristic sizes of crystals. The nonlinear population balance equations were solved numerically over the bi-dimensional size domain using the so-called method of classes. An effort was made to improve usual simulation studies through the introduction of physical knowledge in the kinetic laws involved during nucleation and growth phenomena of complex organic products. The performances of the simulation algorithm were successfully assessed through the reproduction of two well-known theoretical and experimental features of ideal continuous crystallization processes: the computation of size-independent growth rates from the plot of the steady-state crystal size distribution and the possibility for MSMPR crystallizers to exhibit low-frequency oscillatory behaviours in the case of insufficient secondary nucleation.

Steady-state size distributions for collisional populations:: analytical solution with size-dependent strength

The steady-state population of bodies resulting from a collisional cascade depends on how material strength varies with size. We find a simple expression for the power-law index of the population, given a power law that describes how material strength varies with size. This result is extended to the case relevant for the asteroid belt and Kuiper belt, in which the material strength is described by 2 separate power laws—one for small bodies and one for larger bodies. We find that the power-law index of the small body population is unaffected by the strength law for the large bodies, and vice versa. Simple analytical expressions describe a wave that is superimposed on the large body population because of the transition between the two power laws describing the strength. These analytical results yield excellent agreement with a numerical simulation of collisional evolution. These results will help to interpret observations of the asteroids and KBOs, and constrain the strength properties of those objects.

Particle size distributions and viscosity of suspensions undergoing shear-induced coagulation and fragmentation

The dynamic behavior of concentrated suspensions (up to a solids volume fraction of 20%) of non-spherical particles is investigated theoretically by coupling a rheological law to a population balance model accounting for coagulation and fragmentation of the detailed particle size distribution. In these suspensions, the immobilization of matrix liquid renders the viscosity dependent on the particle aggregation state. The effect of initial solids volume concentration and shear rate on the transient behavior of particle size distribution and suspension viscosity is examined. Power law correlations for the equilibrium flow curves of aggregating suspensions are deduced and compared to experimental data. Steady-state or equilibrium particle size distributions are found to be self-preserving with respect to solids volume fraction and shear rate.

Analytical solutions for steady and unsteady state particle size distributions in FBC and CFBC boilers for non-breaking char particles

Continuous analytical solutions for the particle size distributions of char in steady and unsteady states in fluidized beds, when the inlet fuel feed is presented by monosize, lognormal, Rosin-Rammler or gamma distributions, are derived from a population balance model. The stationary size distribution is directly related to the rate of reduction of the particle size. Combustion and attrition reduce the particle size. Thus, it is possible to extract the dependence of the rate of reduction of radius (affected by a fuel’s reactivity and attrition) on radius from a measured steady-state particle size distribution. Unsteady particle size distributions are derived for impulse, step and square pulse changes in the fuel feed, when the oxygen level in the reactor is maintained constant.

Performance analysis of an AAL multiplexer with dynamic bandwidth allocation in an IP/ATM environment

In this paper, we present a discrete-time queueing model with a Markov modulated batch Bernoulli process (MMBBP) as a bursty input traffic model of internet protocol packets in order to investigate the performance of a ATM adaptation layer (AAL) multiplexer with a finite buffer and a threshold-based dynamic bandwidth allocation scheme (DBAS). As a deterministic service time, we use the segmentation processing time devoted to a cell payload in the AAL multiplexer. This service time consists of several slot times and varies in line with threshold levels of the buffer contents. In an MMBBP process, the arrivals during a slot time occur as batch Bernoulli processes with a general distribution of batch sizes varying according to the phases of a Markov chain. We obtain the loss probability and the mean delay of an arbitrary packet. We present some numerical results to show the effects of the burstiness and thresholds on the performance of the AAL multiplexer with a DBAS. Scope and purpose The model which we present takes its origin in the research of appropriate traffic management schemes for carrying Internet protocol (IP) traffic on ATM networks. Due to the connection-oriented nature of ATM and unpredictable bursty characteristics of IP traffic, we need some dynamic schemes in the AAL multiplexer of an interworking unit such as a router in order to support a bandwidth allocation algorithm for IP traffic over ATM backbone networks. Discrete-time finite queueing model has gained to describe queueing multiplexer in an ATM environment where time is slotted into one cell transmission time. Recently, many works reveal the bursty nature of packet arrivals to the IP packet multiplexer. Therefore, we focus on a discrete-time finite queueing model MMBBP/D/1/K of the AAL multiplexer with a threshold-based DBAS in order to control the bursty IP traffic transferred to ATM networks. We analyze the presented model by using the classical supplementary variable method in queueing theory. As a result, we obtain the steady-state queue size distribution at an arbitrary time point and the loss probability and the mean delay of an arbitrary packet.

Gelation of liquid bridges in spherical agglomeration

Abstract A method is introduced to increase agglomerate strength in an agitated system during spherical agglomeration. Spherical agglomeration is a process wherein the particles of dispersion are held together in aggregates by liquid bridges. The use of surfactants to improve wetting of the particulates by the bridging liquid often leads to lower interfacial tension, and liquid bridges too weak to withstand the levels of agitation required to bring the particles together. The approach suggested herein involves the addition of crosslinkable polymers to the agglomerating agent and effecting gelation as liquid bridge form. The added mechanical strength thus compensates for the reduction in agglomerate strength caused by the wetting agent. The approach is tested in a proof-of-principle practical application, viz., deinking of toner-printed paper. For the cases examined, the gelation of the liquid bridges significantly improves the agglomerate strength, as inferred from measurements of the agglomerate size distributions as functions of time for shear rates commonly encountered in a separation unit. Times required for gelation and for attainment of a steady-state aggregate size distribution are reasonable for this application.

Kinetics of Normal Grain Growth Depending on the Size Distribution of Small Grains

m þ 1. The same relationship is also derived from the analysis of the mean field model for particle growth. The validity of this relationship is confirmed from the consistence with existing grain-growth models and numerical simulations. Experimental consistence is observed for the size distribution on sectional surface. It is found that a log-normal function is not a steady-state size distribution for normal grain growth with n ¼ 2, even though it may approximate experimental size distributions. The obtained relationship is also shown to be applicable to particle coarsening by Ostwald-ripening.

A moment methodology for coagulation and breakage problems: Part 3—generalized daughter distribution functions

Population balances for simultaneous coagulation and breakage (and their analogs, e.g., polymerization and depolymerization) are employed in describing many systems including aerosols, powders and polymers, and many unit operations including reactors, crystallizers, and size reduction/enlargement equipment. The birth term for the breakage process is usually formulated in terms of a distribution of breakage products known as the daughter distribution. There are many daughter distribution forms proposed in the literature in part because these distributions are notoriously difficult to determine experimentally. Here, a generalization of these forms is developed for multi-particle breakup which has the flexibility to represent a wide variety of distribution shapes. The simplicity of the generalized expression renders the population balance equations for simultaneous coagulation and breakage accessible to analytical attack, leading to an analytical expression for the fine end of the steady-state product size distribution. This expression has potential utility in both design and analysis of experiments aimed at measuring daughter distribution parameters.

Performance prediction of loss and delay Markovian queueing model with nopassing and removable additional servers

In this investigation, the loss and delay Markovian queueing system with nopassing is proposed. The customers may balk or renege with certain probability, on finding all servers busy on their arrival. To cope up with the balking and reneging behaviour of the customers, there is provision of removable additional servers apart from permanent servers so as to provide the better grade of service at optimal cost operating conditions. The customers are classified into two classes depending upon whether they can wait or lost when all servers are busy. The customers can also be categorized into two classes from service point of view. Type A customers have zero service time whereas type B customers have exponential service time. The explicit expressions for the average number of customers in the system, the expected waiting time for both types of customers, etc., are derived by using steady-state queue size distribution. Some earlier results are deduced by setting appropriate system parameters. The system behaviour is examined with the help of numerical illustrations by varying different parameters. Scope and purpose The performance prediction of various systems in communication switching network, remote border security check post, jobs processing in computers, etc., are influenced by the customers behaviour, in particular, when nopassing constraints are prevalent. The incorporation of loss and delay phenomena is likely to bring about understanding whether the customers would like to wait in the queue or would be lost in case when all servers are busy. The provision of additional removable servers will be helpful in upgrading the service and to reduce the discouragement behaviour of the customers in such congestion situations.

A two-scale PBM for modeling turbulent flocculation in water treatment processes

A population balance model (PBM) that incorporates two scales of turbulent motion in the breakup frequency function has been presented. The breakup frequency function is designed such that particles smaller than the impeller-region Kolmogoroff microscale will erode according to a critical velocity related to the local energy dissipation rate. Particles larger than the impeller-region Kolmogoroff microscale will fracture according to a critical velocity related to the impeller tip speed. The two-scale model was found to better predict the experimental steady-state particle size distribution in 5, 28, and 560 l tank sizes and with a Rushton turbine and A310 fluid foil impellers. The two-scale PBM was also used to investigate the most appropriate scale-up law for drinking water flocculation processes. In addition, the impact of higher tank average energy dissipation rate, primary particle concentration, and coagulant concentration on the volume mean particle size with increasing tank size and different impeller types was also presented.

General Becker–Döring equations: effect of dimer interactions

The effect of dimer (two-particle) interactions on the Becker–Doring model of nucleation is investigated. Initially we consider the problem with size-independent aggregation and fragmentation coefficients and a constant monomer concentration. Either an equilibrium or a steady-state solution is found: the former when fragmentation is stronger than aggregation, the latter otherwise. By employing asymptotic techniques, the manner in which the system reaches these states is examined. The dimer interaction is found to accelerate the system towards the equilibrium solution, but has little impact on the relaxation time to the steady-state solution. In cases where aggregation is dominant, the steady-state cluster size distribution can only be determined consistently when the manner of approach to steady state is also known. In the terminology of asymptotic methods, one needs to know the first correction term in order to deduce the leading-order solution. We show how this can be derived and so at steady state we find a flux of matter to larger aggregation numbers due to monomer interactions, with a small and decreasing reverse flux due to dimer interactions. We then consider the case of constant density, that is allowing the monomer concentration to vary, and investigate the effect of a strong dimer interaction on the convergence to equilibrium. Two timescales are present and each one is investigated. We determine the intermediate meta-stable state, the final state and the timescales over which the system relaxes into these states. All results are shown to agree with numerical simulations.

CFD Study of Local Crystal Growth Rate in a Continuous Suspension Crystallizer.

The objective of the present work is to determine the local crystal growth rate in an imperfectly mixed suspension crystallizer. Experiments were carried out in pure potassium sulphate-water solution. The steady-state Crystal Size Distributions (CSDs) were measured at four different heights of the crystallizer using an in-situ particle image analyzer. Experimentally obtained CSD was used to determine the overall growth rate. Then, Computational Fluid Dynamics (CFD) was used to calculate the Particle Size Distributions (PSDs) of the same kind of suspension which was used in the experiments. The computational study was focused to simulate the mass transfer coefficient between particles and solution, and supersaturation. These quantities were determined from hydrodynamics of suspension in a crystallizer. The multi-fluid model was applied to describe suspension flow in the crystallizer. One continuous phase (water) and six dispersed phases, which were taken to be particles in six size classes, were used to approximate PSD. The crystal growth rate model was based on the diffusion process and was implemented into a CFD environment. The local growth rates were calculated from the slip velocity or from energy dissipation rate at the local temperature and supersaturation.

Unsteady and steady-state particle size distributions in batch and continuous fluidized bed granulation systems

A two-dimensional balance model of populations distributed by property was developed using the ordinates of particle diameter and particle density. This model makes it possible to calculate the spectrums of granulation changing over time in the fluidized bed during batch operation and in addition to calculate the time-dependent size distribution of the product granulate for continuous operation. A physics-based model was devised for the submechanism of the growth of the granulates. On the one hand, the population balance model is divided into the particle balance around the granulator and, on the other hand, when granulation is continuous, particles are constantly removed from the fluidized bed. To this end, a separator model was used that subdivides the extractor device into two areas of balancing in accordance with the geometry of the apparatus and the properties of the particle. In the case of continuous operation, the results when the product is classified using a classifier tube inside the fluidized bed are discussed. The outcome is the influence of the various particle flows on the particle diameter and the product mass withdrawn. In particular, the problematic nature of the start-up is dealt with, i.e. reaching stable points of steady operation. The simulation results are validated using results from continuous tests on a semi-industrial plant.

A preliminary numerical investigation of agglomeration in a rotary drum

In this paper, the discrete element method (DEM) is employed to analyse the wet granulation process. It is essentially a computer simulation of various types of interactions of particles in a rotating drum that eventually results in tracking of trajectories of each particle and particle agglomerates. Non-linear spring-dashpot type contact models are used to monitor particle–particle and particle–wall interactions. The contact between the particles in the presence of a binder is modelled by the well-known JKR theory. To establish the potential of this numerical approach, a 10-cm rotating drum is simulated with a polydispersed system of particles of top size 1 mm. A scraper is suitably located within the drum to restrict the motion of the adhered particles along the mill shell. A spray zone is located within the drum such that the particles receive the binder dosage only within this spray zone. A drying time is also specified beyond which the particles no longer adhere together. Implementation of these ideas within the computer program makes the simulation more realistic, which in turn allows prediction of the steady-state size distribution of the agglomerates.

Constant effort and constant quota fishing policies with cut-offs in a random environment

ABSTRACT. Consider a population subjected to constant effort or constant quota fishing with a generaldensity-dependence population growth function (because that function is poorly known). Consider environmental random fluctuations that either affect an intrinsic growth parameter or birth/death rates, thus resulting in two stochastic differential equations models. From previous results of ours, we obtain conditions for non-extinction and for existence of a population size stationary density. Constant quota (which always leads to extinction in random environments) and constant effort policies are studied; they are hard to implement for extreme population sizes. Introducing cut-offs circumvents these drawbacks. In a deterministic environment, for a wide range of values, cutting-off does not affect the steady-state yield. This is not so in a random environment and we will give expressions showing how steady-state average yield and population size distribution vary as functions of cut-off choices. We illustrate these general results with function plots for the particular case of logistic growth.

Identical Feed Drops in a Flow Vessel May not Always Be Desirable

In this paper, the effect of feed size and daughter drop size distributions on the steady-state drop size distribution in a continuous flow vessel is discussed. For identical sizes of feed drops, a new criterion has been proposed to ensure the complete breakage of the feed drops giving rise to smooth continuous drop size distributions. This has been predicted on the basis of two competing time scales, namely, breakage and residence times.

Steady-state queue size distribution of discrete-time PH/Geo/1 retrial queues

This paper studies discrete-time single server retrial queues with phase-type interarrival times and geometric service times. A matrix analytical method is applied to derive the analytical solution for the joint steady-state distribution of arrival phases and queue sizes of the system. The necessary and sufficient condition for system stability is also determined. Based on the special form of the matrix analytic solution of the system, upper and lower bounds for the joint steady-state distribution of arrival phases and queue sizes are developed. It is shown that the errors between these upper and lower bounds can be made as small as desired. Finally, an efficient and numerically stable algorithm for computing the joint steady-state probabilities of arrival phases and queue sizes is presented.

Ostwald ripening: an approach with dynamical systems

This approach assumes three functions independently acting on a set of microparticles. The first one, w1, concerns re-distribution of mass to decrease the surface energy. The second one, w2, concerns re-distribution of mass to increase the entropy of the microparticle set. The third one, w3, is a further re-distribution of mass that vanishes a microparticle. Once vanished, its mass is distributed among its neighbors. w1 and w3 release energy, whereas w2 absorbs energy. Part of the energy released should be available to sustain w2. The action frequency of w1, w2, and w3, the amount of mass exchanged in each iteraction, the fraction of released energy available to sustain w2, and the size of a vanishing microparticle can be varied. As the dynamical system formed by w1, w2, and w3 act on an initial microparticle set, it is observed an evolution resembling the Ostwald ripening concerning steady-state size distribution and microparticle growth.