Distribution Of Particle Properties Research Articles

Deep bed filtration and formation damage by particles with distributed properties

Current models for deep bed filtration describe particles with uniform properties. Yet, the sizes, densities, and mineral composition of particles vary significantly in the same injection well. The aim of this work is to provide an effective mathematical model for water injection of particles with distributed properties and formation damage prediction. We average the set of traditional population balance equations for single-property particles and obtain one upscaled equation. The upscaled equation for particle retention rate contains a non-linear function of suspended concentration, which we call the 'suspension function'. We derive analytical solutions for the upscaled equation for linear (coreflood) and radial (well injectivity) flows. Then we treat lab coreflood data to determine the model suspension function and provide a model for well injectivity prediction. The retention profile for the flow of uniform particles has an exponential form. Frequently reported in the literature, hyper-exponential forms have been hypothetically explained by multiple particle properties. The inverse solution allows revealing the individual filtration coefficients for binary mixtures from total breakthrough concentrations during coreflood. Treatment of the data from lab experiments reveals individual filtration coefficients that belong to common intervals. For the first time, deep bed filtration of particles with distributed properties is upscaled and presented using a single equation that reflects the particle property distribution. This equation provides an effective mathematical model for tuning lab coreflood data, determines the model function, and uses it for injectivity decline prediction.

External field-driven property localization in liquids of responsive macromolecules.

We explore theoretically the effects of external potentials on the spatial distribution of particle properties in a liquid of explicitly responsive macromolecules. In particular, we focus on the bistable particle size as a coarse-grained internal degree of freedom (DoF, or "property"), σ, that moves in a bimodal energy landscape, in order to model the response of a state-switching (big-to-small) macromolecular liquid to external stimuli. We employ a mean-field density functional theory (DFT) that provides the full inhomogeneous equilibrium distributions of a one-component model system of responsive colloids (RCs) interacting with a Gaussian pair potential. For systems confined between two parallel hard walls, we observe and rationalize a significant localization of the big particle state close to the walls, with pressures described by an exact RC wall theorem. Application of more complex external potentials, such as linear (gravitational), osmotic, and Hamaker potentials, promotes even stronger particle size segregation, in which macromolecules of different size are localized in different spatial regions. Importantly, we demonstrate how the degree of responsiveness of the particle size and its coupling to the external potential tune the position-dependent size distribution. The DFT predictions are corroborated by Brownian dynamics simulations. Our study highlights the fact that particle responsiveness can be used to localize liquid properties and therefore helps to control the property- and position-dependent function of macromolecules, e.g., in biomedical applications.

An LES-PBE-PDF approach for modeling particle formation in turbulent reacting flows

Many chemical and environmental processes involve the formation of a polydispersed particulate phase in a turbulent carrier flow. Frequently, the immersed particles are characterized by an intrinsic property such as the particle size, and the distribution of this property across a sample population is taken as an indicator for the quality of the particulate product or its environmental impact. In the present article, we propose a comprehensive model and an efficient numerical solution scheme for predicting the evolution of the property distribution associated with a polydispersed particulate phase forming in a turbulent reacting flow. Here, the particulate phase is described in terms of the particle number density whose evolution in both physical and particle property space is governed by the population balance equation (PBE). Based on the concept of large eddy simulation (LES), we augment the existing LES-transported probability density function (PDF) approach for fluid phase scalars by the particle number density and obtain a modeled evolution equation for the filtered PDF associated with the instantaneous fluid composition and particle property distribution. This LES-PBE-PDF approach allows us to predict the LES-filtered fluid composition and particle property distribution at each spatial location and point in time without any restriction on the chemical or particle formation kinetics. In view of a numerical solution, we apply the method of Eulerian stochastic fields, invoking an explicit adaptive grid technique in order to discretize the stochastic field equation for the number density in particle property space. In this way, sharp moving features of the particle property distribution can be accurately resolved at a significantly reduced computational cost. As a test case, we consider the condensation of an aerosol in a developed turbulent mixing layer. Our investigation not only demonstrates the predictive capabilities of the LES-PBE-PDF model but also indicates the computational efficiency of the numerical solution scheme.

A geometrical texture model for multi-mineral liberation prediction

We present a novel methodology for numerically reconstructing a three-dimensional (3D) texture for simulating multi-mineral rock and estimating the particle properties and liberation characteristics of its breakage products. A stochastic geometrical model has been developed to describe the intact structure of ores at any scale, potentially containing several mineral types, each with unique grade and grain size distribution parameters. Additional controls over the intact texture include the ability to vary the proportion of grain inclusions within grains of a different mineral phase and by varying the degree of spatial clustering of grains within the texture. Repeated random 2D and 3D sectioning of the 3D texture to simulate particle production and to calculate their compositions is used to generate outputs comparable to reports from mineral liberation analysis systems; including density distributions, liberation distributions, and particle class distributions. These outputs enable texture parameters to be fitted to corresponding measured liberation data and is demonstrated for a four-mineral sulphide ore, yielding a realistic distribution of particle properties, essential for accurately modelling particle separations.

An explicit adaptive grid approach for the numerical solution of the population balance equation

Many engineering applications, such as the formation of soot in hydrocarbon combustion or the precipitation of nanoparticles from aqueous solutions, encompass a polydispersed particulate phase that is immersed in a reacting carrier flow. From a Eulerian perspective, the evolution of the particulate phase both in physical and in particle property space can be described by the population balance equation (PBE). In this article, we present an explicit solution-adaptive numerical scheme for discretizing the spatially inhomogeneous and unsteady PBE along a one-dimensional particle property space. This scheme is based on a space and time dependent coordinate transformation which redistributes resolution in particle property space according to the shapes of recent solutions for the particle property distribution. In particular, the coordinate transformation is marched in time explicitly. In comparison to many existing moving or adaptive grid approaches, this has the advantage that the semi-discrete PBE does not need to be solved in conjunction with an additional system of equations governing the movement of nodes in particle property space.By design, our adaptive grid technique is able to accurately capture sharp features such as peaks or near-discontinuities, while maintaining the semi-discrete system size and adhering to a uniform fixed grid discretization in transformed particle property space. This is particularly advantageous if the PBE is combined with a spatially and temporally fully resolved flow model and a standard Eulerian solution scheme is applied in physical space. In order to accommodate localized source terms and to control the grid stretching, we develop a robust scheme for modifying the coordinate transformation such that constraints on the resolution in physical particle property space are obeyed.As an example, we consider the precipitation of BaSO4 particles from an aqueous solution in a plug flow reactor. Our findings demonstrate that for a given accuracy of the numerical solution the explicit adaptive grid technique requires over an order of magnitude fewer grid points than a comparable fixed grid discretization scheme.

Copula-based approximation of particle breakage as link between DEM and PBM

Abstract In process engineering, the breakage behavior of particles is needed for the modeling and optimization of comminution processes. A popular tool to describe (dynamic) processes is population balance modeling (PBM), which captures the statistical distribution of particle properties and their evolution over time. It has been suggested previously to split up the description of breakage into a machine function (modeling of loading conditions) and a material function (modeling of particle response to mechanical stress). Based on this idea, we present a mathematical formulation of machine and material functions and a general approach to compute them. Both functions are modeled using multivariate probability distributions, where in particular so-called copulas are helpful. These can be fitted to data obtained by the discrete element method (DEM). In this paper, we describe the proposed copula-based breakage model, and we construct such a model for an artificial dataset that allows to evaluate the prediction quality.

Experimental investigation and correlation of the Bodenstein number in horizontal fluidized beds with internal baffles

In many branches of industry, there is a high demand for the formation or the conditioning of particulate products. In many cases the fluidized bed technology is applied due to its robustness and intensity of heat and mass transfer. Furthermore, many producers look for a continuous process for economic reasons instead of a batch process with high labor and dead-time. These continuous processes, however, hold the disadvantage of a particle residence time distribution (RTD) and therefore a distribution of particle properties. To minimize the spread of the RTD vertical baffles can be installed into the system, causing an influence on the particle transport behavior. To investigate this influence several experiments have been conducted, varying relevant parameters, such as superficial gas velocity, particle diameter, the kind and placement as well as the number of internal baffles, the height of the exit weir and the apparatus itself (pilot and lab-scale plant). From the experiments different influences of the varied parameters on the Bodenstein number, as a measure of convective to dispersive transport, are shown. Based on these trends an empirical correlation for the Bodenstein number is developed and compared to the experimentally determined Bodenstein numbers. Furthermore a correlation for apparent bed porosity is proposed, which uses the set exit weir height instead of the unknown bed height to predict the particle holdup mass and thus the particle mean residence time. It is shown, that both correlations fit well with the experimentally determined values. Finally a prediction of the RTD is shown from the correlated Bodenstein number and the mean particle residence time.

Adaptive discrepancy based control of particulate processes

This contribution is concerned with a relatively new approach to particulate process control. The main idea is to use of a generalized distance measure, the discrepancy, in order to describe deviations from a desired particle property distribution. Applying, the associated stability theory, i.e. stability theory with respect to two discrepancies, a stabilizing control law can be derived. One of the main advantages of the proposed discrepancy based control method is that no model reduction is required. In this contribution an adaptive extension of the discrepancy based control is proposed in order to control a crystallization process in the presence of parameter uncertainties.

Model-based control of particle properties in fluidised bed spray granulation

In this contribution different control approaches, ranging from standard linear control to non-linear model predictive control, are applied to fluidised bed spray granulation processes with internal and external product classification. These processes exhibit sustained non-linear oscillations in the particle property distribution, i.e. size distribution, that have negative influence on steady-state operation, for example a constant product mass flow with constant properties. The controllers are applied to stabilise these open-loop unstable steady-states.

Spatial distribution of particle composition and microbial activity in benthic boundary layer (BBL) of the Northeast Water Polynya

Heterotrophic activity and related measures of pelagic microorganisms were studied in the Northeast Water Polynya, (NEWP 77–82°N, 10–15°W) in 1993 during a FS Polarstern cruise. A bottom water sampler (BWS), which enabled collection of distinct 12 1 water samples at variable heights above the seabed, was used to study the distribution of various biological parameters with high vertical resolution in the benthic boundary layer (BBL). Samples were taken in the water column from varying depths in the intermediate water column (IWC) down to 5 m above bottom using a CTD as well as at 40, 20, 12 and 7 cm above the sea bed using the bottom water sampler. Total suspended matter, particulate organic carbon and nitrogen (POC, PON), chlorophyll a equivalents (chl. a equiv.), bacterial abundance and size distribution and microbial activity were determined in each of the samples. Most of the deep samples (IWC and BBL, maximum water depth 479 m) came from warmer Atlantic influenced water that fills a circular trough system on the northeast Greenland shelf. In general the activity of the microbial community, cell-specific activity and concentrations of POC and chl. a equiv. were significantly higher in the BBL compared to the IWC. Microbial activity appeared to be independent of the quantity of available POC, indicating the nutritional importance of dissolved organic matter for microorganisms. The spatial distribution of particle composition and microbial activity in the BBL was strongly determined by the anticyclonic current pattern and the spatial pattern of primary production in the polynya. Close coupling of biological processes in the surface waters and particle composition and microbial activity in the BBL were found below areas of fluctuating ice cover and above 260 m water depth in the northeastern Westwind Trough. Highest microbial activity occurred in the heavily ice covered Belgica Trough, although only remnants of freshly produced organic matter were present. In the open polynya near-bottom particle properties were mainly determined by lateral advection. The benthic boundary layer is shown to represent a distinct environment, with small scale vertical distribution of particle properties and stimulated microbial activity, indicating that rapid modification of organic matter occurs in the BBL before its final incorporation into the sediment.

On the distribution of physical and chemical particle properties in the atmospheric aerosol

Consideration of sources and growth dynamics of aerosols has led to the conclusion that there may be a distribution or variation of chemical composition and physical structure among atmospheric aerosol particles as a function of size, and within a narrow size range as well. A mathematical representation of these particle properties in terms of an additional dimension to the number size distribution is described. Examples of the relevance of this aspect of aerosol characterization for physical and chemical processes in the atmosphere are discussed. A review of the available techniques shows that several methods are available which can and have provided quantitative results on the distribution of particle properties. Examples of data from the literature have been selected and are presented as three-dimensional distributions illustrating the wide range of particle properties which may exist in narrow size intervals. An evaluation of these results reiterates the value of taking the distribution of particle properties in the atmosphere into account for sampling and modeling purposes.