R. Dennis Vigil

Quadrature method of moments for aggregation-breakage processes.

Investigation of particulate systems often requires the solution of a population balance, which is a continuity statement written in terms of the number density function. In turn, the number density function is defined in terms of an internal coordinate (e.g., particle length, particle volume) and it generates integral and derivative terms. Different methods exist for numerically solving the population balance equation. For many processes of industrial significance, due to the strong coupling between particle interactions and fluid dynamics, the population balance must be solved as part of a computational fluid dynamics (CFD) simulation. Such an approach requires the addition of a large number of scalars and the associated transport equations. This increases the CPU time required for the simulation, and thus it is clear that it is very important to use as few scalars as possible. In this work the quadrature method of moments (QMOM) is used. The QMOM has already been validated for crystal growth and aggregation; here the method is extended to include breakage. QMOM performance is tested for 10 different cases in which the competition between aggregation and breakage leads to asymptotic solutions.

Implementation of the quadrature method of moments in CFD codes for aggregation-breakage problems

In this work the quadrature method of moments (QMOM) is implemented in a commercial computational fluid dynamics (CFD) code (FLUENT) for modeling simultaneous aggregation and breakage. Turbulent and Brownian aggregation kernels are considered in combination with different breakage kernels (power law and exponential) and various daughter distribution functions (symmetric, erosion, uniform). CFD predictions are compared with experimental data taken from other work in the literature and conclusions about CPU time required for the simulations and the advantages of this approach are drawn.

Random sequential adsorption of unoriented rectangles onto a plane

Random sequential adsorption of nonoverlapping rectangles of arbitrary orientation onto a continuous plane was investigated by computer simulation. The approach to the jamming limit was found to obey Feder’s law for a wide range of rectangle aspect ratios. The coverage fraction at the jamming limit was found to depend upon the aspect ratio of the adsorbed rectangles, with a maximum in the jamming coverage occurring at aspect ratios ≊2.

On the stability of coagulation—fragmentation population balances

Abstract When coagulation and fragmentation both occur in a system, the competition between these processes may lead to a steady-state size distribution. We consider some specific moment solutions to a generalized coagulation-fragmentation population balance equation (in which multiple breakup is allowed) in order to determine when it is may be possible for such steady states to exist. Steady states occur for systems with homogeneous rate kernels of order β (fragmentation) and λ (coagulation) that satisfy β − λ + 1 > 0. Finally, we discuss the applicability of scaling to this generalized coagulation-fragmentation population balance.

Turing patterns in a simple gel reactor

We introduce a very simple open two-dimensional gel reactor for studying chemical patterns that arise in reaction-diffusion systems. The reactor consists of a thin disk-shaped layer of polyvinyl alcohol gel with one face in direct contact with a well-stirred flow reactor containing reactants of the chlorite-iodide-malonic acid system; the other face of the gel is in contact with a piece of transparent plexiglass. We have used this reactor to study the transition from a uniform state to a hexagonal pattern; for other concentrations, striped patterns were observed. The wavelength of these structures is greater than the thickness of the gel layer, which indicates that the patterns are two-dimensional (a single layer). This reactor provides a promising new tool for studying chemical patterns since the diffusion time of reactants into and out of the gel can be made small compared to the total residence time in the stirred flow reactor. This feature facilitates the comparison between theory and experiment since the chemical concentrations leading to pattern formation are close to those in the stirred flow reactor and hence can be directly measured. Pattern formation in reaction-diffusi on systems has attracted the interest of experimentalists and theorists alike during the last few decades. Until recently experimental work centered on spatio-temporal patterns in closed systems. Typically, a thin layer of reactive solution was placed in a petri dish and patterns such as spiral waves and target patterns were observed. However, since these appeared in a closed system, they were only short-lived transients. Several open chemical reactor designs have been introduced in order to allow the study of chemical patterns at conditions maintained far from equilibrium. Most of these designs employ an inert gel reaction medium in which convection is suppressed; such a reactor has been called a CFUR (continuous flow unstirred reactor), in analogy with the acronym CSTR used for continuous flow stirred tank reactors. In 1987 Noszticzius et al. [1] used a CFUR with an annular gel that was fed at the inner and outer rims to study traveling waves in the Belousov-Zhaboti nskii (BZ) reaction. Another CFUR design consists of a thin gel layer that is sandwiched between two thin porous glass plates [2]. The face of each porous glass plate opposite to the gel is in contact with a stirred

Axial dispersion during low Reynolds number Taylor-Couette flow: intra-vortex mixing effects

Abstract Tracer experiments were employed to study axial mass transport in a Taylor-Couette device with a radius ratio of 0.875. The inner cylinder was rotated (with the outer cylinder fixed) at circumferential Reynolds numbers ranging from 1.05 to 7.50 times the critical value for the onset of laminar Taylor vortex flow. Apparent axial dispersion coefficients were found by fitting a three parameter model to tracer-response data. The model consisted of a network of CSTRs (each with an associated exchange volume) connected in series. The CSTRs corresponded to the well mixed outer layers of the Taylor vortices; the exchange volumes represented poorly mixed vortex cores. For 1.05 ≤ Re Re c , the apparent axial dispersion coefficient increased nonmonotonically with increasing Reynolds numbers. At inner cylinder rotation rates Re Re c > 5.0 , the model recovered the commonly used discrete approximation of the one-dimensional diffusion equation.

Kinetics of random sequential adsorption of rectangles and line segments

Random sequential adsorption of nonoverlapping rectangles of arbitrary orientation onto a continuous plane was investigated by computer simulation. It is shown that the approach to the jamming limit of rectangles with various aspect ratios obeys the scaling law θJ−θ(τ)∼τ−1/3. Furthermore, the adsorption kinetics of rectangles approach those of line segments as the rectangle aspect ratio becomes large.

Simulation of photosynthetically active radiation distribution in algal photobioreactors using a multidimensional spectral radiation model.

A numerical method for simulating the spectral light distribution in algal photobioreactors is developed by adapting the discrete ordinate method for solving the radiative transport equation. The technique, which was developed for two and three spatial dimensions, provides a detailed accounting for light absorption and scattering by algae in the culture medium. In particular, the optical properties of the algal cells and the radiative properties of the turbid culture medium were calculated using a method based on Mie theory and that makes use of information concerning algal pigmentation, shape, and size distribution. The model was validated using a small cylindrical bioreactor, and subsequently simulations were carried out for an annular photobioreactor configuration. It is shown that even in this relatively simple geometry, nontrivial photon flux distributions arise that cannot be predicted by one-dimensional models.

Enhanced algal growth rate in a Taylor vortex reactor

The rate of production of algal biomass in optically dense photobioreactors depends crucially on the temporal light exposure of microorganisms, which in turn is determined by fluid flow patterns and the quantity and spatial distribution of photosynthetically active radiation. In this report it is demonstrated that highly organized and robust toroidal flow structures known as Taylor vortices cause significant increases in the rate of biomass production, efficiency of light utilization, and CO2 uptake, and these effects become more pronounced at higher Reynolds numbers. In light of these findings and previously reported experiments using Taylor vortex flow to culture algae, it is argued that the flashing light effect, rather than mass transport effects, is responsible for the observed increases in the rate of photosynthesis. Biotechnol. Bioeng. 2013; 110: 2140–2149.

On the scaling theory of two-component aggregation

The behavior of a closed system undergoing irreversible aggregation and containing two distinct monomer types is considered. For large cluster sizes and at large times, the size-composition distribution is described by a two-parameter scaling law, which is valid for homogeneous non-gelling kernels. This scaling law is composed of the product of a normal distribution in particle composition and the one-parameter scaling function for the corresponding homogeneous aggregation problem. It is shown that this scaling law also applies to a kernel that depends linearly (and not simply as the total particle size) upon particle composition.