Design of a Semi-analytical Method to Describe Plasma-Wall Interactions

Electric probes are objects immersed in the plasma with sharp boundaries which collect of emit charged particles. Consequently, the nearby plasma evolves under abrupt imposed and/or naturally emerging conditions. There could be localized currents, different time scales for plasma species evolution, charge separation and absorbing-emitting walls. The traditional numerical schemes based on differences often transform these disparate boundary conditions into computational singularities. This is the case of models using advection-diffusion differential equations with source-sink terms (also called Fokker-Planck equations). These equations are used in both, fluid and kinetic descriptions, to obtain the distribution functions or the density for each plasma species close to the boundaries. We present a resolution method grounded on an integral advancing scheme by using approximate Green's functions, also called short-time propagators. All the integrals, as a path integration process, are numerically calculated, what states a robust grid-free computational integral method, which is unconditionally stable for any time step. Hence, the sharp boundary conditions, as the current emission from a wall, can be treated during the short-time regime providing solutions that works as if they were known for each time step analytically. The form of the propagator (typically a multivariate Gaussian) is not unique and it can be adjusted during the advancing scheme to preserve the conserved quantities of the problem. The effects of the electric or magnetic fields can be incorporated into the iterative algorithm. The method allows smooth transitions of the evolving solutions even when abrupt discontinuities are present. In this work it is proposed a procedure to incorporate, for the very first time, the boundary conditions in the numerical integral scheme. This numerical scheme is applied to model the plasma bulk interaction with a charge-emitting electrode, dealing with fluid diffusion equations combined with Poisson equation self-consistently. It has been checked the stability of this computational method under any number of iterations, even for advancing in time electrons and ions having different time scales. This work establishes the basis to deal in future work with problems related to plasma thrusters or emissive probes in electromagnetic fields.

From point defects to plate tectonic faults

Understanding and explaining emergent constitutive laws in the multi-scale evolution from point defects, dislocations and two-dimensional defects to plate tectonic scales is an arduous challenge in condensed matter physics. The Earth appears to be the only planet known to have developed stable plate tectonics as a means to get rid of its heat. The emergence of plate tectonics out of mantle convection appears to rely intrinsically on the capacity to form extremely weak faults in the top 100 km of the planet. These faults have a memory of at least several hundred millions of years, yet they appear to rely on the effects of water on line defects. This important phenomenon was first discovered in laboratory and dubbed ``hydrolytic weakening''. At the large scale it explains cycles of co-located resurgence of plate generation and consumption (the Wilson cycle), but the exact physics underlying the process itself and the enormous spanning of scales still remains unclear. We present an attempt to use the multi-scale non-equilibrium thermodynamic energy evolution inside the deforming lithosphere to move phenomenological laws to laws derived from basic scaling quantities, develop self-consistent weakening laws at lithospheric scale and give a fully coupled deformation-weakening constitutive framework. At meso- to plate scale we encounter in a stepwise manner three basic domains governed by the diffusion/reaction time scales of grain growth, thermal diffusion and finally water mobility through point defects in the crystalline lattice. The latter process governs the planetary scale and controls the stability of its heat transfer mode.

Simulation process of biodiesel production over heterogeneous catalysts

Biodiesel production is a very promising area due to the relevance that it is an environmental-friendly diesel fuel alternative to fossil fuel derived diesel fuels. Nowadays, most industrial applications of biodiesel production are performed by the transesterification of renewable biological sources based on homogeneous acid catalysts, which requires downstream neutralization and separation leading to a series of technical and environmental problems. However, heterogeneous catalyst can solve these issues, and be used as a better alternative for biodiesel production. Thus, a heuristic diffusion-reaction kinetic model has been established to simulate the transesterification of alkyl ester with methanol over a series of heterogeneous Cs-doped heteropolyacid catalysts. The novelty of this framework lies in detailed modeling of surface reacting kinetic phenomena and integrating that with particle-level transport phenomena all the way through to process design and optimisation, which has been done for biodiesel production process for the first time. This multi-disciplinary research combining chemistry, chemical engineering and process integration offers better insights into catalyst design and process intensification for the industrial application of Cs-doped heteropolyacid catalysts for biodiesel production. A case study of the transesterification of tributyrin with methanol has been demonstrated to establish the effectiveness of this methodology.

Simulation process of biodiesel production over heterogeneous catalysts

Biodiesel production is a very promising area due to the relevance that it is an environmental-friendly diesel fuel alternative to fossil fuel derived diesel fuels. Nowadays, most industrial applications of biodiesel production are performed by the transesterification of renewable biological sources based on homogeneous acid catalysts, which requires downstream neutralization and separation leading to a series of technical and environmental problems. However, heterogeneous catalyst can solve these issues, and be used as a better alternative for biodiesel production. Thus, a heuristic diffusion-reaction kinetic model has been established to simulate the transesterification of alkyl ester with methanol over a series of heterogeneous Cs-doped heteropolyacid catalysts. The novelty of this framework lies in detailed modeling of surface reacting kinetic phenomena and integrating that with particle-level transport phenomena all the way through to process design and optimisation, which has been done for biodiesel production process for the first time. This multi-disciplinary research combining chemistry, chemical engineering and process integration offers better insights into catalyst design and process intensification for the industrial application of Cs-doped heteropolyacid catalysts for biodiesel production. A case study of the transesterification of tributyrin with methanol has been demonstrated to establish the effectiveness of this methodology.

Monte Carlo Study of Particle Transport Problem in Air Pollution

2002 Mathematics Subject Classification: 65C05.

Seawater carbonate chemistry, chlorophyll a and phosphate during experiments with Trichodesmium erythraeum IMS101 (CCMP1985), 2010

In this laboratory study, we monitored the buildup of biomass and concomitant shift in seawater carbonate chemistry over the course of a Trichodesmium bloom under different phosphorus (P) availability. During exponential growth, dissolved inorganic carbon (DIC) decreased, while pH increased until maximum cell densities were reached. Once P became depleted, DIC decreased even further and total alkalinity (TA) dropped, accompanied by precipitation of aragonite. Under P-replete conditions, DIC increased and TA remained constant in the postbloom phase. A diffusion-reaction model was employed to estimate changes in carbonate chemistry of the diffusive boundary layer. This study demonstrates that Trichodesmium can induce precipitation of aragonite from seawater and further provides possible explanations about underlying mechanisms.

Variationelle Zeitdiskretisierungen höherer Ordnung der Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängigen Gebieten

Wir betrachten zeitabhängige Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängi- gen Gebieten, wobei die Bewegung des Gebietsrandes bekannt ist. Die zeitliche Entwicklung des Gebietes wird durch die ALE-Formulierung behandelt, die die Nachteile der klassischen Euler- und Lagrange-Betrachtungsweisen behebt. Die Position des Randes und seine Geschwindigkeit werden dabei so in das Gebietsinnere fortgesetzt, dass starke Gitterdeformationen verhindert werden. Als Zeitdiskretisierungen höherer Ordnung werden stetige Galerkin-Petrov-Verfahren (cGP) und unstetige Galerkin-Verfahren (dG) auf Probleme in zeitabhängigen Gebieten angewendet. Weiterhin werden das C 1 -stetige Galerkin-Petrov-Verfahren und das C 0 -stetige Galerkin- Verfahren vorgestellt. Deren Lösungen lassen sich auch in zeitabhängigen Gebieten durch ein einfaches einheitliches Postprocessing aus der Lösung des cGP-Problems bzw. dG-Problems erhalten. Für Problemstellungen in festen Gebieten und mit zeitlich konstanten Konvektions- und Reaktionstermen werden Stabilitätsresultate sowie optimale Fehlerabschätzungen für die nachbereiteten Lösungen der cGP-Verfahren und der dG-Verfahren angegeben. Für zeitabhängige Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängigen Gebieten präsentieren wir konservative und nicht-konservative Formulierungen, wobei eine besondere Aufmerksamkeit der Behandlung der Zeitableitung und der Gittergeschwindigkeit gilt. Stabilität und optimale Fehlerschätzungen für die in der Zeit semi-diskretisierten konservativen und nicht-konservativen Formulierungen werden vorgestellt. Abschließend wird das volldiskretisierte Problem betrachtet, wobei eine Finite-Elemente-Methode zur Ortsdiskretisierung der Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängigen Gebieten im ALE-Rahmen einbezogen wurde. Darüber hinaus wird eine lokale Projektionsstabilisierung (LPS) eingesetzt, um der Konvektionsdominanz Rechnung zu tragen. Weiterhin wird numerisch untersucht, wie sich die Approximation der Gebietsgeschwindigkeit auf die Genauigkeit der Zeitdiskretisierungsverfahren auswirkt.

Um problema de dispersão de poluentes em rios e canais por meio do método de separação de variáveis

Neste trabalho obtém-se uma solução analítica para a equação de advecção-difusão aplicada a problemas de dispersão de poluentes em rios e canais. Para tanto, consideram-se os casos unidimensionais e bidimensionais em regime transiente com coeficientes de difusividade e velocidades constantes. A abordagem utilizada para a resolução deste problema é o método de Separação de Variáveis. Os modelos resolvidos foram simulados utilizando o MatLab. Apresentam-se os resultados das simulações numéricas em formato gráfico. Os resultados de algumas simulações numéricas existem na literatura e puderam ser comparados. O modelo proposto mostrou-se coerente em relação aos dados considerados. Para outras simulações não foram encontrados comparativos na literatura, todavia esses problemas governados por equações diferenciais parciais, mesmo lineares, não são de fácil solução analítica. Sendo que, muitas delas representam importantes problemas de matemática e física, com diversas aplicações na engenharia. Dessa forma, é de grande importância a disponibilidade de um maior número de problemas-teste para avaliação de desempenho de formulações numéricas, cada vez mais eficazes, já que soluções analíticas oferecem uma base mais segura para comparação de resultados.

Canards in advection-reaction-diffusion systems in one spatial dimension

This thesis contains a mathematical investigation of the existence of travelling wave solutions to singularly perturbed advection-reaction-diffusion models of biological processes. An enhanced mathematical understanding of these solutions and models is gained via the identification of canards (special solutions of fast/slow dynamical systems) and their role in the existence of the most biologically relevant, shock-like solutions. The analysis focuses on two existing models. A new proof of existence of a whole family of travelling waves is provided for a model describing malignant tumour invasion, while new solutions are identified for a model describing wound healing angiogenesis.

A bistable reaction-diffusion system in a stretching flow

We examine the evolution of a bistable reaction in a one-dimensional stretching flow, as a model for chaotic advection. We derive two reduced systems of ordinary differential equations (ODEs) for the dynamics of the governing advection-reaction-diffusion partial differential equations (PDE), for pulse-like and for plateau-like solutions, based on a non-perturbative approach. This reduction allows us to study the dynamics in two cases: first, close to a saddle-node bifurcation at which a pair of nontrivial steady states are born as the dimensionless reaction rate (Damkoehler number) is increased, and, second, for large Damkoehler number, far away from the bifurcation. The main aim is to investigate the initial-value problem and to determine when an initial condition subject to chaotic stirring will decay to zero and when it will give rise to a nonzero final state. Comparisons with full PDE simulations show that the reduced pulse model accurately predicts the threshold amplitude for a pulse initial condition to give rise to a nontrivial final steady state, and that the reduced plateau model gives an accurate picture of the dynamics of the system at large Damkoehler number. Published in Physica D (2006)

Electron diffusion and back reaction in dye-sensitized solar cells : the effect of nonlinear recombination kinetics

The electron collection efficiency in dye-sensitized solar cells (DSCs) is usually related to the electron diffusion length, L = (Dτ)^1/2, where D is the diffusion coefficient of mobile electrons and τ is their lifetime, which is determined by electron transfer to the redox electrolyte. Analysis of incident photon-to-current efficiency (IPCE) spectra for front and rear illumination consistently gives smaller values of L than those derived from small amplitude methods. We show that the IPCE analysis is incorrect if recombination is not first-order in free electron concentration, and we demonstrate that the intensity dependence of the apparent L derived by first-order analysis of IPCE measurements and the voltage dependence of L derived from perturbation experiments can be fitted using the same reaction order, γ ≈ 0.8. The new analysis presented in this letter resolves the controversy over why L values derived from small amplitude methods are larger than those obtained from IPCE data.

Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions

We consider time-space fractional reaction diffusion equations in two dimensions. This equation is obtained from the standard reaction diffusion equation by replacing the first order time derivative with the Caputo fractional derivative, and the second order space derivatives with the fractional Laplacian. Using the matrix transfer technique proposed by Ilic, Liu, Turner and Anh [Fract. Calc. Appl. Anal., 9:333--349, 2006] and the numerical solution strategy used by Yang, Turner, Liu, and Ilic [SIAM J. Scientific Computing, 33:1159--1180, 2011], the solution of the time-space fractional reaction diffusion equations in two dimensions can be written in terms of a matrix function vector product $f(A)b$ at each time step, where $A$ is an approximate matrix representation of the standard Laplacian. We use the finite volume method over unstructured triangular meshes to generate the matrix $A$, which is therefore non-symmetric. However, the standard Lanczos method for approximating $f(A)b$ requires that $A$ is symmetric. We propose a simple and novel transformation in which the standard Lanczos method is still applicable to find $f(A)b$, despite the loss of symmetry. Numerical results are presented to verify the accuracy and efficiency of our newly proposed numerical solution strategy.