Quasi-separation of the biharmonic partial differential equation

In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

Study of phase separation of secondary dispersions

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Magneto-vibratory separation of glass and bronze granular mixtures immersed in a paramagnetic liquid

A fluid-immersed granular mixture may spontaneously separate when subjected to vertical vibration, separation occurring when the ratio of particle inertia to fluid drag is sufficiently different between the component species of the mixture. Here, we describe how fluid-driven separation is influenced by magneto-Archimedes buoyancy, the additional buoyancy force experienced by a body immersed in a paramagnetic fluid when a strong inhomogeneous magnetic field is applied. In our experiments glass and bronze mixtures immersed in paramagnetic aqueous solutions of MnCl2 have been subjected to sinusoidal vertical vibration. In the absence of a magnetic field the separation is similar to that observed when the interstitial fluid is water. However, at modest applied magnetic fields, magneto-Archimedes buoyancy may balance the inertia/fluid-drag separation mechanism, or it may dominate the separation process. We identify the vibratory and magnetic conditions for four granular configurations, each having distinctive granular convection. Abrupt transitions between these states occur at well-defined values of the magnetic and vibrational parameters. In order to gain insight into the dynamics of the separation process we use computer simulations based on solutions of the Navier-Stokes' equations. The simulations reproduce the experimental results revealing the important role of convection and gap formation in the stability of the different states.

The development of a novel Bio-MEMS filtration chip for the separation of specific cells in fluid suspension

In the clinical/microbiological laboratory there are currently several ways of separating specific cells from a fluid suspension. Conventionally cells can be separated based on size, density, electrical charge, light-scattering properties, and antigenic surface properties. Separating cells using these parameters can require complex technologies and specialist equipment. This paper proposes new Bio-MEMS (microelectromechanical systems) filtration chips manufactured using deep reactive ion etching (DRIE) technology that, when used in conjunction with an optical microscope and a syringe, can filter and grade cells for size without the requirement for additional expensive equipment. These chips also offer great versatility in terms of design and their low cost allows them to be disposable, eliminating sample contamination. The pumping mechanism, unlike many other current filtration techniques, leaves samples mechanically and chemically undamaged. In this paper the principles behind harnessing passive pumping are explored, modelled, and validated against empirical data, and their integration into a microfluidic device to separate cells from a mixed population suspension is described. The design, means of manufacture, and results from preliminary tests are also presented. © IMechE 2007.

Noncovalent coatings for the separation of synthetic polypeptides by nonaqueous capillary electrophoresis

Recently, we demonstrated the possibility to extend the range of capillary electrophoresis (CE) applications to the separation of non-water-soluble synthetic polymers. This work focuses on the control of the electro-osmotic flow (EOF) and on the limitation of the solute adsorption in nonaqueous electrolytes. For these purposes, different strategies were investigated. For the initial, a viscous additive (ethylene glycol or glycerol) was used in the electrolyte in order to decrease the EOF magnitude and, possibly, to compete with solute adsorption. A second strategy was to modify, before separation, the fused-silica capillary wall by the adsorption of poly(ethylene oxide) (PEO) via hydrogen bonding. The influence of the molecular mass of the adsorbed PEO on the EOF magnitude and direction was studied in electrolytes based on methanol/acetonitrile mixtures containing ammonium ions. For PEO molecular masses above 1000 g/mol, reversed (anodic) EOF were reported in accordance with previous results obtained with PEO covalently bonded capillaries. The influence of the nature and the concentration of the background electrolyte cation on the EOF magnitude and direction were also investigated. A third strategy consisted in modifying the capillary wall by the adsorption of a cationic polyelectrolyte layer. Advantageously, this polyelectrolyte layer suppressed the adsorption of the polymer solutes onto the capillary wall. The results obtained in this work confirm the high potential and the versatility of CE for the characterization of ionizable organic polymers in nonaqueous media.

Directed phase separation of PFO:PS blends during spin-coating using feedback controlled in situ stroboscopic fluorescence microscopy

Uniform thin-films of polymer blends can be produced through spin-coating, which is used on an industrial scale for the production of light emitting diodes, and more recently organic photovoltaic devices. Here, we present the results of the direct observation, and control, over the phase separation of polystyrene and poly(9,9′-dioctylfluorene) during spin-coating using high speed stroboscopic fluorescence microscopy. This new approach, imaging the fluorescence, from a blend of fluorescent + non-fluorescent polymers allows for intensity to be directly mapped to composition, providing a direct determination of composition fluctuations during the spin-coating process. We have studied the compositional development and corresponding structural development for a range of compositions, which produce a range of different phase separated morphologies. We initially observe domains formed by spinodal decomposition, coarsening via Ostwald Ripening until an interfacial instability causes break-up of the bicontinuous morphology. Ostwald ripening continues, and depending upon composition a bicontinuous morphology is re-established. By observing compositional and morphological development in real-time, we are able to direct and control morphological structure development through control of the spin coating parameters via in situ feedback. © 2013 The Royal Society of Chemistry.

Optimisation of chromatographic separation of oxidised phospholipids and detection with targeted approaches provides greater coverage of the oxidised lipidome

Phospholipid oxidation by adventitious damage generates a wide variety of products with potentially novel biological activities that can modulate inflammatory processes associated with various diseases. To understand the biological importance of oxidised phospholipids (OxPL) and their potential role as disease biomarkers requires precise information about the abundance of these compounds in cells and tissues. There are many chemiluminescence and spectrophotometric assays available for detecting oxidised phospholipids, but they all have some limitations. Mass spectrometry coupled with liquid chromatography is a powerful and sensitive approach that can provide detailed information about the oxidative lipidome, but challenges still remain. The aim of this work is to develop improved methods for detection of OxPLs by optimisation of chromatographic separation through testing several reverse phase columns and solvent systems, and using targeted mass spectrometry approaches. Initial experiments were carried out using oxidation products generated in vitro to optimise the chromatography separation parameters and mass spectrometry parameters. We have evaluated the chromatographic separation of oxidised phosphatidylcholines (OxPCs) and oxidised phosphatidylethanolamines (OXPEs) using C8, C18 and C30 reverse phase, polystyrene – divinylbenzene based monolithic and mixed – mode hydrophilic interaction (HILIC) columns, interfaced with mass spectrometry. Our results suggest that the monolithic column was best able to separate short chain OxPCs and OxPEs from long chain oxidised and native PCs and PEs. However, variation in charge of polar head groups and extreme diversity of oxidised species make analysis of several classes of OxPLs within one analytical run impractical. We evaluated and optimised the chromatographic separation of OxPLs by serially coupling two columns: HILIC and monolith column that provided us the larger coverage of OxPL species in a single analytical run.

Computationsl modelling of dimethyl ether separation and steam reforming in fluidized bed reactors

This study presents a computational fluid dynamic (CFD) study of Dimethyl Ether (DME) gas adsorptive separation and steam reforming (DME-SR) in a large scale Circulating Fluidized Bed (CFB) reactor. The CFD model is based on Eulerian-Eulerian dispersed flow and solved using commercial software (ANSYS FLUENT). Hydrogen is currently receiving increasing interest as an alternative source of clean energy and has high potential applications, including the transportation sector and power generation. Computational fluid dynamic (CFD) modelling has attracted considerable recognition in the engineering sector consequently leading to using it as a tool for process design and optimisation in many industrial processes. In most cases, these processes are difficult or expensive to conduct in lab scale experiments. The CFD provides a cost effective methodology to gain detailed information up to the microscopic level. The main objectives in this project are to: (i) develop a predictive model using ANSYS FLUENT (CFD) commercial code to simulate the flow hydrodynamics, mass transfer, reactions and heat transfer in a large scale dual fluidized bed system for combined gas separation and steam reforming processes (ii) implement a suitable adsorption models in the CFD code, through a user defined function, to predict selective separation of a gas from a mixture (iii) develop a model for dimethyl ether steam reforming (DME-SR) to predict hydrogen production (iv) carry out detailed parametric analysis in order to establish ideal operating conditions for future industrial application. The project has originated from a real industrial case problem in collaboration with the industrial partner Dow Corning (UK) and jointly funded by the Engineering and Physical Research Council (UK) and Dow Corning. The research examined gas separation by adsorption in a bubbling bed, as part of a dual fluidized bed system. The adsorption process was simulated based on the kinetics derived from the experimental data produced as part of a separate PhD project completed under the same fund. The kinetic model was incorporated in FLUENT CFD tool as a pseudo-first order rate equation; some of the parameters for the pseudo-first order kinetics were obtained using MATLAB. The modelling of the DME adsorption in the designed bubbling bed was performed for the first time in this project and highlights the novelty in the investigations. The simulation results were analysed to provide understanding of the flow hydrodynamic, reactor design and optimum operating condition for efficient separation. Bubbling bed validation by estimation of bed expansion and the solid and gas distribution from simulation agreed well with trends seen in the literatures. Parametric analysis on the adsorption process demonstrated that increasing fluidizing velocity reduced adsorption of DME. This is as a result of reduction in the gas residence time which appears to have much effect compared to the solid residence time. The removal efficiency of DME from the bed was found to be more than 88%. Simulation of the DME-SR in FLUENT CFD was conducted using selected kinetics from literature and implemented in the model using an in-house developed user defined function. The validation of the kinetics was achieved by simulating a case to replicate an experimental study of a laboratory scale bubbling bed by Vicente et al [1]. Good agreement was achieved for the validation of the models, which was then applied in the DME-SR in the large scale riser section of the dual fluidized bed system. This is the first study to use the selected DME-SR kinetics in a circulating fluidized bed (CFB) system and for the geometry size proposed for the project. As a result, the simulation produced the first detailed data on the spatial variation and final gas product in such an industrial scale fluidized bed system. The simulation results provided insight in the flow hydrodynamic, reactor design and optimum operating condition. The solid and gas distribution in the CFB was observed to show good agreement with literatures. The parametric analysis showed that the increase in temperature and steam to DME molar ratio increased the production of hydrogen due to the increased DME conversions, whereas the increase in the space velocity has been found to have an adverse effect. Increasing temperature between 200 oC to 350 oC increased DME conversion from 47% to 99% while hydrogen yield increased substantially from 11% to 100%. The CO2 selectivity decreased from 100% to 91% due to the water gas shift reaction favouring CO at higher temperatures. The higher conversions observed as the temperature increased was reflected on the quantity of unreacted DME and methanol concentrations in the product gas, where both decreased to very low values of 0.27 mol% and 0.46 mol% respectively at 350 °C. Increasing the steam to DME molar ratio from 4 to 7.68 increased the DME conversion from 69% to 87%, while the hydrogen yield increased from 40% to 59%. The CO2 selectivity decreased from 100% to 97%. The decrease in the space velocity from 37104 ml/g/h to 15394 ml/g/h increased the DME conversion from 87% to 100% while increasing the hydrogen yield from 59% to 87%. The parametric analysis suggests an operating condition for maximum hydrogen yield is in the region of 300 oC temperatures and Steam/DME molar ratio of 5. The analysis of the industrial sponsor’s case for the given flow and composition of the gas to be treated suggests that 88% of DME can be adsorbed from the bubbling and consequently producing 224.4t/y of hydrogen in the riser section of the dual fluidized bed system. The process also produces 1458.4t/y of CO2 and 127.9t/y of CO as part of the product gas. The developed models and parametric analysis carried out in this study provided essential guideline for future design of DME-SR at industrial level and in particular this work has been of tremendous importance for the industrial collaborator in order to draw conclusions and plan for future potential implementation of the process at an industrial scale.