The J-Box Problem

The study group was presented with the problem of determining the behaviour of a web of wet cellulose fibres - called a tow - as it passes through part of the manufacturing process. The name of the problem derives from the fact that on its passage down the production line the tow passes through a J-shaped box, whose purpose is to provide a buffer where the tow is stored long enough and hot enough for certain chemical reactions to take place, mainly concerned with giving the right quality to the fibre surface. (The production line in fact involves two J-boxes, one containing wet tow, the other dry, and we are here entirely concerned with the first of these, the wet J-box.) Three aspects of the tow behaviour were proposed for investigation: 1. What is the mechanical behaviour of the tow within the J-box ? Specifically, how do the time spent within the J-box and the shape of the tow outlet column depend on the J- box geometry, tow density and compressibility, flow rate, friction coefficients etc. 2. What is the temperature history, and hence the chemical history, of the tow within the J-box? 3. Why do dislocations and loops occur within the tow? We first give typical parameters and other details of the process, and then further details of these three questions

Asymptotic solution of a model for bilayer organic diodes and solar cells

The current voltage characteristics of an organic semiconductor diode made by placing together two materials with dissimilar electron affinities and ionisation potentials is analysed using asymptotic methods. An intricate boundary layer structure is examined. We find that there are three regimes for the total current passing through the diode. For reverse bias and moderate forward bias the dependency of the voltage on the current is similar to the behaviour of conventional inorganic semiconductor diodes predicted by the Shockley equation and are governed by recombination at the interface of the materials. There is then a narrow range of currents where the behaviour undergoes a transition. Finally for large forward bias the behaviour is different with the current being linear in voltage and is primarily controlled by drift of charges in the organic layers. The size of the interfacial recombination rate is critical in determining the small range of current where there is rapid transition between the two main regimes. The extension of the theory to organic solar cells is discussed and the analogous current voltage curves derived in the regime of interest

Filter Cleaning Using Gas Injection

A filter cleaning process using gas injection is considered. An estimate for the minimum mass flow rate out of the gas injector and the corresponding injector/filter geometry is found. The estimates are based on a similarity solution for a free turbulent jet. The minimum mass flow rate and geometry is worked out for a specific example

Combining mechanical and chemical effects in the deformation and failure of a cylindrical electrode particle in a Li-ion battery

A general framework to study the mechanical behaviour of a cylindrical silicon anode particle in a lithium ion battery as it undergoes lithiation is presented. The two-way coupling between stress and concentration of lithium in silicon, including the possibility of plastic deformation, is taken into account and two particular cases are considered. First, the cylindrical particle is assumed to be free of surface traction and second, the axial deformation of the cylinder is prevented. In both cases plastic stretches develop through the entire cylinder and not just near the surface as is commonly found in spherical anode particles. It is shown that the stress evolution depends both on the lithiation rate and the external constraints. Furthermore, as the cylinder expands during lithiation it can develop a compressive axial stress large enough to induce buckling, which in turn may lead to mechanical failure. An explicit criterion for swelling-induced buckling obtained as a modification of the classical Euler buckling criterion shows the competition between the stabilising effect of radius increase and the destabilising effect of axial stress.Comment: 24 pages, 8 figure

Multiscale modelling and analysis of lithium-ion battery charge and discharge

A microscopic model of a lithium battery is developed, which accounts for lithium diffusion within particles, transfer of lithium from particles to the electrolyte and transport within the electrolyte assuming a dilute electrolyte and Butler–Volmer reaction kinetics. Exploiting the small size of the particles relative to the electrode dimensions, a homogenised model (in agreement with existing theories) is systematically derived and studied. Details of how the various averaged quantities relate to the underlying geometry and assumptions are given. The novel feature of the homogenisation process is that it allows the coefficients in the electrode-scale model to be derived in terms of the microscopic features of the electrode (e.g. particle size and shape) and can thus be used as a systematic way of investigating the effects of changes in particle design. Asymptotic methods are utilised to further simplify the model so that one-dimensional behaviour can be described with relatively simpler expressions. It is found that for low discharge currents, the battery acts almost uniformly while above a critical current, regions of the battery become depleted of lithium ions and have greatly reduced reaction rates leading to spatially nonuniform use of the electrode. The asymptotic approximations are valid for electrode materials where the OCV is a strong function of intercalated lithium concentration, such as Li x C6, but not for materials with a flat discharge curve, such as LiFePO4

Need a Lift? An Elevator Queueing Problem

Various aspects of the behavior and dispatching of elevators (lifts) were studied. Monte Carlo simulation was used to study the statistics of the several models for the peak demand at uppeak times. Analytical models problems were used to prove or disprove whether schemes were optimal. A mostly integer programming problem was formulated but not studied further

Shape optimization of pressurized air bearings

Use of externally pressurized air bearings allows for the design of mechanical systems requiring extreme precision in positioning. One application is the fine control for the positioning of mirrors in large-scale optical telescopes. Other examples come from applications in robotics and computer hard-drive manufacturing. Pressurized bearings maintain a finite separation between mechanical components by virtue of the presence of a pressurized flow of air through the gap between the components. An everyday example is an air hockey table, where a puck is levitated above the table by an array of vertical jets of air. Using pressurized bearings there is no contact between “moving parts” and hence there is no friction and no wear of sensitive components. This workshop project is focused on the problem of designing optimal static air bearings subject to given engineering constraints. Recent numerical computations of this problem, done at IBM by Robert and Hendriks, suggest that near-optimal designs can have unexpected complicated and intricate structures. We will use analytical approaches to shed some light on this situation and to offer some guides for the design process. In Section 2 the design problem is stated and formulated as an optimization problem for an elliptic boundary value problem. In Section 3 the general problem is specialized to bearings with rectangular bases. Section 4 addresses the solutions of this problem that can be obtained using variational formulations of the problem. Analysis showing the sensitive dependence to perturbations (in numerical computations or manufacturing constraints) of near-optimal designs is given in Section 5. In Section 6, a restricted class of “groove network” designs motivated by the original results of Robert and Hendriks is examined. Finally, in Section 7, we consider the design problem for circular axisymmetric air bearings