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

Spin coating of an evaporating polymer solution

We consider a mathematical model of spin coating of a single polymer blended in a solvent. The model describes the one-dimensional development of the thin layer of the mixture as the layer thins due to flow created by a balance of viscous forces and centrifugal forces and due to evaporation of the solvent. In the model both the diffusivity of the solvent in the polymer and the viscosity of the mixture are very rapidly varying functions of the solvent volume fraction. Guided by numerical solutions an asymptotic analysis reveals a number of different possible behaviours of the thinning layer dependent on the nondimensional parameters describing the system.\ud \ud The main practical interest is in controlling the appearance and development of a ``skin'' on the polymer where the solvent concentration reduces rapidly on the outer surface leaving the bulk of the layer still with high concentrations of solvent. The critical parameters controlling this behaviour are found to be $\epsilon$ the ratio of the diffusion to advection time scales, $\delta$ the ratio of the evaporation to advection time scales and $\exp(\gamma)$, the ratio of the diffusivity of the initial mixture and the pure polymer. In particular, our analysis shows that for very small evaporation with $\delta << \exp(-3/(4\gamma)) \epsilon^{3/4}$ skin formation can be prevented

Mathematical issues in city traffic

[Chinese

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

Furnace problem

[Chinese