1,721,057 research outputs found
Modelling moisture uptake in a cereal grain
No full textRecent experimental data have revealed the spatial and temporal structure of moisture content within a cereal grain immersed in boiling water. A simple model of the water's motion is presented, guided by the observed behaviour, which allows for nonlinear (exponential) diffusion within the grain and a constant mass-transfer coefficient to represent the pericarp on the outer surface. Numerical results are presented illustrating the close relationship of the predictions to the experimental results, with the mass-transfer coefficient as a fitting parameter. The model is studied using asymptotic analysis, in the limit of large activation energy in the diffusion coefficient and large mass transfer. The analysis gives insight into the three phases of the process, consisting of initial linear diffusion, linear motion of the moisture front into the grain, and slow filling of the grain in a relatively uniform manner. The problem is also studied using mean-action-time analysis to derive simple expressions for the time for the grain to saturate
Multiscale modelling and analysis of lithium-ion battery charge and discharge
No full textA 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
Gas propagation in cracks from cylindrical boreholes during blasting in mines
No full textDuring mine blasts a large number of boreholes are drilled and filled with explosives. The pattern of boreholes and the timing of the explosions is chosen to maximize the fracturing of the rock. To help understand the complex interaction between the explosive and the rock a single borehole is considered. The very high pressure gas created by the explosion propagates out from the cylindrical borehole into pre-stressed rock. Analysis of the rock stresses uses matched asymptotic methods assuming the gas propagates slowly. The resulting equation for the gas movement are derived and various limiting cases considered. The model is a nonlinear integro-differential equation which behaves as a singular parabolic problem in some cases
Tumour dynamics and necrosis: surface tension and stability
No full textA model is developed for the motion of cells within a multicell spherical tumour. The model allows either for the intercellular forces to be in compression and cells to be compacted to a fixed number density, or for the cell number density to fall and cells to become isolated from each other. The model develops necrotic regions naturally due to force balances rather than being directly attributable to a critical oxygen concentration. These necrotic regions may result in a gradual reduction in local cell density rather than jump to a completely dead region.Numerical and analytical analysis of the spherically symmetric model shows that the long time behaviour of the spheroid depends on any surface tension effects created by cells on the outer surface. For small surface tension the spheroid grows linearly in time developing a large necrotic region, while for larger surface tension the growth can be halted. The linear stability to spherically symmetric perturbations of all the possible resulting steady states is revealed
On the separation of coconuts: a modeling week study
No full textThe modus operandi of "modeling weeks" is explained and illustrated using a classical modeling week problem of subjecting coconuts to an external pressure before splitting them open, in the hope that the coconut meat will then separate easily from the shell. A range of modeling approaches is considered, and a poroelastic model is posed that allows the key problem parameters to be identified and calculated. We conclude that separation by this means is likely to be possible and relatively easy to accomplish
Asymptotic analysis of the growth of cake layers in filters
No full textThe problem of fluid flow in a two-dimensional pleated filter is considered. Of particular interest is the change in the flow due to cake build-up on the surface of the filter material. The flow is taken to be Darcy flow in the cake and the filter material, with Stokes' flow outside the cake. The particles in the flow are taken to be transported with the flow and to stick to the cake without slippage or resuspension, and the cake is taken to be incompressible. The flow is considered in various geometries, particularly long thin filters and corners. The main parameter in the problem is the ratio of the filter-material resistance to the cake resistance, and limiting cases are considered. Travelling waves of cake build-up are found for arbitrary time-dependent variations in the inflow conditions. The time taken for the filter to become clogged by the cake is also considered
Modelling the cell cycle and cell movement in multicellular tumour spheroids
No full textThis paper analyses a recent mathematical model of avascular tumour spheroid growth which accounts for both cell cycle dynamics and chemotactic driven cell movement. The model considers cells to exist in one of two compartments: proliferating and quiescent, as well as accounting for necrosis and apoptosis. One particular focus of this paper is the behaviour created when proliferating and quiescent cells have different chemotactic responses to an extracellular nutrient supply. Two very different steady-state behaviours are identified corresponding to those cases where proliferating cells move either more quickly or more slowly than quiescent cells in response to a gradient in the extracellular nutrient supply. The case where proliferating cells move more rapidly leads to the commonly accepted spheroid structure of a thin layer of proliferating cells surrounding an inner quiescent core. In the case where proliferating cells move more slowly than quiescent cells the model predicts an interesting structure of a thin layer of quiescent cells surrounding an inner core of proliferating and quiescent cells. The sensitivity of this tumour structure to the cell cycle model parameters is also discussed. In particular variations in the steady-state size of the tumour and the types of transient behaviour are explored. The model reveals interesting transient behaviour with sharply delineated regions of proliferating and quiescent cells
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