1,721,088 research outputs found
Modelling the spreading and draining of viscous films
No full textThe focus of the work in this thesis is to gain new insight into the fluid behaviour observed in a float glass furnace by means of simplified mathematical models. In particular, the models explore the dynamics of films of foam, known as logs, that spread across the surface of a pool of molten glass. The model employed throughout is the two dimensional Navier-Stokes equations, in the limit of zero Reynolds number, together with appropriate conditions at moving boundaries. Throughout the thesis, the slender geometry of the films is exploited using asymptotic techniques to simplify the models. In the introductory chapter, the motivating float glass manufacturing process is described, then the mathematical techniques and modelling assumptions that are used throughout the thesis are introduced. In the first technical chapter a model for the spreading of viscous films on the surface of a deep viscous pool is considered. Although this model neglects the effects of drainage it enables analytical progress to be made. As such, insight is gained into how the spreading logs interact with one another as they spread across the surface of the underlying pool. Analytic expressions for the evolution of a single spreading film, two spreading films and an infinite array of films are obtained. In addition, some comments on a general configuration of films are made. In the next technical chapter a model for the spreading and draining of a viscous film on a flat surface is considered. Although the model is simplistic, and neglects the interaction of logs via the underlying pool, it does allow some initial ideas on the effects of drainage to be explored. The model is systematically reduced to a nonlinear diffusion PDE. The subsequent analysis is applicable to a broad family of PDEs, hence the analysis is presented in some generality. Solutions to the PDEs under consideration exhibit an interesting behaviour in which the front of a compactly supported solution changes its direction of propagation. To explore this phenomenon, the behaviour of the front of the film as it advances (due to gravity driven spreading) and recedes (due to drainage) is examined. In particular, asymptotic solutions local to a time at which the front of the film changes its direction of propagation are obtained and their implications discussed. In the final technical chapter, the ideas from the previous two chapters are drawn together. A model is considered that incorporates both drainage, and allows the spreading logs to interact via the molten glass pool. It is shown that the model can be systematically reduced to a singular integro-differential equation (SIDE). In a special case, a steady state solution to this SIDE is obtained using a combination of asymptotic and numerical techniques. To complement the analysis in the previous chapters, advancing and receding fronts of solutions to the model are also examined. In the final chapter, the results of the previous chapters are summarised, and the practical implementation of the modelling is discussed. The work not only gives rise to a number of novel mathematical results, but also provides new understanding on the behaviour of spreading viscous films and the industrial float glass proces
Modelling in vivo action potential propagation along a giant axon
Full text linkA partial differential equation model for the three-dimensional current flow in an excitable, unmyelinated axon is considered. Where the axon radius is significantly below a critical value Rcrit (that depends upon intra- and extra-cellular conductivity and ion channel conductance) the resistance of the intracellular space is significantly higher than that of the extracellular space, such that the potential outside the axon is uniformly small whilst the intracellular potential is approximated by the transmembrane potential. In turn, since the current flow is predominantly axial, it can be shown that the transmembrane potential is approximated by a solution to the one-dimensional cable equation. It is noted that the radius of the squid giant axon, investigated by (Hodgkin and Huxley 1952e), lies close to Rcr i t . This motivates us to apply the three-dimensional model to the squid giant axon and compare the results thus found to those obtained using the cable equation. In the context of the in vitro experiments conducted in (Hodgkin and Huxley 1952e) we find only a small difference between the wave profiles determined using these two different approaches and little difference between the speeds of action potential propagation predicted. This suggests that the cable equation approximation is accurate in this scenario. However when applied to the it in vivo setting, in which the conductivity of the surrounding tissue is considerably lower than that of the axoplasm, there are marked differences in both wave profile and speed of action potential propagation calculated using the two approaches. In particular, the cable equation significantly over predicts the increase in the velocity of propagation as axon radius increases. The consequences of these results are discussed in terms of the evolutionary costs associated with increasing the speed of action potential propagation by increasing axon radius
Interface behaviour of the slow diffusion equation with strong absorption: intermediate-asymptotic properties
No full textThe dynamics of interfaces in slow-diffusion equations with strong absorption are studied. Asymptotic methods are used to give descriptions of the behaviour local to a comprehensive range of possible singular events that can occur in any evolution. These events are: when an interface changes its direction of propagation (reversing and anti-reversing), when an interface detaches from a absorbing obstacle (detaching), when two interfaces are formed by film rupture (touchdown) and when the solution undergoes extinction. Our account of extinction and self-similar reversing and anti-reversing is built upon previous work; results on non-self-similar reversing and anti-reversing and on the various types of detachment and touchdown are developed from scratch. In all cases, verification of the asymptotic results against numerical solutions to the full PDE are provided. Self-similar solutions, both of the full equation and of its asymptotic limits, play a central role in the analysis
A model for the operation of perovskite based hybrid solar cells: formulation, analysis, and comparison to experiment
Full text linkThis work is concerned with the modeling of perovskite based hybrid solar cells formed by sandwiching a slab of organic lead halide perovskite (CH3NH3PbI3?xClx) photo-absorber between (n-type) acceptor and (p-type) donor materials—typically titanium dioxide and spiro. A model for the electrical behavior of these cells is formulated based on drift-diffusion equations for the motion of the charge carriers and Poisson’s equation for the electric potential. It is closed by (i) internal interface conditions accounting for charge recombination/generation and jumps in charge carrier densities arising from differences in the electron affinity/ionization potential between the materials and (ii) ohmic boundary conditions on the contacts. The model is analyzed by using a combination of asymptotic and numerical techniques. This leads to an approximate—yet highly accurate—expression for the current-voltage relationship as a function of the solar induced photo- current. In addition, we show that this approximate current-voltage relation can be interpreted as an equivalent circuit model consisting of three diodes, a resistor, and a current source. For sufficiently small biases the device’s behavior is diodic and the current is limited by the recombination at the internal interfaces, whereas for sufficiently large biases the device acts like a resistor and the current is dictated by the ohmic dissipation in the acceptor and donor. The results of the model are also compared to experimental current-voltage curves, and good agreement is shown
A fast and robust numerical scheme for solving models of charge carrier transport and ion vacancy motion in perovskite solar cells
No full textDrift-diffusion models that account for the motion of ion vacancies and electronic charge carriers are important tools for explaining the behaviour, and guiding the development, of metal halide perovskite solar cells. Computing numerical solutions to such models in realistic parameter regimes, where the short Debye lengths give rise to boundary layers in which the solution varies extremely rapidly, is challenging. Two suitable numerical methods, that can effectively cope with the spatial stiffness inherent to such problems, are presented and contrasted (a finite element scheme and a finite difference scheme). Both schemes are based on an appropriate choice of non-uniform spatial grid that allows the solution to be computed accurately in the boundary layers. An adaptive time step is employed in order to combat a second source of stiffness, due to the disparity in timescales between the motion of the ion vacancies and electronic charge carriers. It is found that the finite element scheme provides significantly higher accuracy, in a given compute time, than both the finite difference scheme and some previously used alternatives (Chebfun and pdepe). An example transient sweep of a current-voltage curve for realistic parameter values can be computed using this finite element scheme in only a few seconds on a standard desktop computer
Charge transport modelling of lithium ion batteries
No full textThis paper presents the current state of mathematical modelling of the electrochemical behaviour of lithium-ion batteries as they are charged and discharged. It reviews the models developed by Newman and co-workers, both in the cases of dilute and moderately- concentrated electrolytes and indicates the modelling assumptions required for their de- velopment. Particular attention is paid to the interface conditions imposed between the electrolyte and the active electrode material; necessary conditions are derived for one of these, the Butler-Volmer relation, in order to ensure physically realistic solutions. Insight into the origin of the differences between various models found in the literature is revealed by considering formulations obtained by using different measures of the electric potential. Materials commonly used for electrodes in lithium ion batteries are considered and the various mathematical models used to describe lithium transport in them discussed. The problem of up-scaling from models of behaviour at the single electrode particle scale to the cell scale is addressed using homogenisation techniques resulting in the pseudo 2D model commonly used to describe charge transport and discharge behaviour in lithium-ion cells. Numerical solution to this model is discussed and illustrative results for a common device are computed.<br/
Parametrisation and use of a predictive DFN model for a highenergy NCA/GrSiOx battery
No full textWe demonstrate the predictive power of a parametrised DoyleFullerNewman (DFN) model of a commercial cylindrical (21700) lithiumion cell with NCA/GrSiOx chem istry. Model parameters result from the deconstruction of a fresh commercial cell to deter mine/confirm chemistry and microstructure, and also from electrochemical experiments with halfcells built from electrode samples. The simulations predict voltage profiles for (i) galvanostatic discharge and (ii) drivecycles. Predicted voltage responses deviate from measured ones by <1% throughout at least ∼95% of a full galvanostatic discharge, whilst the drive cycle discharge is matched to a ∼13% error throughout. All simulations are performed using the online computational tool DandeLiion, which rapidly solves the DFN model using only modest computational resource. The DFN results are used to quantify the irreversible energy losses occurring in the cell and deduce their location. In addition to demonstrating the predictive power of a properly validated DFN model, this work pro vides a novel simplified parametrisation workflow that can be used to accurately calibrate an electrochemical model of a cell
DandeLiion v1: an extremely fast solver for the Newman model of lithium-ion battery (dis)charge
Full text linkDandeLiion (available at dandeliion.com) is a robust and extremely fast solver for the Doyle Fuller Newman (DFN) model, the standard electrochemical model for (dis)charge of a planar lithium-ion cell. DandeLiion conserves lithium, uses a second order spatial discretisation method (enabling accurate computations using relatively coarse discretisations) and is many times faster than its competitors. The code can be used "in the cloud"and does not require installation before use. The difference in compute time between DandeLiion and its commercial counterparts is roughly a factor of 100 for the moderately-sized test case of the discharge of a single cell. Its linear scaling property means that the disparity in performance is even more pronounced for bigger systems, making it particularly suitable for applications involving multiple coupled cells. The model is characterised by a number of phenomenological parameters and functions, which may either be provided by the user or chosen from DandeLiion's library. This library contains data for the most commonly used electrolyte (LiPF6) and a number of common active material chemistries including graphite, lithium iron phosphate (LFP), nickel cobalt aluminum (NCA), and a variant of nickel cobalt manganese (NMC).</p
Generalised single particle models for high-rate operation of graded lithium-ion electrodes: Systematic derivation and validation
Full text linkA derivation of the single particle (SP) model is made from a Doyle-Fuller-Newman (DFN) model for electrodes composed of uniformly sized spherical electrode particles of one chemistry. The derivation uses a formal asymptotic method based on the disparity between the size of the thermal voltage and that of the characteristic change in overpotential that occurs during (de)lithiation. Comparison is made be- tween the SP model and the DFN model for electrodes made from: lithium nickel manganese cobalt oxide (NMC), graphite and lithium iron phosphate (LFP). These are used to identify regimes where the SP model is accurate. For most chemistries, even at moderate rates, there are discrepancies between the DFN model and the SP model due to spatial non-uniformities in the electrolyte. Motivated by these discrepancies a correction term to the SP model is derived. Incorporating this into the SP model gives the corrected SP (cSP) model whose accuracy is very significantly improved over the SP model. Generalised versions of the cSP model for graded electrodes containing multiple electrode particle sizes (or chem- istries) in different regions of the electrode, are also derived. The results of this generalisation to the cSP model compare favourably to the full DFN model, even at relatively high discharge rates, where the active electrode material is either graphite or a particular NMC variant
IonMonger: a free and fast planar perovskite solar cell simulator with coupled ion vacancy and charge carrier dynamics
Full text linkDetails of an open-source planar perovskite solar cell simulator, that includes ion vacancy migration within the perovskite layer coupled to charge carrier transport throughout the perovskite and adjoining transport layers in one dimension, are presented. The model equations are discretised in space using a finite element scheme and temporal integration of the resulting system of differential-algebraic equations is carried out in MATLAB. The user is free to modify device parameters, as well as the incident illumination and applied voltage. Time-varying voltage and/or illumination protocols can be specified, e.g. to simulate current-voltage sweeps, or to track the open-circuit conditions as the illumination is varied. Typical simulations, e.g. current-voltage sweeps, only require computation times of seconds to minutes on a modern personal computer. An example set of hysteretic current-voltage curves is presented
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