Method Of Lines Technique Research Articles

Numerical simulation of one-dimensional perovskite solar cell model

Perovskite solar cell (PSC) is one of the advanced third-generation solar cells that have rapid efficiencies but instability issues in terms of air, moisture, and UV light sensitivity becomes a barrier to commercialization. The instability issue is due to the charge accumulation at the interface of the PSC reducing its efficiency. This research focuses on the operation of PSCs through a one-dimensional (1D) drift-diffusion model for planar heterojunction PSCs. The model also accounts for the electric potential by Poisson’s equation, the ion generation, and the recombination rate. The method of lines technique is applied to solve the model for the perovskite layer numerically using the finite difference method which is then solved forward in time using the ‘ode15s’ solver in MATLAB. It is highlighted that the comparison with the experimental data from the reference shows good agreement. The effect of parameter thickness variation of the perovskite layer upon the efficiency of PSCs is analyzed. The result shows that the best efficiency obtained is 19.77% obtained at thickness 0.25 𝜇𝑚. The results may prove useful for a guideline of the cell thickness that predicts the perovskites cell performance.

Lithium Ion Battery Performance for Different Size of Electrode Particles and Porosity

Nowadays, the interest of high proficiency of Lithium ion batteries is increasing as they provide high volumetric energy densities and to meet the demand of exponentially growth of electronic devices. Furthermore, LIB has shown their potential to offer high performance rechargeable battery in the research of electric vehicles. Electrochemical process of Lithium ion batteries encompasses a complex ion transport between the anode and cathode within an electrolyte. The multiscale LIB model consists of charge transport within electrode particle and in electrolyte and the reaction rate at the electrolyte-electrode particle interface which directly relating the geometry of microstructure (the size of particles, about 1nm) to the behaviour in macroscopic model (within the thickness of electrode, about 1 μm). Thus, the geometry of cell and the interfacial behaviour are significantly control the rate of reaction rate. This study concerns about the effect of geometry variations of cell upon the discharge curve of LiFePO4 cathode material. The electrochemical model is solved using Method of Lines technique by discretising the spatial variable using Finite Difference Method. The simulation result is verified with experimental data of LiFePO4 cell by Yu et. al. [14]. The effect of different sizes of particles and volume fractions upon the cell performance are examined. It has been shown that decreasing the size of electrode particles produce high cell potential but slightly low capacity. On the other hand, the optimal volume fraction is shown to be provided that all the particles are spherical and of the same size. Smaller volume fractions resulted in low capacity.

Numerical Simulation of Leakage Current on Conductive Insulator Surface

The outdoor insulator is commonly exposed to environmental pollution. The presence of water like raindrops and dew on the contaminant surface can lead to surface degradation due to leakage current. However, the physical process of this phenomenon is not well understood. Hence, in this study we develop a mathematical model of leakage current on the outdoor insulator surface using the Nernst Planck theory which accounts for the charge transport between the electrodes (negative and positive electrode) and charge generation mechanism. Meanwhile the electric field obeys Poisson’s equation. Method of Lines technique is used to solve the model numerically in which it converts the PDE into a system of ODEs by Finite Difference Approximations. The numerical simulation compares reasonably well with the experimental conduction current. The findings from the simulation shows that the conduction current is affected by the electric field distribution and charge concentration. The rise of the conduction current is due to the distribution of positive ion while the dominancy of electron attachment with neutral molecule and recombination with positive ions has caused a significant reduction of electron and increment of negative ions.

Numerical Treatment of the Modified Burgers’ Equation via Backward Differentiation Formulas of Orders Two and Three

Abstract We present an efficient numerical method for solving the nonlinear modified Burgers’ equation (MBE) using the multi-step method. The nonlinear MBE is first discretized along the spatial direction alone by using the method of lines technique, and this method converts the MBE to a nonlinear system of ordinary differential equations. Multistep methods are employed to solve the nonlinear system of ordinary differential equations. Applicability of the proposed numerical techniques is established through test examples. Discrete root mean square error norm ( L 2 ) $(L_{2})$ and maximum error norm ( L ∞ ) $(L_{\infty})$ are computed and presented for demonstrating the accuracy of the present numerical method. Numerical experiments supported by figures shows that the proposed numerical scheme shows excellent agreement with exact solution and is superior to some existing numerical methods.

Burgers equation : A study using method of lines

Burgers Equation has been the center of study for many researchers because of its vast applications in engineering problems particularly in Fluid Dynamics. Various methods have been formulated for solving this equation both exactly and numerically. In this work Burgers Equation is studied numerically by using Method of Lines technique. The approximate numerical solution is compared with the available Exact solution. The phenomenon of Shock wave formation is displayed through graph.

An efficient method-of-lines simulation procedure for organic semiconductor devices

We describe an adaptive grid method-of-lines (MOL) solution procedure for modelling charge transport and recombination in organic semiconductor devices. The procedure we describe offers an efficient, robust and versatile means of simulating semiconductor devices that allows for much simpler coding of the underlying equations than alternative simulation procedures. The MOL technique is especially well-suited to modelling the extremely stiff (and hence difficult to solve) equations that arise during the simulation of organic-and some inorganic-semiconductor devices. It also has wider applications in other areas, including reaction kinetics, combustion and aero- and fluid dynamics, where its ease of implementation also makes it an attractive choice. The MOL procedure we use converts the underlying semiconductor equations into a series of coupled ordinary differential equations (ODEs) that can be integrated forward in time using an appropriate ODE solver. The time integration is periodically interrupted, the numerical solution is interpolated onto a new grid that is better matched to the solution profile, and the time integration is then resumed on the new grid. The efficacy of the simulation procedure is assessed by considering a single layer device structure, for which exact analytical solutions are available for the electric potential, the charge distributions and the current-voltage characteristics. Two separate state-of-the-art ODE solvers are tested: the single-step Runge-Kutta solver Radau5 and the multi-step solver ODE15s, which is included as part of the Matlab ODE suite. In both cases, the numerical solutions show excellent agreement with the exact analytical solutions, yielding results that are accurate to one part in 1 x 10(4). The single-step Radau5 solver, however, is found to provide faster convergence since its efficiency is not compromised by the periodic interruption of the time integration when the grid is updated.

Symbolic semi-discretization of partial differential equation systems

Abstract A symbolical computation approach to the preprocessing of distributed parameter systems for preparing a numerical solution by a method of lines technique is investigated. A first prototype of a toolbox for the semi-discretization of partial differential equation systems on simple spatial domains has been implemented in MACSYMA. The equations are classified first in order to support the selection of a particular type of finite difference or spectral methods provided. The resulting differential-algebraic system is either represented symbolically in linear implicit form or in a recursive notation to be passed to an automatic code generation module. Additionally, the semi-discrete system can be analyzed numerically to check its stability properties prior to numerical integration. This way, unsuitable discretizations can be detected and discarded without any simulation experiment. The tool mainly addresses the needs which arise in the analysis of the dynamics of chemical engineering processes.

Numerical simulation of the power density distribution generated in a multimode cavity by using the method of lines technique to solve directly for the electric field

In this paper, a new numerical method is presented in order illustrate how the Method of Lines technique can be used to obtain the power density distribution in a dielectric material by solving directly for the electric field in three-dimensional space. A detailed analysis of the treatment of the boundary conditions at the interfaces that exist between air and the material, as well as at absorbing boundary and input planes, are also given in this paper. The method is tested and verified on some simple waveguide examples for which analytic solutions are available. The technique is subsequently applied to the more complicated cavity problem and the solutions for the power density distribution are compared directly with those obtained in previous research using the finite-difference time-domain (FDTD) method. The results of all tests conducted in this research indicate that the Method of Lines technique is a robust numerical tool which can be used to readily handle the hyperbolic nature of the Maxwell equations. Finally, in order to demonstrate the versatility of the developed model, the power density distribution generated inside a dielectric material loaded in a cavity that has multiple input waveguides is presented. The chosen examples exhibit the complicated electromagnetic phenomena which arise inside the cavity and provides some idea of the effect of multiple waveguide input on the power density distribution.

Domain-adaptive finite difference methods for collapsing annular liquid jets

A domain-adaptive technique which maps a time-dependent, curvilinear geometry into a unit square is used to determine the steady state mass absorption rate and the collapse of annular liquid jets. A method of lines is used to solve the one-dimensional fluid dynamics equations written in weak conservation-law form, and upwind differences are employed to evaluate the axial convective fluxes. The unknown, time-dependent, axial location of the downstream boundary is determined from the solution of an ordinary differential equation which is nonlinearly coupled to the fluid dynamics and gas concentration equations. The equation for the gas concentration in the annular liquid jet is written in strong conservation-law form and solved by means of a method of lines at high Peclet numbers and a line Gauss-Seidel method at low Peclet numbers. The effects of the number of grid points along and across the annular jet, time step, and discretization of the radial convective fluxes on both the steady state mass absorption rate and the jet's collapse rate have been analyzed on staggered and non-staggered grids. The steady state mass absorption rate and the collapse of annular liquid jets are determined as a function of the Froude, Peclet and Weber numbers, annular jet's thickness-to-radius ratio at the nozzle exit, initial pressure difference across the annular jet, nozzle exit angle, temperature of the gas enclosed by the annular jet, pressure of the gas surrounding the jet, solubilities at the inner and outer interfaces of the annular jet, and gas concentration at the nozzle exit. It is shown that the steady state mass absorption rate is proportional to the inverse square root of the Peclet number except for low values of this parameter, and that the possible mathematical incompatibilities in the concentration field at the nozzle exit exert a great influence on the steady state mass absorption rate and on the jet collapse. It is also shown that the steady state mass absorption rate increases as the Weber number, nozzle exit angle, gas concentration at the nozzle exit, and temperature of the gases enclosed by the annular liquid jet are increased, but it decreases as the Froude and Peclet numbers, and annular liquid jet's thickness-to-radius ratio at the nozzle exit are increased. It is also shown that the annular liquid jet's collapse rate increases as the Weber number, nozzle exit angle, temperature of the gases enclosed by the annular liquid jet, and pressure of the gases which surround the jet are increased, but decreases as the Froude and Peclet numbers, and annular liquid jet's thickness-toradius ratio at the nozzle exit are increased. It is also shown that both the ratio of the initial pressure of the gas enclosed by the jet to the pressure of the gas surrounding the jet and the ratio of solubilities at the annular liquid jet's inner and outer interfaces play an important role on both the steady state mass absorption rate and the jet collapse. If the product of these ratios is greater or less than one, both the pressure and the mass of the gas enclosed by the annular liquid jet decrease or increase, respectively, with time. It is also shown that the numerical results obtained with the conservative, domain-adaptive method of lines technique presented in this paper are in excellent agreement with those of a domain-adaptive, iterative, non-conservative, block-bidiagonal, finite difference method which uncouples the solution of the fluid dynamics equations from that of the convergence length.