4th Order Runge-Kutta Method Research Articles

Optimization of an auto-thermal ammonia synthesis reactor using cyclic coordinate method

The ammonia synthesis system is an important chemical process used in the manufacture of fertilizers, chemicals, explosives, fibers, plastics, refrigeration. In the literature, many works approaching the modeling, simulation and optimization of an auto-thermal ammonia synthesis reactor can be found. However, they just focus on the optimization of the reactor length while keeping the others parameters constant. In this study, the other parameters are also considered in the optimization problem such as the temperature of feed gas enters the catalyst zone, the initial nitrogen proportion. The optimal problem requires the maximization of an objective function which is multivariable function and subject to a number of equality constraints involving the solution of coupled differential equations and also inequality constraint. The cyclic coordinate search was applied to solve the multivariable-optimization problem. In each coordinate, the golden section method was applied to find the maximum value. The inequality constraints were treated using penalty method. The coupled differential equations system was solved using Runge-Kutta 4th order method. The results obtained from this study are also compared to the results from the literature.

Large Elaso-plastic Deflection of Micro-beams Using Strain Gradient Plasticity Theory

Metallic micro-scale structures are observed to display more stiffness unlike the predictions of the classical plasticity theories, thus showing size-dependency. Hence, for accurate predictions of the behavior of micro-scale structures, non-classical theories are developed with intrinsic material length in their constitutive relation. Out of the many developed models, strain gradient plasticity theory is the one which is extensively followed. Power law based moment curvature relationship obtained from strain gradient plasticity is available in literature which models rigid-plastic behavior of micro-beam bending. In this article, elastic deformation is appended to the power law based moment curvature relationship. The derived law is employed to obtain the governing differential equation of a micro-cantilever under a normal terminal follower load undergoing large deflection. The non-linear differential equation is then solved by employing an efficient semi-incremental approach. In this approach the equation is solved by using Runge-Kutta 4th order method considering load to be at the current value but the material modulus at its previous value. Adoption of such a technique gave considerable computational advantage over the completely incremental method. The elasto-plastic force-displacement response curves of micro-beams showed stiffer results as compared to their classical counterpart. Also the elasto-plastic divergence from their corresponding elastic response curves are observed to happen at the same displacement for different micro-beams with same length to thickness ratio.

Predictive Study of Velocimetry in the Coronary Artery after Iodinated Contrast Agent Injection

This study aims to determine the retention time of iodinated contrast agents (ICA) in the coronary artery. The mechanical aspect of ICA displacement was studied in order to better understand the effect of these products on the inner wall of the artery of patients with coronary artery stenosis, undergoing repeated imaging examinations with iodinated contrast agents. ICA flow, ICA and blood flow in the artery were modelled. The fluid was regarded to be viscous, incompressible and Newtonian. Blood flow was presumed to be unidirectional, laminar and unstationary. Iodine flow velocity and retention time were calculated using the Runge-Kutta 4th order method programmed in C++ and MatLab R2013a language. The results showed that for coronary artery CT-scans, ICA retention time is 1 minute 40 seconds and for coronary arteriography it is between 2.41 and 3.61 seconds. The values calculated were compared to theoretical values and to clinical observations. The results enabled us to validate our model.

Complex for Simulation Modeling of the Dynamics of Dosing Systems

The design of dosing systems with regard to the engineering and design requirements based on the system approach is a relevant and important issue for various industries. Stage modeling of such systems is one of the main issues in the design process. The purpose of work is development of complex for simulation modeling of systems of volumetric dosing of liquids. Realization of this goal is based on a combination of fundamental and computational schemes of functioning of dosing systems; mathematical models taking into account the dynamic relationships of their elements in different modes of operation and software for the study of these models. The structure of the complex is given. Using the diagrams the interaction of the elements of the metering system is shown and the relevant mathematical relationships present. As an example, dynamic phenomena processes dosing for hydraulic and mechanical drive of the executive body of the metering system are described. The algorithm of calculation of dynamic characteristics of dosing systems is developed on the basis of which the proposed software for mathematical modeling in the environment of Matlab using integrated numerical algorithms for solving systems of differential equations by the Runge-Kutta method of 4th and 5th order. Some results of calculations of the dynamic characteristics of the dispensing process for cases hydraulic, and the Cam actuator of the Executive body of the dispenser are shown. A graphical illustration of the results obtained is given and the comparison of the dynamic characteristics of dosing systems for the types of drives that allows their evaluation and selection of one or another construction in accordance with specified requirements is given. System for simulation modeling allows you to implement a directed study of the dynamics of dosing systems with access to the solution of problems of optimal design and synthesis.

Dynamics of a Frozen Sea Buckhorn Berry Motion in a Separator

After removing frozen sea buckthorn berries from the branches of sea buckthorn tree in a separator (patent EE05717B1), the berries fall onto separator's moving conveyer belt under a certain angle of inclination, thus guiding the berries in the collector of separator. This paper presents the mathematical model of the berry motion along the conveyer and after falling from it. The nonlinear differential equations describing the berry motion on the conveyer and after falling from it take into account air resistance depending on the square value of berry's absolute velocity. All these equations were solved numerically by using Runge-Kutta method of 4th order on Mathcad worksheet. This paper focuses on berry motion depending on belt velocity, friction coefficient between belt and berry, and inclination angle of the belt. Composed video clip visualizes berry motion and demonstrates how the berry moves from the initial position in the same direction as conveyer belt velocity, down to lower edge of the belt and falls from conveyer. Results of this paper can be used by engineers designing conveyors and by educators teaching agricultural machinery.

Higher order explicit solutions for nonlinear dynamic model of column buckling using variational approach and variational iteration algorithm-II

This paper deals with the determination of approximate solutions for a model of column buckling using two efficient and powerful methods called He’s variational approach and variational iteration algorithm-II. These methods are used to find analytical approximate solution of nonlinear dynamic equation of a model for the column buckling. First and second order approximate solutions of the equation of the system are achieved. To validate the solutions, the analytical results have been compared with those resulted from Runge-Kutta 4th order method. A good agreement of the approximate frequencies and periodic solutions with the numerical results and the exact solution shows that the present methods can be easily extended to other nonlinear oscillation problems in engineering. The accuracy and convenience of the proposed methods are also revealed in comparisons with the other solution techniques.

A simplified steady-state model for air, water and steam curtains

Curtain mitigation systems are modeled here since they have experimentally shown their efficiency in reducing the concentration of certain toxic gases within dense gas clouds. Air, water and steam are analyzed in a model as the physical barriers to decrease the gas concentration. The model, developed for a steady-state mitigation process, is based on the mass, energy and momentum conservation laws. Concentration estimations during the dispersion before and after the mitigation are performed with a SLAB type model. A sensitivity analysis for each model is given to detect which variables have bigger effects. A release of chlorine is used as an example and the results are calculated in a prototype developed in Visual C++, where the model is solved using the Runge–Kutta 4th order method. The results include the effects of composition, speed, temperature and height of the releasing point as well as a comparison with CFD simulations. The proposed model is simplified and it cannot reproduce eddy effects but it is fast and robust enough. The model provides a set of equations that can be used in numerical problems where explicit derivatives are required, e.g. optimizations procedures.

Prediction of Critical Clearing Time of Java-Bali 500 kv Power System Under Multiple Bus Load Changes Using Neural Network Based Transient Stability Model

A transient stability model based on back propagation neural network is used to analyze transient stability of Java-Bali electricity system, especially in calculating the critical clearing time. The real and the active load changes on each bus that shows the real load pattern of the system used as neural network input, while the target is the Critical Clearing Time (CCT). By using the load pattern as input, it is hoped that the robustness of the proposed method against load changes at multiple bus can be achieved. Data of target critical clearing time used for the training was calculated from the concept of One Machine Infinite Bus (OMIB), by reducing the multi-machine system using a combination of methods of Equal Area Criterion (EAC) through the Trapezoidal method and the Runge-Kutta 4th order method. To analyze transient stability, a three phase ground fault was conducted at one bus and assumed not changed during the simulation. The proposed method will be implemented at Java-Bali 500 kv power system. The simulation results show the calculation of critical clearing time from the proposed method has a minimum error of 0.0016% and a maximum error of 0.0419% compared with CCT by OMIB.

Modelling the intra-particle transport phenomena and chemical reactions of olive kernel fast pyrolysis

In the present study, the development of a mathematical model for the description of the pyrolysis of a single solid olive kernel particle and the prediction of the fast pyrolysis product yields, is presented. Kinetic model is coupled with heat transfer model. The global degradation of biomass is based on Koufopanos et al. mechanism and described by two parallel 1-order reactions. The analysis is focused on primary degradation for small particle and simulations have been carried out for a spherical particle, with radius of 175 μm. The model has been validated against experiments carried out in a laboratory wire mesh reactor, for temperature range from 573 K to 873 K and a heating rate of 200 K/s. The results of the simulation are in good agreement with the experimental data, regarding temperature, conversion histories and product distribution of olive kernel fast pyrolysis. The numerical method applied was finite difference for the heat transfer model and Runge-Kutta 4th order method for chemical kinetics model equations.

Most sensitive parameters in pyrolysis of shrinking biomass particle

In the present study, the impact of shrinking and non-shrinking biomass particles on pyrolysis is studied employing a kinetic model coupled with a heat transfer model using a practically significant kinetic scheme consisting of physically measurable parameters. The numerical model is used to predict the effects of the most important physical and thermal properties (thermal conductivity, heat transfer coefficient, emissivity, reactor temperature and heat of reaction number) considering cylindrical geometry. A finite difference pure implicit scheme utilizing the tri-diagonal matrix algorithm (TDMA) is employed for solving the heat transfer model equation. The Runge-Kutta 4th order method is used for the chemical kinetics model equations. The computer simulations are performed for wide ranges of particle size and temperatures. The results obtained are in excellent agreement with many experimental studies, much better than the agreement with earlier models reported in the literature. The most dominant design variable is reactor temperature and exothermic reaction. The applications of these findings are important and useful for optimum design of biomass gasifiers and pyrolysis reactors.

Modeling for pyrolysis of solid particle: kinetics and heat transfer effects

In the present study, a mathematical model to describe the pyrolysis of a single solid particle of biomass is developed by incorporating improvements in the existing model reported in literature. It couples the heat transfer equation with the chemical kinetics equations. The pyrolysis rate has been simulated by a kinetic scheme involving three reactions (primary and secondary): two parallel reactions and a third for the secondary interactions between the volatile and gaseous products and the char. The dependence of convective heat transfer coefficient on Reynolds number and Prandtl number is incorporated in the model. A finite difference method using a pure implicit scheme is used for solving the heat transfer equation and the Runge–Kutta 4th order method for the chemical kinetics equations. The model equation is solved for cylindrical pellets, spheres and slab geometries of equivalent radius ranging from 0.00025 to 0.013 m and temperature ranging from 303 to 1000 K. The simulated results obtained using the present model are in excellent agreement with the experimental data, much better than the agreement with the earlier models reported in the literature.

Optimization of number of effects for higher yield from an inverted absorber solar still using the Runge-Kutta method

This paper presents a computer simulation model based on the Runge-Kutta 4th-order method to solve the energy balance equations for a multi-basin inverted absorber distiller unit. The effect of reuse of latent heat from vaporization from respective lower basins on daily yield has been studied for optimization of number of effects. It has been observed that when the number of basins is increased beyond seven, there is only a marginal increase in the yield. This results is in accordance with that reported by Fernandez and Chargoy [1].

Solution of parabolic partial differential equations by a double collocation method

A method for solution of parabolic PDE by means of interpolation of the differential operators in two dimensions is described. The method is developed on the basis of previously published methods for solution of ordinary differential equations by orthogonal collocation. It is shown to be highly economical and very stable in comparison with the conventional Crank—Nicolson or with explicit methods such as the Runge—Kutta 4th order method. The linear heat equation is used to illustrate the principle of the method and to discuss its convergence properties.