Featuring Pig Movement in Two-Phase Gas Pipelines

2012 ◽  
Author(s):  
David E. G. P. Bueno ◽  
Aline Barbosa Figueiredo ◽  
Renan Martins Baptista ◽  
Felipe B. F. Rachid ◽  
Gustavo C. R. Bodstein
Keyword(s):  
Numerical Scheme ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
General Purpose ◽  
Gas Pipelines ◽  
Two Phase ◽  
Hydrodynamic Friction ◽  
Friction Forces ◽  
Different Types ◽  
Initial Boundary Value

This paper presents a mechanical model, along with a numerical scheme for obtaining approximating solutions for the resulting initial-boundary-value problem, for describing the pig movement in transient two-phase gas pipelines. By taking advantage of the best features of the existing models presented in the literature so far, an idealized general purpose pig model is proposed, contemplating the possibility of representing, within a same context, different types of pigs or pig functions. Both mechanical and hydrodynamic friction forces at the interface of the pig and the pipe wall, as well as by-pass flow rates for the liquid and gaseous phases, are naturally incorporated in the modeling in a coherent mechanical context. The governing equations of the two-phase flow model are intentionally written in a general form, so that different existing models can be used within the framework presented herein. Following this same strategy, a detailed numerical scheme is presented in which the discretization of the flux terms are left open, so that different numerical strategies of first or higher orders can be accommodated without any additional difficulties.

On the Computations of Gas-Solid Mixture Two-Phase Flow

2014 ◽  
Vol 6 (01) ◽  
pp. 49-74 ◽  
Author(s):  
D. Zeidan ◽  
R. Touma
Keyword(s):  
Numerical Methods ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Test Cases ◽  
Two Phase ◽  
Solid Mixture ◽  
Flow Problems ◽  
Model Equations ◽  
Initial Boundary Value

AbstractThis paper presents high-resolution computations of a two-phase gas-solid mixture using a well-defined mathematical model. The HLL Riemann solver is applied to solve the Riemann problem for the model equations. This solution is then employed in the construction of upwind Godunov methods to solve the general initial-boundary value problem for the two-phase gas-solid mixture. Several representative test cases have been carried out and numerical solutions are provided in comparison with existing numerical results. To demonstrate the robustness, effectiveness and capability of these methods, the model results are compared with reference solutions. In addition to that, these results are compared with the results of other simulations carried out for the same set of test cases using other numerical methods available in the literature. The diverse comparisons demonstrate that both the model equations and the numerical methods are clear in mathematical and physical concepts for two-phase fluid flow problems.

scholarly journals Numerical implementation of the initial-boundary value problem for nonlinear the onedimensional equations of poroelasticity for the water-ice system

2018 ◽  
Vol 64 (3) ◽  
pp. 337-343
Author(s):  
P. V. Korobov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Homogeneous Medium ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Partial Density ◽  
Sea Waves ◽  
Two Phase ◽  
Water Ice ◽  
Initial Boundary Value

Summary This article is devoted to the problem of propagation of elastic transverse oscillations in a two-phase medium consisting of water and ice (ice impregnated with water). If we consider ice as a kind of porous homogeneous medium with constant partial density, then it becomes possible to apply the problems of the theory of filtration to the water-ice medium. In this paper, we consider one of the possible formulations of the direct problem modeling the propagation of a signal in this medium is considered. The initial-boundary value problem for a one-dimensional nonlinear system of poroelasticity equations is solved by numerical method on the basis of an explicit-difference scheme. A series of numerical calculations for a trial model of the media is presented.The aim of the paper is to describe the approach to the study of water-ice media using the equations of filtration theory. The object of the study is the propagation of wave oscillations in such media. Such fluctuations can have different nature (seismic, acoustic, etc.). For example, it is of interest to use this approach to model the propagation of sea waves in the ice of the initial stage of ice formation.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

RESEARCH OF THE IMPACT DEVICE MODEL OF THE ROD TYPE BY DIFFERENCE METHOD

2020 ◽  
pp. 73-80
Author(s):  
А.М. Слиденко ◽  
В.М. Слиденко
Keyword(s):  
Boundary Value Problem ◽  
Difference Method ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cross Sectional ◽  
Impact Device ◽  
Initial Boundary Value ◽  
The Impact

Приводится анализ механических колебаний элементов ударного устройства с помощью модели стержневого типа. Ударник и инструмент связаны упругими и диссипативными элементами, которые имитируют их взаимодействие. Аналогично моделируется взаимодействие инструмента с рабочей средой. Сформулирована начально-краевая задача для системы двух волновых уравнений с учетом переменных поперечных сечений стержней. Площади поперечных сечений определяются параметрическими формулами при сохранении объемов стержней. Параметрические формулы позволяют получать различного вида зависимости площади поперечного сечения стержня от его длины. Начальные условия отражают физическую картину взаимодействия инструмента с ударником и рабочей средой. Краевые условия описывают контактные взаимодействия ударника с инструментом и последнего с рабочей средой. В качестве модельной задачи рассматривается соударение ударника и инструмента через элемент большой жесткости. Начально-краевая задача исследуется разностным методом. Проводится сравнение решений задачи, полученных с помощью двухслойной и трехслойной разностных схем. Такие схемы реализованы в общей компьютерной программе в системе Mathcad. Показано, что при вычислениях распределения нормальных напряжений по длине стержня лучшими свойствами относительно устойчивости обладает двухслойная схема The article gives the analysis of mechanical vibrations of the impact device elements using the model of the rod type. The hammer and the tool are connected by elastic and dissipative elements that simulate their interaction. The interaction of the tool with the processing medium is simulated in a similar way. An initial boundary-value problem is formulated for a system of two wave equations taking into account the variable cross sections of the rods. Cross-sectional areas are determined by parametric formulas maintaining the volume of the rods. Parametric formulas allow one to obtain various dependence types of the cross-sectional area of the rod on its length. The initial and boundary conditions reflect the physical phenomenon of the tool interaction with the processing medium, and also describe the contact interactions of the hammer with the tool. The impacting of the hammer and the tool through an element of high rigidity is considered as a model problem. To control the limiting values, the solution of the model problem by the Fourier method is used. The initial-boundary-value problem is investigated by the difference method. A comparison of solutions obtained for the two-layer and three-layer difference schemes is given. Such schemes are realized in a common computer program in the Mathcad. It is shown that the two-layer scheme has the best properties in relation to stability while calculating the distribution of normal voltage along the length of the rod

Asymptotics of the eigenmodes and stability of an elastic structure with general feedback matrix

2019 ◽  
Vol 84 (5) ◽  
pp. 873-911 ◽  
Author(s):  
Marianna A Shubov ◽  
Laszlo P Kindrat
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Control Input ◽  
Vibrational Modes ◽  
Boundary Feedback ◽  
Feedback Matrix ◽  
Dissipative Boundary ◽  
Initial Boundary Value ◽  
The Right ◽  
Euler Bernoulli Beam

Abstract The distribution of natural frequencies of the Euler–Bernoulli beam subject to fully non-dissipative boundary conditions is investigated. The beam is clamped at the left end and equipped with a 4-parameter ($\alpha ,\beta ,k_1,k_2$) linear boundary feedback law at the right end. The $2 \times 2$ boundary feedback matrix relates the control input (a vector of velocity and its spatial derivative at the right end), to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space equipped with the energy norm. The dynamics generator has a purely discrete spectrum (the vibrational modes) denoted by $\{\nu _n\}_{n\in \mathbb {Z}^{\prime}}$. The role of the control parameters is examined and the following results have been proven: (i) when $\beta \neq 0$, the set of vibrational modes is asymptotically close to the vertical line on the complex $\nu$-plane given by the equation $\Re \nu = \alpha + (1-k_1k_2)/\beta$; (ii) when $\beta = 0$ and the parameter $K = (1-k_1 k_2)/(k_1+k_2)$ is such that $\left |K\right |\neq 1$ then the following relations are valid: $\Re (\nu _n/n) = O\left (1\right )$ and $\Im (\nu _n/n^2) = O\left (1\right )$ as $\left |n\right |\to \infty$; (iii) when $\beta =0$, $|K| = 1$, and $\alpha = 0$, then the following relations are valid: $\Re (\nu _n/n^2) = O\left (1\right )$ and $\Im (\nu _n/n) = O\left (1\right )$ as $\left |n\right |\to \infty$; (iv) when $\beta =0$, $|K| = 1$, and $\alpha>0$, then the following relations are valid: $\Re (\nu _n/\ln \left |n\right |) = O\left (1\right )$ and $\Im (\nu _n/n^2) = O\left (1\right )$ as $\left |n\right |\to \infty$.

scholarly journals On the flow of MHD generalized maxwell fluid via porous rectangular duct

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 989-1002
Author(s):  
Aamir Farooq ◽  
Muhammad Kamran ◽  
Yasir Bashir ◽  
Hijaz Ahmad ◽  
Azeem Shahzad ◽  
...  
Keyword(s):  
Fluid Model ◽  
Maxwell Fluid ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Rectangular Duct ◽  
Modified Darcy’S Law ◽  
Initial Boundary Value ◽  
Generalized Maxwell Fluid ◽  
Limiting Case ◽  
The Impact

Abstract The purpose of this proposed investigation is to study unsteady magneto hydrodynamic (MHD) mixed initial-boundary value problem for incompressible fractional Maxwell fluid model via oscillatory porous rectangular duct. Considering the modified Darcy’s law, the problem is simplified by using the method of the double finite Fourier sine and Laplace transforms. As a limiting case of the general solutions, the same results can be obtained for the classical Maxwell fluid. Also, the impact of magnetic parameter, porosity of medium, and the impact of various material parameters on the velocity profile and the corresponding tangential tensions are illuminated graphically. At the end, we will give the conclusion of the whole paper.

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