Search results for: Convection-diffusion equations

Authors: A. Pashayev, D. Askerov, R. Sadiqov, A. Samedov, C. Ardil

Abstract:

In contrast to existing methods which do not take into account multiconnectivity in a broad sense of this term, we develop mathematical models and highly effective combination (BIEM and FDM) numerical methods of calculation of stationary and quasi-stationary temperature field of a profile part of a blade with convective cooling (from the point of view of realization on PC). The theoretical substantiation of these methods is proved by appropriate theorems. For it, converging quadrature processes have been developed and the estimations of errors in the terms of A.Ziqmound continuity modules have been received. For visualization of profiles are used: the method of the least squares with automatic conjecture, device spline, smooth replenishment and neural nets. Boundary conditions of heat exchange are determined from the solution of the corresponding integral equations and empirical relationships. The reliability of designed methods is proved by calculation and experimental investigations heat and hydraulic characteristics of the gas turbine first stage nozzle blade.

Keywords: Gas turbine, cooled blade, nozzle blade, temperature field.

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Authors: Mohamed Zakaulla, A. R. Anwar Khan

Abstract:

Al6061 alloy base matrix, reinforced with particles of silicon carbide (10 wt %) and Graphite powder (1wt%), known as hybrid composites have been fabricated by liquid metallurgy route (stir casting technique) and optimized at different parameters like applied load, sliding speed and sliding distance by taguchi method. A plan of experiment generated through taguchi technique was used to perform experiments based on L27 orthogonal array. The developed ANOVA and regression equations are used to find the optimum coefficient of friction and wear under the influence of applied load, sliding speed and sliding distance. On the basis of “smaller the best” the dry sliding wear resistance was analysed and finally confirmation tests were carried out to verify the experimental results.

Keywords: Analysis of variance, dry sliding wear, Hybrid composite, orthogonal array, Taguchi technique.

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Authors: M.R.Alinaghizadehand, M.R.Isvandzibaei

Abstract:

Study of the vibration cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel is important. Material properties are graded in the thickness direction of the shell according to volume fraction power law distribution. The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of boundary conditions on the natural frequencies of the FG cylindrical shell. The study is carried out using third order shear deformation shell theory. The governing equations of motion of FG cylindrical shells are derived based on shear deformation theory. Results are presented on the frequency characteristics, influence of constituent volume fractions and the effects of clampedclamped boundary conditions.

Keywords: Vibration, FGM, Cylindrical shell, Hamilton's principle.

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Authors: A. Errasti, F. Doffagne, O. Foucrier, S. Kao, A. Meigne, H. Pellae, T. Rouland

Abstract:

Renewable energy recovery is an important domain of research in past few years in view of protection of our ecosystem. Several industrial companies are setting up widespread recovery systems to exploit wave energy. Most of them have a large size, are implanted near the shores and exploit current flows. However, as oceans represent 70% of Earth surface, a huge space is still unexploited to produce energy. Present analysis focuses on surface small scale wave energy recovery. The principle is exactly the opposite of wheel damper for a car on a road. Instead of maintaining the car body as non-oscillatory as possible by adapted control, a system is designed so that its oscillation amplitude under wave action will be maximized with respect to a boat carrying it in view of differential potential energy recuperation. From parametric analysis of system equations, interesting domains have been selected and expected energy output has been evaluated.

Keywords: Small scale wave, potential energy, optimized energy recovery, auto-adaptive system.

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Authors: G.H. Farrahi, S.A. Faghidian, D.J. Smith

Abstract:

A thin layer on the component surface can be found with high tensile residual stresses, due to turning operations, which can dangerously affect the fatigue performance of the component. In this paper an analytical approach is presented to reconstruct the residual stress field from a limited incomplete set of measurements. Airy stress function is used as the primary unknown to directly solve the equilibrium equations and satisfying the boundary conditions. In this new method there exists the flexibility to impose the physical conditions that govern the behavior of residual stress to achieve a meaningful complete stress field. The analysis is also coupled to a least squares approximation and a regularization method to provide stability of the inverse problem. The power of this new method is then demonstrated by analyzing some experimental measurements and achieving a good agreement between the model prediction and the results obtained from residual stress measurement.

Keywords: Residual stress, Limited measurements, Inverse problems, Turning process.

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Authors: Tariq T. Darabseh

Abstract:

The transient thermoelastic response of thick hollow cylinder made of functionally graded material under thermal loading is studied. The generalized coupled thermoelasticity based on the Green-Lindsay model is used. The thermal and mechanical properties of the functionally graded material are assumed to be varied in the radial direction according to a power law variation as a function of the volume fractions of the constituents. The thermal and elastic governing equations are solved by using Galerkin finite element method. All the finite element calculations were done by using commercial finite element program FlexPDE. The transient temperature, radial displacement, and thermal stresses distribution through the radial direction of the cylinder are plotted.

Keywords: Finite element method, thermal stresses, Green-Lindsay theory, functionally graded material.

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Authors: R.Goudarzizadeh, N.Hedayat, S.H.Mousavi Jahromi

Abstract:

Channel junctions can be analyzed in two ways of division (lateral intake) and combined flows (confluence). The present paper investigates 3D flow pattern at lateral intake using Navier-Stokes equation and κ -ε (RNG) turbulent model. The equations are solved by Finite-Volume Method (FVM) and results are compared with the experimental data of (Barkdoll, B.D., 1997) to test the validity of the findings. Comparison of the results with the experimental data indicated a close proximity between the two sets of data which suggest a very close simulation. Results further indicated an inverse relation between the effects of discharge ratio ( r Q ) on the length and width of the separation zone. In other words, as the discharge ration increases, the length and width of separation zone decreases.

Keywords: 900 junction, flow division, turbulent flow, numerical modeling, flow separation zone.

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Authors: Mário C. Ricci

Abstract:

A new, rapidly convergent, numerical procedure for internal loading distribution computation in statically loaded, singlerow, angular-contact ball bearings, subjected to a known combined radial and thrust load, which must be applied so that to avoid tilting between inner and outer rings, is used to find the load distribution differences between a loaded unfitted bearing at room temperature, and the same loaded bearing with interference fits that might experience radial temperature gradients between inner and outer rings. For each step of the procedure it is required the iterative solution of Z + 2 simultaneous nonlinear equations – where Z is the number of the balls – to yield exact solution for axial and radial deflections, and contact angles.

Keywords: Ball, Bearing, Static, Load, Iterative, Numerical, Method, Temperature, Fit.

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Authors: Mansour Al Qubeissi, Sergei S. Sazhin, Cyril Crua, Morgan R. Heikal

Abstract:

This paper presents the application of the Discrete Component Model for heating and evaporation to multi-component biodiesel fuel droplets in direct injection internal combustion engines. This model takes into account the effects of temperature gradient, recirculation and species diffusion inside droplets. A distinctive feature of the model used in the analysis is that it is based on the analytical solutions to the temperature and species diffusion equations inside the droplets. Nineteen types of biodiesel fuels are considered. It is shown that a simplistic model, based on the approximation of biodiesel fuel by a single component or ignoring the diffusion of components of biodiesel fuel, leads to noticeable errors in predicted droplet evaporation time and time evolution of droplet surface temperature and radius.

Keywords: Heat/Mass Transfer, Biodiesel, Multi-component Fuel, Droplet, Evaporation.

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Authors: Z.D. Yang, E.S.S. Lam

Abstract:

Considering the merits and limitations of energy dissipation system, seismic isolation system and suspension system, a new earthquake resistant system is proposed and is demonstrated numerically through a frame-core structure. Base isolators and story isolators are installed in the proposed system. The former “isolates" the frame from the foundation and the latter “separates" the frame from the center core. Equations of motion are formulated to study the response of the proposed structural system to strong earthquake motion. As compared with the fixed-base building system, the proposed structural system shows substantial reduction on structural response.

Keywords: Base Isolator, Core-tube, Isolated frame, Seismic Mitigation, Story Isolator

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Authors: R. Suzana, T. Wardah, A.B. Sahol Hamid

Abstract:

The use of radar in Quantitative Precipitation Estimation (QPE) for radar-rainfall measurement is significantly beneficial. Radar has advantages in terms of high spatial and temporal condition in rainfall measurement and also forecasting. In Malaysia, radar application in QPE is still new and needs to be explored. This paper focuses on the Z/R derivation works of radarrainfall estimation based on rainfall classification. The works developed new Z/R relationships for Klang River Basin in Selangor area for three different general classes of rain events, namely low (<10mm/hr), moderate (>10mm/hr, <30mm/hr) and heavy (>30mm/hr) and also on more specific rain types during monsoon seasons. Looking at the high potential of Doppler radar in QPE, the newly formulated Z/R equations will be useful in improving the measurement of rainfall for any hydrological application, especially for flood forecasting.

Keywords: Radar, Quantitative Precipitation Estimation, Z/R development, flood forecasting

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Authors: I. Abuiziah, A. Oulhaj, K. Sebari, D. Ouazar

Abstract:

The primary objectives of transient analysis are to determine the values of transient pressures that can result from flow control operations and to establish the design criteria for system equipment and devices (such as control devices and pipe wall thickness) so as to provide an acceptable level of protection against system failure due to pipe collapse or bursting. Because of the complexity of the equations needed to describe transients, numerical computer models are used to analyze transient flow hydraulics. An effective numerical model allows the hydraulic engineer to analyze potential transient events and to identify and evaluate alternative solutions for controlling hydraulic transients, thereby protecting the integrity of the hydraulic system. This paper presents the influence of using the protection devices to control the adverse effects due to excessive and low pressure occurs in the transient.

Keywords: Flow Transient, Water hammer, Pipeline System, Surge Tank, Simulation Model, Protection Devices.

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Authors: N. Thongjub, B. Puangkird, V. Ngamaramvaranggul

Abstract:

The numerical simulation of the slip effect via vicoelastic fluid for 4:1 contraction problem is investigated with regard to kinematic behaviors of streamlines and stress tensor by models of the Navier-Stokes and Oldroyd-B equations. Twodimensional spatial reference system of incompressible creeping flow with and without slip velocity is determined and the finite element method of a semi-implicit Taylor-Galerkin pressure-correction is applied to compute the problem of this Cartesian coordinate system including the schemes of velocity gradient recovery method and the streamline-Upwind / Petrov-Galerkin procedure. The slip effect at channel wall is added to calculate after each time step in order to intend the alteration of flow path. The result of stress values and the vortices are reduced by the optimum slip coefficient of 0.1 with near the outcome of analytical solution.

Keywords: Slip effect, Oldroyd-B fluid, slip coefficient, time stepping method.

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Authors: S. Chauhan, T. P. K. Grewal, S. K. Agrawal, V. K. Srivastava

Abstract:

Combustion of fuels in industrial and transport sector has lead to an alarming release of polluting gases to the atmosphere. Carbon monoxide is one such pollutant, which is formed as a result of incomplete oxidation of the fuel. In order to analyze the effect of catalyst on the reduction of CO emissions to the atmosphere, two catalysts Mn2O3 and Hopcalite are considered. A model was formed based on mass and energy balance equations. Results show that Hopcalite catalyst as compared to Mn2O3 catalyst helped in faster conversion of the polluting gas as the operating temperature of the hopcalite catalyst is much lower as compared to the operating temperature of Mn2O3 catalyst.

Keywords: Carbon monoxide, modeling, hopcalite, manganese oxide.

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Authors: O. Ansari, H. Hafs, A. Bah, M. Asbik, M. Malha, M. Bakhouya

Abstract:

Water is one of the most important and vulnerable natural resources due to human activities and climate change. Water-level continues declining year after year and it is primarily caused by sustained, extensive, and traditional usage methods. Improving water utilization becomes an urgent issue in order satisfy the increasing population needs. Desalination of seawater or brackish water could help in increasing water potential. However, a cost-effective desalination process is required. The most appropriate method for performing this desalination is solar-driven distillation, given its simplicity, low cost and especially the availability of the solar energy source. The main objective of this paper is to demonstrate the influence of coupling integrated basin plate by fins with preheating by solar collector on the performance of solar still. The energy balance equations for the various elements of the solar still are introduced. A numerical example is used to show the efficiency of the proposed solution.

Keywords: Active solar still, Brackisch water, desalination, fins, solar collector.

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Authors: А. Pashayev, C. Ardil, D. Askerov, R. Sadiqov, A. Samedov

Abstract:

In contrast to existing methods which do not take into account multiconnectivity in a broad sense of this term, we develop mathematical models and highly effective combination (BIEM and FDM) numerical methods of calculation of stationary and quasistationary temperature field of a profile part of a blade with convective cooling (from the point of view of realization on PC). The theoretical substantiation of these methods is proved by appropriate theorems. For it, converging quadrature processes have been developed and the estimations of errors in the terms of A.Ziqmound continuity modules have been received. For visualization of profiles are used: the method of the least squares with automatic conjecture, device spline, smooth replenishment and neural nets. Boundary conditions of heat exchange are determined from the solution of the corresponding integral equations and empirical relationships. The reliability of designed methods is proved by calculation and experimental investigations heat and hydraulic characteristics of the gas turbine first stage nozzle blade.

Keywords: Mathematical Modeling, Gas Turbine Blade Cooling, Neural Networks, BIEM and FDM.

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Authors: S. Sarkar, R. Mandal

Abstract:

The three-dimensional incompressible flow past a rectangular open cavity is investigated, where the aspect ratio of the cavity is considered as 4. The principle objective is to use large-eddy simulation to resolve and control the large-scale structures, which are largely responsible for flow oscillations in a cavity. The flow past an open cavity is very common in aerospace applications and can be a cause of acoustic source due to hydrodynamic instability of the shear layer and its interactions with the downstream edge. The unsteady Navier-stokes equations have been solved on a staggered mesh using a symmetry-preserving central difference scheme. Synthetic jet has been used as an active control to suppress the cavity oscillations in wake mode for a Reynolds number of ReD = 3360. The effect of synthetic jet has been studied by varying the jet amplitude and frequency, which is placed at the upstream wall of the cavity. The study indicates that there exits a frequency band, which is larger than a critical value, is effective in attenuating cavity oscillations when blowing ratio is more than 1.0.

Keywords: Cavity oscillation, Large Eddy Simulation, Synthetic Jet, Flow Control, Turbulence

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Authors: B. K. Gill, Himani Sharma, V. K. Rattan

Abstract:

Experimental data of refractive index, excess molar volume and viscosity of binary mixture of morpholine with cumene over the whole composition range at 298.15 K, 303.15 K, 308.15 K and normal atmospheric pressure have been measured. The experimental data were used to compute the density, deviation in molar refraction, deviation in viscosity and excess Gibbs free energy of activation as a function of composition. The experimental viscosity data have been correlated with empirical equations like Grunberg- Nissan, Herric correlation and three body McAllister’s equation. The excess thermodynamic properties were fitted to Redlich-Kister polynomial equation. The variation of these properties with composition and temperature of the binary mixtures are discussed in terms of intermolecular interactions.

Keywords: Cumene, excess Gibbs free energy, excess molar volume, morpholine.

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Authors: Jeong-Hyeon Kim, Ki-Yeob Kang, Myung-Hyun Kim, Jae-Myung Lee

Abstract:

Primary barrier of membrane type LNG containment system consist of corrugated 304L stainless steel. This 304L stainless steel is austenitic stainless steel which shows different material behaviors owing to phase transformation during the plastic work. Even though corrugated primary barriers are subjected to significant amounts of pre-strain due to press working, quantitative mechanical behavior on the effect of pre-straining at cryogenic temperatures are not available. In this study, pre-strain level and pre-strain temperature dependent tensile tests are carried to investigate mechanical behaviors. Also, constitutive equations with material parameters are suggested for a verification study.

Keywords: Constitutive equation, corrugated sheet, pre-strain effect, elasto-visco-plastic-damage model, 304L stainless steel.

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Authors: V. D. Thi, M. Li, M. Khelifa, M. El Ganaoui, Y. Rogaume

Abstract:

This paper presents a two-dimensional model to study the heat and moisture transfer through porous building materials. Dynamic and static coupled models of heat and moisture transfer in porous material under low temperature are presented and the coupled models together with variable initial and boundary conditions have been considered in an analytical way and using the finite element method. The resulting coupled model is converted to two nonlinear partial differential equations, which is then numerically solved by an implicit iterative scheme. The numerical results of temperature and moisture potential changes are compared with the experimental measurements available in the literature. Predicted results demonstrate validation of the theoretical model and effectiveness of the developed numerical algorithms. It is expected to provide useful information for the porous building material design based on heat and moisture transfer model.

Keywords: Finite element method, heat transfer, moisture transfer, porous materials, wood.

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Authors: Evgeniy Burlutskiy

Abstract:

The paper presents a one-dimensional transient mathematical model of compressible thermal multi-component gas mixture flows in pipes. The set of the mass, momentum and enthalpy conservation equations for gas phase is solved. Thermo-physical properties of multi-component gas mixture are calculated by solving the Equation of State (EOS) model. The Soave-Redlich-Kwong (SRK-EOS) model is chosen. Gas mixture viscosity is calculated on the basis of the Lee-Gonzales-Eakin (LGE) correlation. Numerical analysis on rapid decompression in conventional dry gases is performed by using the proposed mathematical model. The model is validated on measured values of the decompression wave speed in dry natural gas mixtures. All predictions show excellent agreement with the experimental data at high and low pressure. The presented model predicts the decompression in dry natural gas mixtures much better than GASDECOM and OLGA codes, which are the most frequently-used codes in oil and gas pipeline transport service.

Keywords: Mathematical model, Rapid Gas Decompression

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Authors: M. Fakharian, M. I. Khodakarami

Abstract:

In this paper, a new trend for improvement in semianalytical method based on scale boundaries in order to solve the 2D elastodynamic problems is provided. In this regard, only the boundaries of the problem domain discretization are by specific subparametric elements. Mapping functions are uses as a class of higherorder Lagrange polynomials, special shape functions, Gauss-Lobatto- Legendre numerical integration, and the integral form of the weighted residual method, the matrix is diagonal coefficients in the equations of elastodynamic issues. Differences between study conducted and prior research in this paper is in geometry production procedure of the interpolation function and integration of the different is selected. Validity and accuracy of the present method are fully demonstrated through two benchmark problems which are successfully modeled using a few numbers of DOFs. The numerical results agree very well with the analytical solutions and the results from other numerical methods.

Keywords: 2D Elastodynamic Problems, Lagrange Polynomials, G-L-Lquadrature, Decoupled SBFEM.

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Authors: Mohamed M. Mousa, Aidarkhan Kaltayev

Abstract:

The equations governing the flow of an electrically conducting, incompressible viscous fluid over an infinite flat plate in the presence of a magnetic field are investigated using the homotopy perturbation method (HPM) with Padé approximants (PA) and 4th order Runge–Kutta method (4RKM). Approximate analytical and numerical solutions for the velocity field and heat transfer are obtained and compared with each other, showing excellent agreement. The effects of the magnetic parameter and Prandtl number on velocity field, shear stress, temperature and heat transfer are discussed as well.

Keywords: Electrically conducting elastico-viscous fluid, symmetry solution, Homotopy perturbation method, Padé approximation, 4th order Runge–Kutta, Maple

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Authors: Rabah Haoui

Abstract:

This work presents a numerical simulation of the interaction of an incident shock wave propagates from the left to the right with a cone placed in a tube at shock. The Mathematical model is based on a non stationary, viscous and axisymmetric flow. The Discretization of the Navier-stokes equations is carried out by the finite volume method in the integral form along with the Flux Vector Splitting method of Van Leer. Here, adequate combination of time stepping parameter, CFL coefficient and mesh size level is selected to ensure numerical convergence. The numerical simulation considers a shock tube filled with air. The incident shock wave propagates to the right with a determined Mach number and crosses the cone by leaving behind it a stationary detached shock wave in front of the nose cone. This type of interaction is observed according to the time of flow.

Keywords: Supersonic flow, viscous flow, finite volume, cone body

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Authors: Yeongyu Choi, Ju H. Park, Ho-Youl Jung

Abstract:

In this paper, we propose labeling based RANSAC algorithm for lane detection. Advanced driver assistance systems (ADAS) have been widely researched to avoid unexpected accidents. Lane detection is a necessary system to assist keeping lane and lane departure prevention. The proposed vision based lane detection method applies Canny edge detection, inverse perspective mapping (IPM), K-means algorithm, mathematical morphology operations and 8 connected-component labeling. Next, random samples are selected from each labeling region for RANSAC. The sampling method selects the points of lane with a high probability. Finally, lane parameters of straight line or curve equations are estimated. Through the simulations tested on video recorded at daytime and nighttime, we show that the proposed method has better performance than the existing RANSAC algorithm in various environments.

Keywords: Canny edge detection, k-means algorithm, RANSAC, inverse perspective mapping.

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Authors: Liu Yi, Bao Liyong

Abstract:

Aiming at the problem of low efficiency of pipelined scheduling in periodic query-qualified service, this paper proposes a system service resource scheduling strategy with parallel optimized qualified service polling control. The paper constructs the polling queuing system and its mathematical model; firstly, the first-order and second-order characteristic parameter equations are obtained by partial derivation of the probability mother function of the system state variables, and the complete mathematical, analytical expressions of each system parameter are deduced after the joint solution. The simulation experimental results are consistent with the theoretical calculated values. The system performance analysis shows that the average captain and average period of the system have been greatly improved, which can better adapt to the service demand of delay-sensitive data in the dense data environment.

Keywords: Polling, parallel scheduling, mean queue length, average cycle time.

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Authors: Rabah Haoui

Abstract:

The aim of this work is to analyze a viscous flow in the axisymmetric nozzle taken into account the mesh size both in the free stream and into the boundary layer. The resolution of the Navier- Stokes equations is realized by using the finite volume method to determine the supersonic flow parameters at the exit of convergingdiverging nozzle. The numerical technique uses the Flux Vector Splitting method of Van Leer. Here, adequate time stepping parameter, along with CFL coefficient and mesh size level is selected to ensure numerical convergence. The effect of the boundary layer thickness is significant at the exit of the nozzle. The best solution is obtained with using a very fine grid, especially near the wall, where we have a strong variation of velocity, temperature and shear stress. This study enabled us to confirm that the determination of boundary layer thickness can be obtained only if the size of the mesh is lower than a certain value limits given by our calculations.

Keywords: Supersonic flow, viscous flow, finite volume, nozzle

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Authors: Makhlouf Mourad, Medkour Mihoub, Bouchher Omar, Messabih Sidi Mohamed, Benrachedi Khaled

Abstract:

This work is the modeling and simulation of fluid flow (liquid) through porous media. This type of flow occurs in many situations of interest in applied sciences and engineering, fluid (oil) consists of several individual substances in pure, single-phase flow is incompressible and isothermal. The porous medium is isotropic, homogeneous optionally, with the rectangular format and the flow is two-dimensional. Modeling of hydrodynamic phenomena incorporates Darcy's law and the equation of mass conservation. Correlations are used to model the density and viscosity of the fluid. A finite volume code is used in the discretization of differential equations. The nonlinearity is treated by Newton's method with relaxation coefficient. The results of the simulation of the pressure and the mobility of liquid flowing through porous media are presented, analyzed, and illustrated.

Keywords: Darcy equation, middle porous, continuity equation, Peng Robinson equation, mobility.

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Authors: A.Memari

Abstract:

In this article, a simulation method called the Homotopy Perturbation Method (HPM) is employed in the steady flow of a Walter's B' fluid in a vertical channel with porous wall. We employed Homotopy Perturbation Method to derive solution of a nonlinear form of equation obtained from exerting similarity transforming to the ordinary differential equation gained from continuity and momentum equations of this kind of flow. The results obtained from the Homotopy Perturbation Method are then compared with those from the Runge–Kutta method in order to verify the accuracy of the proposed method. The results show that the Homotopy Perturbation Method can achieve good results in predicting the solution of such problems. Ultimately we use this solution to obtain the other terms of velocities and physical discussion about it.

Keywords: Steady flow; Walter's B' Fluid;, vertical channel;porous media, Homotopy Perturbation Method (HPM), Numerical Solution (NS).

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Authors: Azizollah Khormali, Seyyed Shahab Tabatabaee Moradi, Dmitry Petrakov

Abstract:

One of the most important parameters in petroleum reservoirs is the pressure distribution along the reservoir, as the pressure varies with the time and location. A popular method to determine the pressure distribution in a reservoir in the unsteady state regime of flow is applying Darcy’s equation and solving this equation numerically. The numerical simulation of reservoirs is based on these numerical solutions of different partial differential equations (PDEs) representing the multiphase flow of fluids. Pressure profile has obtained in a one dimensional system solving Darcy’s equation explicitly. Changes of pressure profile in three situations are investigated in this work. These situations include section length changes, step time changes and time approach to infinity. The effects of these changes in pressure profile are shown and discussed in the paper.

Keywords: Explicit solution, Numerical simulation, Petroleum reservoir, Pressure distribution.

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