Mathematical Model Of Gas Flow Research Articles

EN Fluid dynamic design of centrifugal compressor diffusers

Diffusers are used to convert the kinetic energy of the gas flow into potential energy, i.e. to reduce the velocity and to increase the pressure. The most common types of diffusers are vaneless, vaned, and channel ones. Each type of diffuser has its design features and its performances. Improving the operational cha­racteristics of vaneless diffusers is possible by using stepped diffusers with flow injection. The developed mathematical model of gas flow in a vaneless diffuser with flow injection provides design of the stage of centrifugal compressor with the wider range of stable operation. Traditional methods of designing vaned and channel diffusers of turbomachines are focused on the use of simple geometric lines and surfaces, such as a straight line, arc of a circle, plane, cylindrical surface, etc. The design is performed according to recommendations based on experimental data. The other way of designing the vaned and channel diffusers of turbomachines is related to solving the inverse problem of fluid dynamics, when the shape of the surfaces is determined by the given distribution of velocities along the surfaces of the channel. This paper describes the design principles for the centrifugal compressor diffusers based on physical and mathema­tical models of the flow of swirling viscous compressible fluid. According to the presented method, the designing diffusers are based on the preseparation condition of the boundary layer along one of the surfa­ces. Such a design ensures a reduction in separation zones in the channels of the diffusers and, accor­dingly, a reduction in total pressure losses and an expansion of the range of stable operation. A new method of designing the vaned and channel diffusers provides an improvement in the gasdynamic characteristics of diffusers compared to traditional geometry diffusers

Research on Gas Flow in Vertical Fissure Field of Coal Mine Goaf

The gas in the goaf of a coal mine is difficult to extract compared to the gas in the roadway. Studying the flow pattern of gas in the goaf fracture field is of great significance for optimizing gas extraction measures and ensuring coal mine production safety. Taking a coal mining face in Shaanxi Province, China as the research object, a mathematical model of gas flow was constructed. Through finite element numerical simulation, the pressure and velocity distribution of gas in the goaf fracture field were obtained, and the influence of vertical fracture field on gas flow was analyzed. The research results provide a theoretical basis for coal mine gas control.

Parallel fully implicit chemical potential-based modeling of unconventional shale gas reservoirs

The design of novel mathematical models and state-of-the-art simulators for unconventional shale gas reservoirs plays an increasingly important role in the current energy mix of the world’s growing energy demand. In this paper, we present a robust and scalable framework, which is based on the chemical potential-based modeling and the fully implicit method (FIM), to model and simulate this highly nonlinear flow transport problem on a large scale. In the proposed approach, a thermodynamically consistent mathematical model of gas flow in shale formations, which employs the gas density and the chemical potential gradient, is developed to satisfy the second law of thermodynamics and meanwhile guarantee the energy dissipation property. And then the family of fully implicit finite element algorithms is utilized to accurately capture the complicated flow physics behind the transport process in shale media on high resolution grids. In particular, our approach further enhances the numerical formulation by proposing the family of Newton–Krylov methods for efficiently computing, and the parallel implementation of the simulator is achieved by using the domain decomposition technique. Numerical experiments are presented to demonstrate the robustness and parallel scalability of the solution strategies for several interesting shale gas flow problems in two or three dimensions. With the proposed parallel method, large-scale reservoir simulation can be obtained on the Shaheen-II supercomputer with up to 40960 processors, which enables to achieve a good strong scalability by saving more than 90 percent of computing time, when the problem size is enlarged to hundreds of millions of degrees of freedom.

Building a mathematical model to prevent hydrate formation in gas pipelines

Development of mathematical models of laminar gas flow in certainty and uncertainty conditions were considered. All factors that influence to character of flow of gas in pipeline and conditions of arising of hydrate inside of pipeline wall are analyzed. Results of analyze were used for development mathematical model of gas flow in pipeline that allow to calculate main parameters of hydrate deposition process. Model of gas flow consist of three non-linear differential equations that can be solved by exist soft wares. Two and three-dimension characteristic obtained, that describe of quantity of hydrate deposited at given process depending on time.
 Besides, the effectiveness of using DELPHI 7 software to determine the preparation of gas for transportation and the hydrates formed during transportation and its prevention based on the results of the application software was discussed. As a result, the change in cross-sectional area of the pipeline of hydrates formed on the inner surface of the pipeline is shown in 3D. Hydrate formation and elimination are visually represented by graphs. The results of theoretical and practical studies of changes in the inner surface of the pipeline depending on temperature and pressure have been identified. All this was investigated during quasi-stationary gas flow in the pipelines and the results were obtained.
 The assumes regarding calculation of parameters of gas flow were determined and necessary recommendations for applying of developed model in different conditions with taking account of temperature and pressure variation and depending on time of hydrate deposition were presented. The diagram of gas-hydrate separation boundary considered for detailed analysis of process

Transport Efficiency of a Homogeneous Gaseous Substance in the Presence of Positive and Negative Gaseous Sources of Mass and Momentum

In this article, a theoretical mathematical model of gas flow through a duct in the case of local mass and momentum sources and sinks is presented. The continuity equation and motion equation with one-dimensional, density-stable gas flows were used to create this model. The size of sources and sinks and their locations have an effect on the size of gas stream flows in the duct, gas energy losses, and the parameters of the mechanical source energy that is causing the flow. In the traditional approach to describing the gas flow in the duct, the concept of resistivity and the equivalent resistance of the conduit is used. In the case of flow in the duct with local mass and momentum sources and sinks, the transport resistance depends on a bigger number of parameters than the concept of specific resistance usage. The location and size of the source flux or mass and momentum sinks and the fan work (suction, blowing) were taken into account in the presented model. The model gives the opportunity to determine the mechanical energy losses and efficiency of gas transport in the duct.

Chemical Potential-Based Modeling of Shale Gas Transport

Shale gas plays an increasingly important role in the current energy industry. Modeling of gas flow in shale media has become a crucial and useful tool to estimate shale gas production accurately. The second law of thermodynamics provides a theoretical criterion to justify any promising model, but it has been never fully considered in the existing models of shale gas. In this paper, a new mathematical model of gas flow in shale formations is proposed, which uses gas density instead of pressure as the primary variable. A distinctive feature of the model is to employ chemical potential gradient rather than pressure gradient as the primary driving force. This allows to prove that the proposed model obeys an energy dissipation law, and thus, the second law of thermodynamics is satisfied. Moreover, on the basis of energy factorization approach for the Helmholtz free energy density, an efficient, linear, energy stable semi-implicit numerical scheme is proposed for the proposed model. Numerical experiments are also performed to validate the model and numerical method.

Distribution of reformed coke oven gas in a shaft furnace

In recent years, the reformed coke oven gas (COG) was proposed to be used as reducing gas in a shaft furnace. A mathematical model of gas flow based on the reformed COG was built. The effects of the pressure ratio of reducing gas to cooling gas (k) on the gas distribution in the shaft furnace were investigated. The calculation results show that k is an important operation parameter, which can obviously affect the gas distribution in the shaft furnace. The value of k should be compromised. Both too big and too small k values are not appropriate, and the most reasonable value for k is 1:1.33. Under this condition, the utilization coefficient of reducing gas, the utilization coefficient of cooling gas and the coefficient of upward gas are 0.94, 0.92 and 1.03, respectively. Based on the validation of physical experiments, the calculated values of the model agreed well with the physical experimental data. Thus, the established model can properly describe the reformed COG distribution in an actual shaft furnace.

Multiscale Apparent Permeability Model of Shale Nanopores Based on Fractal Theory

Based on fractal geometry theory, the Hagen–Poiseuille law, and the Langmuir adsorption law, this paper established a mathematical model of gas flow in nano-pores of shale, and deduced a new shale apparent permeability model. This model considers such flow mechanisms as pore size distribution, tortuosity, slippage effect, Knudsen diffusion, and surface extension of shale matrix. This model is closely related to the pore structure and size parameters of shale, and can better reflect the distribution characteristics of nano-pores in shale. The correctness of the model is verified by comparison with the classical experimental data. Finally, the influences of pressure, temperature, integral shape dimension of pore surface and tortuous fractal dimension on apparent permeability, slip flow, Knudsen diffusion and surface diffusion of shale gas transport mechanism on shale gas transport capacity are analyzed, and gas transport behaviors and rules in multi-scale shale pores are revealed. The proposed model is conducive to a more profound and clear understanding of the flow mechanism of shale gas nanopores.

A mathematical model of gas flow during coal outburst initiation

A proposed concept of outburst initiation examines the release of a large amount of gas from coal seams resulted from disintegrating thermodynamically unstable coal organic matter (COM). A coal microstructure is assumed to getting unstable due to shear component appearance triggered by mining operations and tectonic activities considered as the primary factor while COM disintegration under the impact of weak electric fields can be defined as a secondary one. The energy of elastic deformations stored in the coal microstructure activates chemical reactions to tilt the energy balance in a “coal–gas” system. Based on this concept a mathematical model of a gas flow in the coal where porosity and permeability are changed due to chemical reactions has been developed. Using this model we calculated gas pressure changes in the pores initiated by gas release near the working face till satisfying force and energy criteria of outburst. The simulation results demonstrated forming overpressure zone in the area of intensive gas release with enhanced porosity and permeability. The calculated outburst parameters are well combined with those evaluated by field measurements.

Determination of Particular Endogenous Fires Hazard Zones in Goaf with Caving of Longwall

Hazard of endogenous fires is one of the basic and common presented occupational safety hazards in coal mine in Poland and in the world. This hazard means possibility of coal self-ignition as the result of its self-heating process in mining heading or its surrounding. In underground coal-mining during ventilating of operating longwalls takes place migration of parts of airflow to goaf with caving. In a case when in these goaf a coal susceptible to selfignition occurs, then the airflow through these goaf may influence on formation of favourable conditions for coal oxidation and subsequently to its self-heating and self-ignition. Endogenous fire formed in such conditions can pose a serious hazard for the crew and for continuity of operation of mining plant. From the practical point of view, a very significant meaning has determination of the zone in the goaf with caving, in which necessary conditions for occurrence of endogenous fire are fulfilled. In the real conditions determination of such a zone is practically impossible. Therefore, authors of paper developed a methodology of determination of this zone basing on the results of modelling tests. This methodology includes a development of model of tested area, determination of boundary conditions and carrying out the simulation calculations. Based on the obtained results particular hazardous zone of endogenous fire is determined. A base for development of model of investigated region and selection of boundary conditions are the results of real tests. In the paper fundamental assumption of developed methodology, particularly in a range of assumed hazard criterion and sealing coefficient of goaf with caving were discussed. Also a mathematical model of gas flow through the porous media was characterized. Example of determination of a zone particularly endangered by endogenous fire for real system of mining heading in one of the hard coal mine was presented. Longwall ventilated in the „Y” system was subjected to the tests. For determined mining-geological conditions, the critical value of velocity of airflow and oxygen concentration in goaf, conditioning initiation of coal oxidation process were determined. For calculations ANSYS Fluent software based on finite volume method, which enable very precisely to determine the physical and chemical air and parameters at any point of tested mining heading and goaf with caving was used. Such precisely determination of these parameters on the base of the test in real conditions is practically impossible. Obtained results allowed to take early proper actions in order to limit the occurrence of endogenous fire. One can conclude, that presented methodology creates great possibilities of practical application of modelling tests for improvement of the occupational safety state in mine.

Improving the efficiency of heat supply systems on the basis of plants operating on organic Rankine cycle

Results of experimental and analytical studies of the plant main element – plant turbomachine (turbo-expander) operating on organic Rankine cycle were obtained for facilities of the heat supply systems of small-scale power generation. At simultaneous mathematical modeling and experimental studies it was found that the best working medium to be used in the turbomachines of these plants is Freon R245fa which has the most suitable calorimetric properties to be used in the cycle. The mathematical model of gas flow in the turbomachine was developed. The main engineering dependencies to calculate the optimal design parameters of the turbomachine were obtained. The engineering problems of providing the minimum axial size of the turbomachine impeller were solved and the main design elements were unified.

TO CALCULATION OF GAS FLOW THROUGH RING SEAL SEALS WITH REGULAR DYNAMICS OF PISTON RINGS

The mathematical model of gas flow through the ring seals of the cylinder-piston group which takes into account the throttling effect of the upper piston belt and changes in the incoming sections and volumes of the ring labyrinth caused by the displacement of the piston rings in the piston grooves has been suggested. Tee calculated and experimental data on the displacement of piston rings in the grooves of the piston are presented. There is a significant effect of the behavior of the second compression ring on the dynamic stability of the ring seal. Comparison of the experimental data with the results of calculations shows their satisfactory convergence.

Limit Drainage Radius for Different Types of Wells in a Shale Reservoir

The low porosity and permeability of shale deposits is due to the complex structure of the pores. Due to the threshold pressure gradient, fluid seepage in low-permeable reservoirs does not obey Darcy’s law. Horizontal well drilling and fracturing treatment are effective methods of developing lowpermeable reservoirs, since these methods increase the well drainage area. In the present study a mathematical model of gas flow in different types of wells is obtained on the basis of seepage theory. The pressure distribution and relationship between the threshold pressure gradient and limit well spacing are obtained using the point-source function and Laplace transforms. The limit well spacing increases with decreasing threshold pressure gradient. The method is of importance for determining an appropriate spacing between different combinations of different types of wells in the development of shale gas.

Application of Numerical Laplace Inversion Methods in Chemical Engineering with Maple®

Abstract In this paper, the procedure of choice of the effective numerical inversion method of Laplace transform for solving of the mathematical model of gas flow was shown. Two inversion methods are evaluated in this work: the Gaver-Stehfest method, and the Dubner and Abate method. The numerical tests has been made using test functions. Based on the test results the fast and accurate method was selected and applied for modeling of the gas flow. The Gaver-Stehfest method has proven the most effective. The model was developed using the computer algebra system Maple®. Implementation of the methods were done by authors.

The Application of the Laplace Transform for Modeling of Gas Flow Using Maple¯

AbstractThe Laplace transform is powerful method for solving differential equations. This paper presents the application of Laplace transform to solve the mathematical model of gas flow through the measuring system. The basis of the mathematical model used to describe and simulate the analyzed process is a system of ordinary differential equations. As a result is obtained a single algebraic equation in terms of the complex variable s. In order to study the dynamics of the system these equation should be reverted to the time domain by performing an inverse Laplace transform. Calculations were performed in the environment of Maple?. Determining of mixing in a u-shape vessel is aim of presented calculations.

MATHEMATICAL MODEL AND ANALYTICAL SOLUTIONS FOR UNSTEADY FLOW IN NATURAL GAS RESERVOIRS

The effects of pressure on the gas viscosity and compressibility factor lead to a nonlinear partial differential equation for the flow in a gas reservoir even if the flow process follows Darcy’s law at isothermal conditions. For further study on gas flow performances in gas reservoirs, a mathematical model of gas flow is developed in this article. Exact analytical solutions of one-dimensional unsteady gas flow at low and high pressures in gas reservoirs are obtained by transferring the nonlinear partial differential equation into a nonlinear ordinary differential equation. The numerical solutions obtained by finite difference for two cases of lowand high-pressure condition are given to validate the analytical solutions presented in this work. The key parameters, such as viscosity index, permeability, and porosity, to determine the characteristic of pressure distribution in porous media are analyzed in this work. The solutions at high pressures imply that it leads to obvious errors for prediction pressure distribution when ignoring pressure’s effects on gas viscosity and compressibility factor for gas flow at high pressures in deep gas reservoirs. Both the increase in viscosity index and the decrease in permeability lead to an increase in pressure gradients along the distance.

On a boundary layer problem related to the gas flow in shales

The development of gas deposits in shales has become a significant energy resource. Despite the already active exploitation of such deposits, a mathematical model for gas flow in shales does not exist. Such a model is crucial for optimizing the technology of gas recovery. In the present article, a boundary layer problem is formulated and investigated with respect to gas recovery from porous low-permeability inclusions in shales, which are the basic source of gas. Milton Van Dyke was a great master in the field of boundary layer problems. Dedicating this work to his memory, we want to express our belief that Van Dyke’s profound ideas and fundamental book Perturbation Methods in Fluid Mechanics (Parabolic Press, 1975) will live on—also in fields very far from the subjects for which they were originally invented.

Simulation of gas flow in double-circuit Ranque-Hilsch vortex tube

The mathematical model of gas flow in a Ranque-Hilsch vortex tube implemented in the OpenFOAM environment is described. Six types of one- and two-parametric semi-empirical turbulence models are used to simulate the vortex gas flow. It is shown that there is quantitative and qualitative disagreement between the model characteristics of the vortex tube and the experimental data. The obtained results indicate that the applied turbulence model should be modified.

The influence of gas storage parameters on gas emission rate from borehole

Gas emission rate from borehole is one of the most important indexes for the coal and gas outburst prediction. The mathematical model of gas flow in the coal seam, gas flow into the measuring chamber, gas pressure change in the measuring chamber, and gas flow out of the chamber through the pipe is established. Gas migration in the coal seam, gas pressure in borehole chamber and gas flow in pipe is simulated using the finite difference method. Gas emission rate is obtained under dynamic boundary conditions. The influence of gas storage parameters on gas emission rate from borehole is analyzed. Results show that: the gas pressure and the permeability coefficient have great impacts on the value of gas flow quantity in borehole. The larger the original gas pressure of coal seam and the permeability coefficient of coal seam are, the greater the maximum value of gas emission rate form borehole and the later the maximum appears.

Analyses on 3-D gas flow and heat transfer in ladle furnace lid

In order to verify the reasonableness of off-gas pressure and wall temperature, a mathematical model for gas flow and heat transfer in ladle furnace (LF) lid is developed based on 3-D Navier–Stokes equations and k– ε two equation turbulent models as well as energy conservation equation. The gas velocity vector distribution of skirt clearance between the top edge of ladle and furnace lid and electrode gaps between three graphite electrodes and furnace lid, the gas flow line distribution, pressure and temperature distribution on the furnace lid wall are simulated. Simulation results show that appropriate off-gas pressures are 200 Pa, 200 Pa and 150 Pa when electric arc emerges from molten steel surface and alloy hole is unsealed, electric arc emerges from molten steel surface and alloy hole is closure, electric arc immerges into molten steel surface and alloy hole is closure, respectively. The maximum temperature presents in the middle of LF lid in all of heating conditions, and the temperature value are 563, 603 and 343 K. Finally, the relations between gas volume and off-gas pressure are analyzed in different width of skirt clearance, and some relevant mathematical expressions are obtained. By comparing both simulation results and practical data, the advice on reducing off-gas pressure is proposed, and the maximum temperatures of furnace lid wall have good agreement with actual data.