One-dimensional Gas Flow Research Articles

Analytical Study of Weak Shock Waves in Gas with dust particles

In the present paper, the exact solution of quasilinear hyperbolic system of equations governing the propagation of weak shock waves in a one-dimensional dusty adiabatic gas flow with generalized geometries is derived. Here, the density behind the shock front is assumed to vary according to a power law of the distance. An analytical expression for the total energy carried by weak shock wave in dusty gas is also derived.

Transient pressure behavior for unconventional gas wells with finite-conductivity fractures

Horizontal wells and hydraulic fracturing technologies are the only way that makes commercial production from the low and ultra-low permeability unconventional plays possible. The development of analytical and semi-analytical tools to analyze transient behavior of multi-fractured horizontal gas wells have been largely biased towards solving the simplified problem of one-dimensional (1D) Cartesian gas flow in matrix domain, which are only applicable to the production from infinite-conductivity fractures. This study presents a novel semi-analytical method for transient behavior analysis of horizontal gas wells producing from finite-conductivity fractures. Unlike the available transient analysis tools for finite-conductivity fractures which are developed based on empirical and approximate solution, the proposed solution is rigorously derived by solving the governing gas diffusivity equations written for gas flow in two contiguous regions of matrix and fracture simultaneously. Nonlinear, pressure-dependent gas properties remaining in pseudopressure-based diffusivity equations—namely viscosity and compressibility—are rigorously and straightforwardly captured in both fracture and matrix systems. The validity of proposed solution is verified against commercial numerical simulator (COMSOL Multiphysics®) for a series of synthetic cases considering constant and variable -rate production constraints. The case studies presented in this paper also showcase the applicability of proposed solution to the analysis of transient behavior of multi-fractured horizontal gas wells in shale plays under the following two state-of-the-art interpretations: 1) production from finite-conductivity hydraulic fractures in a single-porosity matrix; and 2) production from infinite-conductivity hydraulic fractures in a naturally-fractured matrix.

Gas dynamic calculation of detonation in constant-cross-section ducts

The paper presents a computational method with the use of gas-dynamic functions of parameters of detonation in a one-dimensional subsonic flow of ideal gas behind the shock wave propagating in chemically active air-and-fuel mixture in a uniform-cross-section duct, where the resultant of normal pressure forces acting on its side surface is equal to zero. Stabilization of the shock wave is provided by the onset of thermal crisis with the air-and-fuel mixture combustion heat supply to the gas behind the wave. In this case the value of the specific speed of the combustion products is equal to the critical one. The solution of the total-impulse equation considering the above mentioned peculiarities of the flow in a uniform-cross-section duct establishes clear correlation of the specific speed of the stabilized shock wave to the rate of rise of the temperature of the gas behind it, which gives an opportunity to determine all detonation parameters. The shock wave can be initiated by the detonation of an explosive substance and carries a huge amount of energy. It is shown that the shock wave can be obtained only if the source of small disturbances itself moves at the supersonic speed. It is shown that total pressure behind the shock wave decreases significantly and the entropy increases due to the wave losses, whereas the static pressure increases significantly. An explanation of this effect is given. A formula for calculating the rate of gas temperature rise was derived as a function of the specific speed of the shock wave, the air-and-fuel mixture heat value and the heat availability factor that designates the dissociation of the combustion products and heat loss through the duct wall under specified initial conditions. The reliability of the method of calculating detonation was experimentally substantiated. The work is currently important for the evaluation of the detonation engine efficiency.

Shock wave structure in a non-ideal gas under temperature and density-dependent viscosity and heat conduction

The structure of a shock wave is investigated using the continuum hypothesis for steady one-dimensional flow of a viscous non-ideal gas under heat conduction. The coefficients of viscosity and heat conductivity are assumed to be directly proportional to a power of the temperature and density of the gas. The simplified van der Waals equation of state for the non-ideal gas has been assumed in this work. Qualitative analysis of shock wave structure has been done in terms of singularity analysis, isoclines, and integral curves. The exact and numerical solutions of shock structure equations are obtained under the quantitative analysis. The validation of solution is established by comparing the results in the literature (Iannelli in Int J Numer Methods Fluids 72(2):157–176, 2013). The variation of normalized gas velocity, viscous stress, heat flux, and shock thickness have been investigated across shock transition zone with the non-idealness of the gas, temperature, and density exponents in the viscosity and heat conductivity of the gas and initial Mach number. It is found that gas velocity decreases significantly with the increase in non-idealness parameter, temperature, and density exponent in the viscosity of the gas. Shock wave thickness decreases with the increase in the non-idealness of the gas under constant viscosity and heat conductivity but increase under variable gas properties. The thickness of a shock wave decreases with the increase in the temperature exponent and increases with the increase in the density exponent.

A Generalized Approach to the Analysis of Feed Unevenness in Displacement Pumps

A generalized approach to the analysis of feed unevenness in positive displacement pumps is considered. A procedure for calculating the working processes in chambers of constant or variable volume in a discharge line as well as a nonstationary one-dimensional gas flow in a discharge pipe is developed. Boundary conditions are defined. Nonstationary one-dimensional flow is calculated by the method of “characteristics.” By means of a numerical experiment, variations in the pressure in the working chambers and along the length of the pipe as a function of the turning angle of the shaft are established. On the basis of pressure fluctuation in a discharge pipe, it is proved that the feed unevenness significantly exceeds the magnitude of the feed unevenness determined when the pressure fluctuation is not taken into account.

Steady flows, nonlinear gravitostatic waves, and Zeldovich pancakes in a Newtonian gas

We show that equations of Newtonian hydrodynamics and gravity describing one-dimensional steady gas flow possess nonlinear periodic solutions. In the case of a zero-pressure gas the solution exhibits hydrodynamic similarity and is universal: it is a lattice of integrable density singularities coinciding with maxima of the gravitational potential. With finite pressure effects included, there exists critical matter density that separates two regimes of behavior. If the average density is below the critical, the solution is a density wave which is in phase with the wave of the gravitational potential. If the average density is above the critical, the waves of the density and potential are out of phase. Traveling plane gravitostatic waves are also predicted and their properties elucidated. Specifically, subsonic wave is made out of two out of phase oscillations of matter density and gravitational potential. If the wave is supersonic, the density-potential oscillations are in phase.

КВАЗИ-ЛАГРАНЖЕВОЕ ИНТЕГРИРОВАНИЕ ДВУМЕРНЫХ УРАВНЕНИЙ ПРАНДТЛЯ И БУССИНЕСКА В НЕВЯЗКОМ ПРЕДЕЛЕ

For the hydrodynamic type systems locally describing incompressible twodimensional fluid flows without dissipation we suggest quasi-Lagrangian approach for their integration. This method is based on application of incomplete Legendre transformation when the independent variables become a Lagrangian invariant (i.e. the constant quantity along the fluid particle trajectory), instead of one of the spatial coordinate, and the rest ones which are another spatial coordinate and time. Thus, this method is based on the inverse transform of one of the spatial coordinate and by this reason differs from the the complete Legendre transformation. The classical example of the complete Legendre transformation is the Hodograph transformation applying to solve the equations for one-dimensional isoentropycal gas flows. In this paper it is shown that equation for the Lagrangian invariant after applying the incomplete Legendre transformation and introducing the stream function transforms into the linear eqation can be resolved by means of the generating function introduction. This method is turned to be effective for solving the inviscid twodimensional Prandtl equation (this equation describes the boundary layer behavior) that allows one to integrate this equation completely. In the case of the constant pressure along the boundary the parallel velocity component represents the Lagrangian invariant. The obtained solution is written through the initial data and satisfies the non-penetrate boundary condition. Analysis of this solution shows the formation of the singularity for the velocity gradient on the wall. This singularity appears as the result of breaking. At the breaking point the velocity gradient tends to infinity according to the power ~(t0–t)-1 where t0 is the singular time. This solution describes the appearance of the folding type singularity. It is shown also that the Prandtl equation admits complete integration for arbitrary dependence of pressure on the longitudinal coordinate. The simplest solution is written for the case of the constant pressure gradient. For the Boussinesq system the equation for density can be resolved by this method that reduces to one equation for the generating function. This work was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».

Riemann problem for non-ideal polytropic magnetogasdynamic flow

The main motive of the present paper is to derive the analytical solution of the Riemann problem for magnetogasdynamic equations governing an inviscid unsteady one-dimensional flow of non-ideal polytropic gas subjected to the transverse magnetic field with infinite electrical conductivity. By using the Lax entropy condition and R–H conditions, we derive the elementary wave solutions i.e. shock wave, simple wave and contact discontinuities without any restriction on the magnitude of initial data states and discussed about their properties. Further, the density and velocity distribution in the flow field for the cases of compressive wave and rarefaction wave is discussed. Here we also compare/contrast the nature of solution in non-ideal magnetogasdynamic flow and ideal gas flow.

Influence of gas compressibility on gas flow through bed of granular phase change material

One-dimensional time-dependent gas flow through a bed of granular phase change material (PCM) is considered. Such process takes place in thermal energy storage (TES) based on granulated or encapsulated PCM. Using a novel numerical model, the influence of gas compressibility on considered processes is investigated. The study is carried out in a wide range of problem parameters for both known gas pressure at the open boundaries of the bed and known gas flow rate at the object inlet. It is shown that for some problem parameters the neglect of the gas compressibility can lead to large errors in the solution of the problem, and it can cause significant inaccuracies in calculating the full heating time and the full melting time of the granular PCM in TES. Moreover, the neglect of the gas compressibility cannot give guaranteed upper or lower bound for the real time of the process. So, for some problem parameters the gas compressibility must be taken into account to model the TES with granulated or encapsulated PCM more adequately and accurately.

Exact solution of shock wave structure in a non-ideal gas under constant and variable coefficient of viscosity and heat conductivity

This paper investigates the structure of a normal shock wave using the continuum model for steady one-dimensional flow of a viscous non-ideal gas under heat conduction. The coefficients of viscosity and heat conductivity are assumed to be directly proportional to a power of the temperature. The simplified van der Waals equation of state for the non-ideal gas has been assumed in this work. The smooth and rough sphere models of the gas molecules in the kinetic theory of gases are used for the viscosity of a non-ideal gas. Assuming the Prandtl number to be 3 / 4, the complete integral of the energy equation, exact velocity, density, pressure, Mach number, change in entropy, viscous stress, and heat flux across the shock transition zone have been obtained in a perfect and a non-ideal gas under both constant and variable properties of the medium. The validity of the continuum hypothesis with respect to Mach number is examined for the study of shock wave structure in both the smooth and rough sphere models of ideal and non-ideal gas molecules. It has been shown that the continuum theory gives reasonably valid results for flows with higher Mach numbers in the case of a non-ideal gas in comparison with an ideal gas. The inverse thickness of the shock wave is calculated and compared for constant and variable properties of the gases. The shock wave thickness is also discussed as a function of mean free path of the gas molecules computed at different points between the boundary states. It is found that the inverse shock thickness decreases with the increase in non-idealness of the gas. In the rough sphere model of gas molecules, the increase in the non-idealness of the gas and the temperature exponent in the coefficients of viscosity and heat conductivity significantly increases the validity limit of the continuum model.

A New Lumped Parameter Model for Natural Gas Pipelines in State Space

Many algorithms and numerical methods, such as implicit and explicit finite differences and the method of characteristics, have been applied for transient flow in gas pipelines. From a computational point of view, the state space model is an effective method for solving complex transient problems in pipelines. However, the impulse output of the existing models is not the actual behavior of the pipeline. In this paper, a new lumped parameter model is proposed to describe the inertial nature of pipelines with inlet/outlet pressure and flow rate as outer variables in the state space. Starting from the basic mechanistic partial differential equations of the general one-dimensional compressible gas flow dynamics under isothermal conditions, the transfer functions are first acquired as the fundamental work. With Taylor-expansion and a transformation procedure, the inertia state space models are derived with proper simplification. Finally, three examples are used to illustrate the effectiveness of the proposed model. With the model, a real-time automatic scheduling scheme of the natural gas pipeline could be possible in the future.

Gas-Flow Perturbation by a High-Power Heat Source

We consider a gas-dynamic model of a one-dimensional flow of non-viscous perfect gas in the absence of heat conduction when a high-power localized heat source is switched on. Within the framework of this model, it is possible to thoroughly analyze the problem of the gas-flow regimes, which occur as a result of switch-on of such a source, in a wide range of values of the energy-deposition power and initial gas-dynamic characteristics of the flow. Eventually, we obtain a sufficiently general insight into the character of the resulting flows and the conditions of their realizations.

On numerical model of one-dimensional time-dependent gas flows through bed of encapsulated phase change material

Mathematical model and numerical method are proposed for investigating the one-dimensional time-dependent gas flows through a packed bed of encapsulated Phase Change Material (PCM). The model is based on the assumption of interacting interpenetrating continua and includes equations of state, continuity, momentum conservation and energy for PCM and gas. The advantage of the method is that it does not require predicting the location of phase transition zone and can define it automatically as in a usual shock-capturing method. One of the applications of the developed numerical model is the simulation of novel Adiabatic Compressed Air Energy Storage system (A-CAES) with Thermal Energy Storage subsystem (TES) based on using the encapsulated PCM in packed bed. Preliminary test calculations give hope that the method can be effectively applied in the future for modelling the charge and discharge processes in such TES with PCM.

Similarity Solutions of Cylindrical Shock Waves in Non-Ideal Magnetogasdynamics with Thermal Radiation

In the present work we have taken one-dimensional unsteady flow of non-ideal gas with magnetic effect under the presence of thermal radiation. The system is hyperbolic in nature and solved by similarity method using Lie Group of Transformations under the assumption that the system is constantly conformally invariant under the transformations. The similarity solutions are investigated behind a cylindrical shock which is a consequence of a sudden explosion or produced by an expanding piston. The shock is assumed to be strong and propagating into the medium which is at rest, with uniform density. The total energy of the shock is assumed to be time dependent and obeying the power law. By means of similarity method our system of PDEs transformed into the system of ordinary differential equations (ODEs), which in general are nonlinear. The effects of thermal radiation on the the flow variables velocity, density, pressure and magnetic field are investigated behind the shock.

Design Curves for Gas Pressure Drop in Excess Flow Safety Valve

AbstractLaboratory experiments were carried out to investigate pressure drop in excess flow safety valves (EFSVs) assuming isothermal, incompressible, and one-dimensional gas flow. A particular foc...

Numerical modeling of the labyrinth seal taking into account vibrations of the gas transmittal unit rotor in aeroelastic formulation

The article deals with the issues related to the mutual influence of vibrations and gas dynamic processes in the labyrinth seals (LS) of gas transmittal unit compressors. The mutual influence of vibrational gas dynamic processes in LS and vibrations of the rotor is studied. Within the framework of the unified algorithm, a solution is obtained for an unsteady aeroelastic one-dimensional gas flow problem in a deformable LS. A new factor (the rotor diameter in the LS region), which affects the pulsation magnitude of the gas dynamic force in the LS, is revealed. Changing the diameter of the rotor, you can reduce vibration. In this case, it is possible to reduce the designated clearances in the LS and to reduce the leakage.

Algebraic Calculation Method of One-Dimensional Steady Compressible Gas Flow

This paper describes a new method of calculation of one-dimensional steady compressible gas flows in channels with possible heat and mass exchange through perforated sidewalls. The channel is divided into small elements of a finite size for which mass, energy and momentum conservation laws are written in the integral form, assuming linear distribution of the parameters along the length. As a result, the calculation is reduced to finding the roots of a quadratic algebraic equation, thus providing an alternative to numerical methods based on differential equations. The advantage of this method is its high tolerance to coarse discretization of the calculation area as well as its good applicability for transonic flow calculations.

Acoustic disturbances in a gas with an axial temperature gradient

Linear analysis acoustic disturbances in one-dimensional gas flow with a longitudinal gradient of the sound speed provided. Known wave equation for the acoustic velocity is used. In the case of linear distribution of the sound speed in the hot part of the flow equation has an exact analytic solution. This allows to define the expression describing the propagation acoustic disturbances in a gas with varying mean temperature and density. The results can be used to calculate the resonance frequencies of the gas oscillations in the laboratory and industrial combustors.

Magnetogasdynamic Shock Waves in Non-ideal Gas Under Gravitational Field-Isothermal Flow

Self-similar solutions are obtained for one-dimensional unsteady isothermal flow of non-ideal gas with gravitational effects behind a spherical shock wave, in the presence of a spatially decreasing azimuthal magnetic field. The shock wave is driven out by a moving piston with time according to power law. The gas is assumed to be non-ideal having infinite electrical conductivity. The medium is under the influence of the gravitational field due to a heavy nucleus at the origin (Roche model). The effects of variation of the parameter of non-idealness of the gas, the Alfven–Mach number and the gravitational parameter on the flow-field behind the shock are investigated. It is shown that the presence of magnetic field, gravitational field and non-idealness of the gas has decaying effect on the shock wave. Further, it is investigated that the shock strength decreases due to an increase in the strength of magnetic field, gravitational and non-idealness parameter. The present study can be important for description of shocks in supernova explosions, in the study of a flare produced shock in the solar wind, central part of star burst galaxies, nuclear explosion, rupture of a pressurized vessel and explosion in the ionosphere etc.

Exact Solutions for One-Dimensional Transient Gas Flow in Porous Media with Gravity and Klinkenberg Effects

Attempts have been made to find exact solutions for the one-dimensional transient gas flow equation in porous media. By introducing a traveling wave variable, a traveling wave solution of the gas flow equation has been found. The traveling wave solution is presented in an explicit form of the space and time variables, and it takes into account both gravity and Klinkenberg effects (pressure-dependent permeability). We investigated the properties of the traveling wave solution and the effect of some parameters such as the Klinkenberg coefficient. A numerical study has been carried out, which confirms the stability of the traveling wave solution. The traveling wave solution is then used to derive two benchmark solutions defined over the semi-infinite domain. The first one assumes uniform initial gas pressure and non-uniform boundary condition, and the second assumes uniform boundary condition and non-uniform initial distribution of the gas pressure. The benchmark solutions are easy to use and are useful for validating numerical solutions. Two illustrative examples are presented in order to compare the benchmark solutions with the numerical solutions. The results show good agreements between the solutions.