One-dimensional Flow Equations Research Articles

Development of a transient analysis code for trans-critical simulation of SCWR

Supercritical water-cooled reactors (SCWR) experience pressure reductions below the critical point during a loss of coolant accident (LOCA). During the reactor depressurization, boiling crises may occur, leading to pressure fluctuations and significant increases in wall temperatures, posing a severe threat to the cladding. In this study, a one-dimensional transient analysis code has been developed to incorporate one-dimensional steady-state flow equations in the fluid domain and transient thermal conductivity equations in the solid domain. The code also includes a wall heat transfer model and a model for the velocity of the moving quench front to simulate transcritical depressurization transient processes. The simulation successfully predicts the typical experimental phenomena. It is reasonable to use the critical temperature as the demarcation point between the dry and wet conditions of the axial wall of the rod bundle under transcritical conditions. By combining the experimental data with the program calculations, the critical interface for the occurrence of boiling crisis under transcritical conditions is determined using mass flow rate, heat flow density and fluid temperature as inputs. The criterion for the occurrence of CHF under transcritical conditions is obtained, which provides a reference to the safety analysis of SCWRs.

Exact Solution of the Riemann Problem for the One-Dimensional Blood Flow Equations with General Constant Momentum Correction Coefficient

Exact Solution of the Riemann Problem for the One-Dimensional Blood Flow Equations with General Constant Momentum Correction Coefficient

Analyses of gas pulsation characteristics in the discharge pipeline of a hyper compressor

For the analysis of gas pulsation characteristics in the pipeline systems of low density polyethylene (LDPE) hyper compressors, a time-domain calculation method for compressor pipeline gas pulsation based on real gas properties is proposed. First the thermophysical property tables are prepared with gas state parameters as independent and dependent variables. Then one-dimensional unsteady flow equations and characteristic equations for pipeline flow are established based on real gas properties. After setting pipeline parameters and boundary conditions, the flow equations are discretized using a two-step method at the inner points of the pipeline. The characteristic equations are discretized using the trapezoidal integration method at the boundary points. Thus, the time-domain solving of pulsating flow field in the pipeline is realized. A self-developed numerical code was used to conduct the time-domain calculation and analysis of gas pulsation in the discharge pipeline of a two-stage hyper compressor. The calculated pressure pulsation curves based on real gas properties are in good agreement with experimental data, with a maximum difference lower than 4% of the local average pressure, which is smaller than the difference between the results calculated based on the plane wave theory and the measured data. The analysis of gas pulsation characteristics shows that the pressure pulsation in the main pipeline is mainly contributed by the fundamental and second harmonics. The pressure pulsation in the safety valve branch is higher than that in the main pipeline, and the harmonic components falling into the acoustic resonance frequency range of this branch have significant amplitudes. The acoustic resonance frequency of the branch pipeline is not affected by the main pipeline. Pressure pulsation varies with changes in rotational speed and back pressure, and the pulsation level is more significantly affected by rotational speed compared to back pressure. The renovation scheme of connecting the safety valve branch and expansion tube in series can effectively reduce the pressure pulsation levels of the main pipeline and branch pipeline.

Cerebral perfusion simulation using realistically generated synthetic trees for healthy and stroke patients

Background and objectiveCerebral vascular diseases are among the most burdensome diseases faced by society. However, investigating the pathophysiology of diseases as well as developing future treatments still relies heavily on expensive in-vivo and in-vitro studies. The generation of realistic, patient-specific models of the cerebrovascular system capable of simulating hemodynamics and perfusion promises the ability to simulate diseased states, therefore accelerating development cycles using in silico studies and opening opportunities for the individual assessment of diseased states, treatment planning, and the prediction of outcomes. By providing a patient-specific, anatomically detailed and validated model of the human cerebral vascular system, we aim to provide the basis for future in silico investigations of the cerebral physiology and pathology. MethodsIn this retrospective study, a processing pipeline for patient-specific quantification of cerebral perfusion was developed and applied to healthy individuals and a stroke patient. Major arteries are segmented from 3T MR angiography data. A synthetic tree generation algorithm titled tissue-growth based optimization (GBO)1 is used to extend vascular trees beyond the imaging resolution. To investigate the anatomical accuracy of the generated trees, morphological parameters are compared against those of 7 T MRI, 9.4 T MRI, and dissection data. Using the generated vessel model, hemodynamics and perfusion are simulated by solving one-dimensional blood flow equations combined with Darcy flow equations. ResultsMorphological data of three healthy individuals (mean age 47 years ± 15.9 [SD], 2 female) was analyzed. Bifurcation and physiological characteristics of the synthetically generated vessels are comparable to those of dissection data. The inability of MRI based segmentation to resolve small branches and the small volume investigated cause a mismatch in the comparison to MRI data. Cerebral perfusion was estimated for healthy individuals and a stroke patient. The simulated perfusion is compared against Arterial-Spin-Labeling MRI perfusion data. Good qualitative agreement is found between simulated and measured cerebral blood flow (CBF)2. Ischemic regions are predicted well, however ischemia severity is overestimated. ConclusionsGBO successfully generates detailed cerebral vascular models with realistic morphological parameters. Simulations based on the resulting networks predict perfusion territories and ischemic regions successfully.

Symmetry and scaling in one-dimensional compressible two-phase flow

Investigations of shock compression of heterogeneous materials often focus on the shock front width and overall profile. The number of experiments required to fully characterize the dynamic response of a material often belie the structure–property relationships governing these aspects of a shock wave. Recent observations measured a pronounced shock-front width on the order of 10 s of ns in particulate composites. Here, we focus on particulate composites with disparate densities and investigate whether the mechanical interactions between the phases are adequate to describe this emergent behavior. The analysis proceeds with a general Mie–Grüneisen equation of state for the matrix material, a general drag force law with general power-law scaling for the particle-matrix coupling of the phases, and a volume fraction-dependent viscosity. Lie group analysis is applied to one-dimensional hydrodynamic flow equations for the self-consistent interaction of particles embedded in a matrix material. The particle phase is characterized by a particle size and volume fraction. The Lie group analysis results in self-similar solutions reflecting the symmetries of the flow. The symmetries lead to well-defined scaling laws, which may be used to characterize the propagation of shock waves in particle composites. An example of the derived scaling laws for shock attenuation and rise time is shown for experimental data on shock-driven tungsten-loaded polymers. A key result of the Lie analysis is that there is a relationship between the exponents characterizing the form of the drag force and the exponent characterizing the shock velocity and its attenuation in a particulate composite. Comparison to recent experiments results in a single exponent that corresponds to a conventional drag force.

Adiabatic wall temperature in the supersonic flow of moist air with spontaneous condensation

The expansion of moist air in supersonic nozzles is accompanied by spontaneous condensation of water vapor due to a sharp decrease in flow temperature and pressure. However, in the near-wall boundary layer due to the viscous dissipation, moist air temperature decreases slightly in comparison with initial one. This leads to the fact that the vapor remains overheated in the near-wall region and condensation does not spread to its entire depth. Droplets formed in the core flow as a result of condensation can penetrate into the near-wall region due to turbulent diffusion and evaporate due to higher temperature in the near-wall region. This process causes cooling of the near-wall gas layers and, consequently, a decrease in the adiabatic wall temperature. A decrease in the adiabatic wall temperature relative to the stagnation temperature leads to a decrease in aerodynamic heating and, consequently, in the heat flux from the gas to the body. In this paper, measurements of the adiabatic temperature of the supersonic nozzle wall for moist air flow with condensation are presented. Temperature measurements were made using an infrared thermal imager. Initial moisture content varied in the range of 1.5–20.2 g/(kg. dry air) and initial relative humidity in the range 20–90%. It is shown that in the region behind the zone of spontaneous condensation (“condensation shock”), adiabatic wall temperature depends on the initial moisture content and can be either greater or less than the value of a single-phase (dry air) flow with identical initial flow parameters. Estimates of the flow parameters behind the condensation region are obtained using one-dimensional diabatic flow equations and experimental static pressure distributions along the nozzle length. The correlation between the initial moisture content, the average droplet size and the temperature recovery factor is shown.

Tissue-growth-based synthetic tree generation and perfusion simulation

Biological tissues receive oxygen and nutrients from blood vessels by developing an indispensable supply and demand relationship with the blood vessels. We implemented a synthetic tree generation algorithm by considering the interactions between the tissues and blood vessels. We first segment major arteries using medical image data and synthetic trees are generated originating from these segmented arteries. They grow into extensive networks of small vessels to fill the supplied tissues and satisfy the metabolic demand of them. Further, the algorithm is optimized to be executed in parallel without affecting the generated tree volumes. The generated vascular trees are used to simulate blood perfusion in the tissues by performing multiscale blood flow simulations. One-dimensional blood flow equations were used to solve for blood flow and pressure in the generated vascular trees and Darcy flow equations were solved for blood perfusion in the tissues using a porous model assumption. Both equations are coupled at terminal segments explicitly. The proposed methods were applied to idealized models with different tree resolutions and metabolic demands for validation. The methods demonstrated that realistic synthetic trees were generated with significantly less computational expense compared to that of a constrained constructive optimization method. The methods were then applied to cerebrovascular arteries supplying a human brain and coronary arteries supplying the left and right ventricles to demonstrate the capabilities of the proposed methods. The proposed methods can be utilized to quantify tissue perfusion and predict areas prone to ischemia in patient-specific geometries.

Methods of channel calculations to improve the operational mode of main canals

The article discusses solutions to improve the operational mode of the canal with an increase in throughput by means of self-washout, while maintaining the morphometric favorable stable cross-section of the canal for further reliable water supply of suspended crop areas. Taking into account the characteristics of the course of channel deformation processes, we have considered the modern dynamics of channel flows for accurate calculation of channel deformations using the systems of equations of one-dimensional flow in the deformable Saint-Venant channel or two-dimensional shallow water equations (planned task), which will give a positive result for the operational mode of the main canals.

The interaction of rarefaction waves for a system of granular flow equations

In this paper, we consider the interaction problem of two centered rarefaction waves for the one-dimensional granular flow equations, which originates from the interaction and impact of two avalanche flows, mud flows or tsunamis in reality. This interaction problem is essentially a Goursat-type boundary value problem in mathematical theory. When there is no vacuum in the initial data, we show that no vacuum appears in the interaction region, while the vacuum occurs when the collision time is sufficiently large. Based on the characteristic method, the a priori estimates are derived to establish the existence and uniqueness of smooth solutions for the Goursat problem on the whole interaction region.

Theoretical research on gas seepage in the formations surrounding bedded gas storage salt cavern

When constructing salt cavern gas or petroleum storage in lacustrine sedimentary salt formations rich in mudstone interlayers, the influence of the sealing performance of interlayers and salt-mud interface on the storage tightness should be considered adequately. In order to reveal the gas seepage in deep formations surrounding bedded salt cavern underground storage, a leakage analysis model was established based on the characteristics of a low dip angle and the interbedded structure of bedded rock salt. The gas seepage governing equations for one-dimensional and plane radial flow were derived and solved. A gas seepage simulation experiment was conducted to demonstrate the accuracy and reliability of the theoretical calculation results. The error of the seepage range was approximately 6.70%, which is acceptable. The analysis and calculation results indicate that the motion equation of gas in deep formations satisfies a non-Darcy's law with a threshold pressure gradient and slippage effect. The sufficient condition for the gas flow to stop is that the pressure gradient is equal to the threshold pressure gradient. The relationship between the leakage range and operating time is a positive power function, that is, the leakage range gradually increases with time and eventually stabilizes. As the seepage range increases, the seepage pressure decreases sharply during the early stage, and then decreases gradually until the flow stops. Combining the research results with engineering applications, three quantitative evaluation indexes named the maximum admissible leakage range, leakage volume and leakage rate are proposed for the tightness evaluation of gas storage salt cavern during their operating stage. These indexes can be used directly in actual engineering applications and can be compared with the key design parameters stipulated in the relevant specifications. This work is expected to provide theoretical and technical support for the gas loss and tightness evaluation of gas storage salt caverns.

Extending the Validity of Basic Equations for One-dimensional Flow in Tubes with Distributed Mass Sources and Varying Cross Sections

Flow problems are solved using so-called fundamental equations and the corresponding initial and boundary conditions. The fundamental equations are the motion equation, the continuity equation, the energy conservation equation, and the state equations. In our paper, we extend the validity of the equation of motion used to describe one-dimensional, steady-state tubular flow to a case in which the mass flow of the medium changes along the tubular axis during the flow. Such flows occur in perforated and/or porous pipes and air ducts. The research in this direction was motivated by the fact that the extension and formulation of the equation of motion in this direction has not been carried out with completely general validity. In the equation of motion used to solve the problems, the isochoric and isotherm nature were assumed. In our paper, we present fundamental equations that formulate differential equations to describe polytrophic and expanding flows.

Theoretical and experimental investigation on the seismic performance of a novel variable-damping viscous fluid damper

This paper presents a novel three-stage variable-damping-coefficient viscous fluid damper (VCVFD) and investigates its effectiveness in mitigating the seismic structural response. The concept and design of the displacement-dependent VCVFD were based on the equations for one-dimensional viscous flow in the damper orifice. A series of performance tests on VCVFDs were conducted to investigate the mechanical behaviour of the dampers with different design parameters under various loading conditions. The damping force increases more significantly than in the conventional viscous fluid damper (VFD) when the operating displacement is larger, and the expected effect of variable damping is achieved. A smaller gap height is more beneficial to energy dissipation in the VCVFD. Moreover, the hysteresis loop of the VCVFD was consistent with the theoretical results. To validate the effectiveness of the VCVFD in structural control, shaking table tests of two-storey steel frames equipped with VCVFDs under different seismic intensities were conducted. VCVFD can slightly increase the natural frequencies of the steel frame and significantly increase the structural damping ratio. The effectiveness of the VCVFD in mitigating the structural acceleration under low earthquake intensities was not prominent, but as the earthquake intensity increased, the acceleration amplification factor decreased. The added damping contributed by VCVFDs can significantly reduce the displacement response. It is concluded that with reasonable damping structural design, the VCVFD can achieve an almost similar mitigation effect as VFD in low-intensity earthquakes, whereas the VCVFD can provide higher damping under high-intensity earthquakes.

Mathematical Modelling and Simulation for One Dimensional - Two-Phase Flow Equation in Petroleum Reservoir: A Matlab Algorithm Approach

The reservoir behaviors described by a set of differential equation those results from combining Darcy’s law and the law of mass conservation for each phase in the system. The one-dimensional two-phase flow equation is implicit in the pressure and saturation and explicit in relative permeability. A mathematical model of a physical system is a set of partial differential equations together with an appropriate set of boundary conditions, which describes the significant physical processes taking place in that system. The processes occurring in petroleum reservoirs are fluid flow and mass transfer. Two immiscible phases (water& oil) flow simultaneously while mass transfer may take place among the phases. Gravity, capillary, and viscous forces play a role in the fluid-flow process. The model equations must account for all these forces and should also take into account an arbitrary reservoir description with respect to heterogeneity and geometry. Finally, one-dimensional two-phase flow equation through porous media is formulated by considering above reservoir parameters and forces. A numerical method based on finite difference scheme is implemented to get the solutions of one-dimensional two-phase flow equation. A MATLAB algorithm is used to solve the equation with mathematical analysis resulting in upper and lower bounds for the ratio of time step to mesh. The MATLAB algorithm is modified as per the model with appropriate initial and boundary conditions. The algorithm is applied to two-phase water flooding problems in laboratory size cores, and resulting saturation and pressure distribution are presented graphically. The saturation and pressure distribution of two-phase flow model is in agreement with the prediction of the Buckley Leveret theory. The numerical solution is used as a base for evaluating the numerical methods with respect to machine time requirement and allowable tie step for fixed mesh spacing.

Numerical Solution and Stability Analysis for a Class of Nonlinear Differential Equations

In this paper, one-dimensional Burgers equation and one-dimensional Laval nozzle flow Euler equation are numerically solved, and the stability of the numerical solutions is analyzed theoretically. In order to satisfy the stability of the numerical solution, the explicit MacCormack scheme is used to obtain the stable solution of the computer numerical simulation. In addition, the numerical and analytical solutions of the Euler equation for one-dimensional Laval nozzle flow are compared, and the results are completely consistent, which verifies the correctness of the numerical solution.

ОПРЕДЕЛЕНИЕ ГИДРАВЛИЧЕСКИХ ПОТЕРЬ НА ОСНОВЕ МОДЕЛИРОВАНИЯ ТЕЧЕНИЯ В ЭЛЕМЕНТАХ КАМЕРЫ СГОРАНИЯ ГАЗОТУРБИННОГО ДВИГАТЕЛЯ

The article presents the results of the study of hydraulic characteristics using the example of the combustion chamber of the aircraft engine PS-90A. A mathematical model of gas dynamics of the combustion chamber based on the equations of one-dimensional stationary flow is considered. In a one-dimensional formulation, the change in parameters along the path of the combustion chamber is analyzed for various modes of engine operation, from idle to takeoff. The boundary conditions for each model were obtained based on the results of thermogasdynamic calculations. All modes of operation of the combustion cham-ber were considered with combustion and without combustion of liquid fuel. The numerical three-dimensional model provided for the calculation of one sector of the combustion chamber containing one flame tube of a tubular-annular type with a concentrical-ly installed two-row swirler. The grid model was selected based on the calculation results from several options according to the condition of grid independence of the selected parameters. The calculation took into account the effects of turbulence, chemical interaction of fuel with air supplied from the engine compressor. For the reliability of the temperature distribution in the area of the flame tube, a calculation was made taking into account the model of radiant heat transfer. The calculation of the combustion of a premixed mixture was carried out in a non-adiabatic formulation using the model of a thin flame front and in the gas-phase approximation. The main features of the flow and changes in pressures and flow rates in the flow path of the combustion cham-ber are described. It has been established that the main losses occur in the diffuser and the frontal arrangement of the combus-tion chamber. A comparison of the results obtained in a one-dimensional formulation with numerical calculations in a three-dimensional formulation was made, which showed their correspondence with each other with the permissible level of deviation.

Modelling foam improved oil recovery: towards a formulation of pressure-driven growth with flow reversal.

The pressure-driven growth model that describes the two-dimensional (2-D) propagation of a foam through an oil reservoir is considered as a model for surfactant-alternating-gas improved oil recovery. The model assumes a region of low mobility, finely textured foam at the foam front where injected gas meets liquid. The net pressure driving the foam is assumed to reduce suddenly at a specific time. Parts of the foam front, deep down near the bottom of the front, must then backtrack, reversing their flow direction. Equations for one-dimensional fractional flow, underlying 2-D pressure-driven growth, are solved via the method of characteristics. In a diagram of position versus time, the backtracking front has a complex double fan structure, with two distinct characteristic fans interacting. One of these characteristic fans is a reflection of a fan already present in forward flow mode. The second fan however only appears upon flow reversal. Both fans contribute to the flow's Darcy pressure drop, the balance of the pressure drop shifting over time from the first fan to the second. The implications for 2-D pressure-driven growth are that the foam front has even lower mobility in reverse flow mode than it had in the original forward flow case.

Analysis of Optimum Performance of Air Vessels Used in Damping Water Hammer Pressure Wave

Transient flow or water hammer is the sudden change of flow conditions due to an unexpected event such as a sudden closing of a valve installed on the pipeline system. Water hammer can pose serious threats to the pipeline system, as the flow's pressure can be changed above or below the limits which the pipeline network can withstand. Controlling the water hammer pressure becomes indispensable, so a transient surge analysis is required to investigate the points of severe changes along with the pipeline system. Compressed air vessel (CAV) is a pressure control device that is used to control the positive and negative pressure changes. There are two parameters affect the sizing of the air vessel; the initial trapped air volume and throttling the aperture of the air vessel. A computer model based on the unsteady one-dimensional flow equations of momentum and continuity is established to examine the proper size of the two parameters. The equations are solved by the method of characteristics while the vessel is modeled as a quasi-one-dimensional flow system. An experimental test rig provided with a rapid closing magnetic valve and pressure sensors are used to validate the model results. Both the experimental and model results show the high capability of the air vessel to dampen water hammer pressure. In addition, the results show that the introduced throttling action has a wide effect on the size of the air vessel and that the diameter of the throttling shall not be less than 0.3 the diameter of the main pipe. Besides, the initial volume of the trapped air should be in the range of 53% to 78% of the total volume of the air vessel, so that the air vessel works effectively with optimum value of 60%.

Linear stability analysis of RELAP5 two-fluid model in nuclear reactor safety results

System thermal-hydraulic code RELAP5 is based on a two-fluid, non-equilibrium, and non-homogeneous hydrodynamic model for simulation of transient two-phase behavior. The code model includes six governing equations to incorporate the mass, energy, and momentum of the two fluids. In this paper, linear stability analysis is performed to check the ill-posedness of the RELAP5 specific two-fluid model (TFM) for all normal and accident conditions of a standard pressurized water reactor (PWR). The analysis gives information about the soundness of the model and identifies the range of parameters where the solutions obtained from the model will be numerically convergent. The linear two-phase fluid dynamic stability (by dispersion analysis) of the RELAP5 one-dimensional two-fluid model and numerical stability of the difference equation formulation (by Von Neumann method) is presented. The present analysis shows that the two-fluid model becomes ill-posed for some fluid conditions, where the results are less accurate, so sensitivity analysis plays an important role. The ill-posed nature implies that results thus obtained (by finite difference method) have to be interpreted carefully because of the sensitive nature of reactor safety analysis. It is also identified that the variation in various parameters (like slip ratio, system pressure, void fraction, and phasic velocities) can affect the error growth rates. It has been demonstrated that the basic system of one-dimensional two-phase flow equations, that possesses complex characteristics, exhibits unbounded instabilities in the short-wavelength limit and constitutes an improperly posed initial value problem. The semi-implicit numerical method, which is unconditionally stable for hyperbolic systems, becomes unstable for non-hyperbolic systems. For some of the fluid conditions, even after the introduction of artificial viscosity terms (in the difference equation formulation) that damp the high-frequency spatial components of the solution, are not sufficient for regularization of the two-fluid model. Thus, there is a need for the addition of newer terms, e.g. bubble collision, so that the existing incomplete model provides better results. It is also demonstrated that the basic TFM of RELAP5 with additional collision term makes the system unconditionally well-posed which are originally conditionally well-posed. Excellent agreement is obtained between the predicted and computed growth rates of harmonic disturbances. It is found that the instability associated with the two-fluid model discretized system of equations is related to the instability associated with the ill-posedness of the two-fluid model but is quantitatively different.

High-Resolution Simulations for Aerogel Using Two-Phase Flow Equations and Godunov Methods

Aerogel is studied numerically using a one-dimensional two-phase flow equations system. A hyperbolic and conservative two-phase flow model is applied to a mixture of porous media containing nanofluids. The application of non-equilibrium mixture behavior between phases is adopted and promoted in this current investigation. By establishing mixture conservation balance laws, finite volume techniques using Godunov methods of centered type are extended to aerogel simulations. Numerical results are compared with other methods providing a remarkable agreement. The computed results demonstrate the key capabilities of this existing mixture model in the resolution of discontinuities in aerogel problems and more reliable than applying a sophisticated single-phase flows with complex property models.

Parameter estimation to study the immediate impact of aortic cross-clamping using reduced order models.

Aortic cross-clamping is a common strategy during vascular surgery, however, its instantaneous impact on hemodynamics is unknown. We, therefore, developed two numerical models to estimate the immediate impact of aortic clamping on the vascular properties. To assess the validity of the models, we recorded continuous invasive pressure signals during abdominal aneurysm repair surgery, immediately before and after clamping. The first model is a zero-dimensional (0D) three-element Windkessel model, which we coupled to a gradient-based parameter estimation algorithm to identify patient-specific parameters such as vascular resistance and compliance. We found a 10% increase in the total resistance and a 20% decrease in the total compliance after clamping. The second model is a nine-artery network corresponding to an average human body in which we solved the one-dimensional (1D) blood flow equations. With a similar parameter estimation method and using the results from the 0D model, we identified the resistance boundary conditions of the 1D network. Determining the patient-specific total resistance and the distribution of peripheral resistances through the parameter estimation process was sufficient for the 1D model to accurately reproduce the impact of clamping on the pressure waveform. Both models gave an accurate description of the pressure wave and had a high correlation (R2 > .95) with experimental blood pressure data.