Method Of Characteristics Research Articles

Evaluation of service pressure regulation strategy on the performance of a rural water network based on pulse demand; using the method of characteristics

Abstract Reducing the occurrence of pipe bursts, reducing leakage, and reducing energy consumption are the three main goals in implementing pressure control programs in water distribution networks. Service pressure regulation strategy is an evolved approach that encompasses all goals of pressure management. This paper has investigated this approach in a rural network with hydraulic complexities as a case study so that some parts of the network have excess pressure and other low pressure. A computer code based on the method of characteristics (MOC) has been developed for network hydraulic analysis. The generated code analyzes unsteady flow, pressure-driven demand analysis, and dynamic adjustment of pressure control valves based on the target node. Also, the experimental results of a laboratory network have been applied to validate and calibrate the numerical simulation. In addition, by measuring the flow rate and pressure of the network and the results of the minimum night flow method, three consumption patterns were used to generate pulsed nodal demands. Studies show that creating pressure-management areas by hydraulic analysis by MOC will determine the best control strategies. The mean pressure decreased 54% by applying this strategy. Furthermore, the average fluctuations of pressure reduced from 9.7 meters to 3.5 meters.

Impact of Angular and Spatial Source Distribution Approximations on Convergence Performance of Nonlinear Acceleration Methods for MOC in Slab Geometry

The convergence performance of nonlinear acceleration methods for the method of characteristics (MOC) with flat source (FS) approximation (FS MOC) or linear source (LS) approximation (LS MOC) is numerically investigated by focusing on the spatial and angular approximations in the acceleration calculations. The convergence of nonlinear acceleration depends on the consistency of the calculation models between the higher-order and lower-order (acceleration) methods. The convergence of four acceleration methods is evaluated to clarify the relationship between model consistency and convergence performance. These methods consist of FS or LS for the spatial source distribution and P1 or discrete angle for the angular distribution, i.e., (1) FS analytic coarse mesh finite difference (ACMFD) acceleration (FS ACMFD), (2) LS ACMFD, (3) FS angular-dependent discontinuity factor MOC (ADMOC) acceleration (FS ADMOC), and (4) LS ADMOC. The ACMFD and ADMOC accelerations are based on P1 and discrete angle approximations, respectively. The FS MOC and LS MOC are considered higher-order methods. The FS MOC and LS MOC with five acceleration methods, i.e., the aforementioned four acceleration methods and the conventional coarse mesh finite difference acceleration method, are used to perform fixed-source calculations in one-group one-dimensional homogeneous slab geometry, and the spectral radii are numerically evaluated. The numerical results indicate that (1) the nonlinear acceleration methods that are unconditionally stable for FS MOC also show similar convergence properties for LS MOC in one-dimensional slab geometry; (2) better convergence is observed when the consistency of higher- and lower-order models is high; and (3) when a coarse mesh is optically thick, the spatial homogenization degrades the convergence performance, even if spatial and angular approximations are consistent between the higher- and lower-order models.

A novel analytical simulation method of utilization of surfactant-active nanoparticles for well injection enhancement in low-permeability reservoirs

In order to study the mechanism of surfactant-active nanoparticles in drag reduction and injection enhancement in low-permeability reservoirs and to simulate the rough condition of the pore surface of the reservoirs, we construct a microtube model with periodic rectangular groove structure and a force balance model of nanoparticles on rough surface to get the limit adhesion concentration fraction model of surfactant-active nanoparticles on pore surface. The method of characteristics (MOC) is used to analyze and simulate the seepage process of the suspension-adhesion-plugging coupling of surfactant-active nanoparticles in pores of low permeability reservoirs. By comprehensively considering reservoir damage caused by the mechanisms such as particle attachment and blockage and the slip effect established by nanoparticle attachment, a theoretical model for resistance reduction and injection enhancement of surfactant-active nanoparticles has been formed. The results show that comparing with the core without nanoparticles flooding, the pressure drop of the core decreases significantly after the injection of nanoparticles, and the pressure drop reaches 40%. At the same time, by comparing the water injection index before and after injection, the core injection capacity is significantly improved after the surfactant-active nanofluid injection. This model provides a certain theoretical basis for the technology of pressure reduction and injection enhancement of surfactant-active nanoparticles in low permeability reservoirs.

Modeling Transient Pipe Flow in Plastic Pipes with Modified Discrete Bubble Cavitation Model

Most of today’s water supply systems are based on plastic pipes. They are characterized by the retarded strain (RS) that takes place in the walls of these pipes. The occurrence of RS increases energy losses and leads to a different form of the basic equations describing the transient pipe flow. In this paper, the RS is calculated with the use of convolution integral of the local derivative of pressure and creep function that describes the viscoelastic behavior of the pipe-wall material. The main equations of a discrete bubble cavity model (DBCM) are based on a momentum equation of two-phase vaporous cavitating flow and continuity equations written initially separately for the gas and liquid phase. In transient flows, another important source of pressure damping is skin friction. Accordingly, the wall shear stress model also required necessary modifications. The final partial derivative set of equations was solved with the use of the method of characteristics (MOC), which transforms the original set of partial differential equations (PDE) into a set of ordinary differential equations (ODE). The developed numerical solutions along with the appropriate boundary conditions formed a basis to write a computer program that was used in comparison analysis. The comparisons between computed and measured results showed that the novel modified DBCM predicts pressure and velocity waveforms including cavitation and retarded strain effects with an acceptable accuracy. It was noticed that the influence of unsteady friction on damping of pressure waves was much smaller than the influence of retarded strain.

Cross sections polynomial axial expansion within the APOLLO3® 3D characteristics method

A polynomial expansion for cross sections has been introduced in the 3D Method Of Characteristics (MOC) solver of the APOLLO3® code, with the aim of applying the method to depleted systems. This paper addresses the suitable adaptation of the MOC equations with regards to the free and DPN-accelerated power iterations, describing also how the conceived algorithms take advantage of a vectorized implementation. An adaptive quadrature technique is presented, necessary for the evaluation of terms that, due to the polynomial description, cannot be solved analytically. The decisive advantage of the new method over a classical step-constant description of nuclear properties is assessed by comparing the results of the depletion study of a PWR assembly.

Design Parameter Influence on Losses and Downstream Flow Field Uniformity in Supersonic ORC Radial-Inflow Turbine Stators

The design of organic Rankine cycle (ORC) turbines often requires dealing with transonic flows due to the cycle efficiency requirements and the matching of the temperature profiles with heat sources and sinks, as well as the nature of organic fluids, often featuring high molecular weight. Consequently, the use of convergent–divergent turbine stators has been widely established as a solution in the published literature for use in both axial- and radial-inflow machines. With respect to the latter layout in particular, the available design guidelines are still limited. The present work shows the results of an investigation into a series of ORC radial-inflow convergent–divergent nozzles that differ with respect to the vane count and the designed metal angle of the outlet. These stators were designed by fitting the divergent portion of a sharp-edged minimum-length nozzle, designed by means of the method of characteristics (MoC) adapted to dense gases, into a radial-inflow turbine stator. The geometries were analysed by means of steady-state RANS CFD calculations, and the results were used to assess the influence of the design parameters on the nozzle losses and downstream flow field uniformity, showing that conflicting trends exist between optimum stator efficiency and optimum downstream flow field uniformity.

Experimental validation of a new type of direct-connect facility for scramjet studies

Combustion chamber design has been important to the development of scramjet engines for a long time. Conventional experimental facilities of scramjet could be divided into two types: free-jet and direct-connect facilities. Free-jet facility can provide real incoming conditions to the combustion chamber, especially the real shock wave train structures formed in inlet and isolator. Unfortunately, it is relatively expensive. While direct-connect facility has a distinct advantage in experimental cost and period, so it has been used widely in relevant researches. However, traditional direct-connect facility usually can only provide a uniform incoming flow, which is much different from the distorted flow field in real scramjet, and this is a non-negligible disadvantage to relevant design. An inverse design method based on method of characteristics (MOC) is presented to solve this problem. The facility designed by this new method can produce the pre-assigned distorted flow field without encountering unstart phenomenon. In this paper, experimental investigation of the new direct-connect facility, which is designed to reproduce a specific distorted flow field with Mach number more than 2, is further presented. Test data of particle image velocimetry (PIV), flow visualization experiments and corresponding numerical simulation results are used to verify the reliability and accuracy of this new facility. Qualitative flow visualization and quantitative measurement results further prove that this new type of direct-connect facility can generate the target distorted flow field as pre-defined.

Water Hammer Simulation Method in Pressurized Pipeline with a Moving Isolation Device

Smart isolation devices (SIDs) are commonly used in pressurized subsea pipelines that need to be maintained or repaired. The sudden stoppage of the SID may cause large water hammer pressures, which may threaten both the pipeline and the SID. This paper proposes a simulation method by using a coupled dynamic mesh technique to simulate water hammer pressures in the pipeline. Unlike other water hammer simulations, this method is the first to be used in the simulation in pipelines with a moving object. The implicit method is applied to model the moving SID since it has the mutual independence between the space step and the time step. The movement of the SID is achieved by updating the size of the computational meshes close to the SID at each time step. To improve the efficiency of the simulation and the ability of handling complex boundary conditions, the pipe sections far away from the SID can also be simulated by using the explicit Method of Characteristics (MOC). Verifications were conducted using the simulated results from the Computational Fluid Dynamics (CFD) numerical simulation. Two scenarios have been studied and the comparisons between the simulated results by using the dynamic meshes in 1D methods and those by the CFD simulation show a high correlation, thus validating the new method proposed in this paper.

Rasterized coarse mesh finite difference acceleration on method of characteristics for a reactor core with a generalized boundary

The method of characteristics (MOC) is widely employed in neutron transport solvers to obtain accurate solution of a direct transport calculation for a whole reactor core model. Coarse mesh finite difference (CMFD) method is often used to speedup the convergence of MOC solution especially for a large geometry. In this work, a rasterized CMFD method is proposed to improve the convergence rate of the high-order neutron transport calculation for reactor cores with generalized boundaries. The rasterized CMFD method is implemented in a 2D neutron transport MOC code ThorMOC which has been developed to perform the high-fidelity criticality calculation for channel type molten salt reactors (MSRs). The rasterized CMFD method is validated by two benchmarks, a C5G7 hexagonal benchmark and a LFMSR (Liquid-Fueled Molten Salt Reactor) benchmark. The computational result obtained from ThorMOC accelerated by the rasterized CMFD method shows a good agreement with that from the Monte Carlo code OpenMC for the two benchmarks. The rasterized CMFD method can reduce the MOC stand-alone solution time of the LFMSR benchmark by more than 100 times.

A linear source scheme for the 2D/1D transport method in SHARK

The 2D/1D transport method has already become the most popular direct transport method these several years considering the balance of efficiency and accuracy. Linear source scheme has been researched and applied in method of characteristics (MOC), which shows great advantages in efficiency and accuracy compared to the traditional flat source approximation. Therefore, to improve the calculation performance of the 2D/1D method further, the linear source scheme is applied in the 2D/1D method with a newly-developed direct transport code SHARK. The equation derivation is conducted and the framework of the linear source scheme in SHARK is introduced. Two macro cases, including C5G7 benchmark and macro BEAVRS benchmark, are adopted to show the performance improvement of the linear source scheme in the 2D/1D method.

Distributed and vectorized method of characteristics for fast transient simulations in water distribution systems

AbstractModeling transient flow in networked dynamical systems characterized by hyperbolic partial differential equations (PDEs) is essential to engineering applications. Solutions of hyperbolic PDEs are commonly found using the method of characteristics (MOC), particularly when modeling the water hammer phenomenon in water distribution systems (WDSs), which is critical for design and operation. For applications that require fast modeling, existing methods for speeding up traditional MOC simulations either trade off accuracy for simulation time, or do not scale properly due to memory restrictions and prolonged computational times. This work proposes a novel parallel implementation of the MOC for networked systems, which relies on vectorization and distributed parallel computing to evaluate the transient dynamics of WDSs. The proposed method, referred to as distributed and vectorized MOC (DV‐MOC), relies on aligned memory allocation for vectorization and distributed‐memory parallelization to further accelerate vectorized operations and ensure scalability for arbitrary network topologies. The algorithm has been applied to a WDS from the battle of the sensor networks (BWSN‐II) composed by 14,824 pipes and nearly solution points. Through rigorous analyses, we show that the performance of DV‐MOC surpasses that of sequential MOC, with speeding up factors in the order of thousands for sufficiently dense numerical grids.

Fine Mesh Finite Difference acceleration method based on the Generalized Equivalence Theory for the 2-D MOC transport calculation

Normally, the pin-cell-based Coarse Mesh Finite Difference (CMFD) method is directly used to accelerate the high-fidelity neutron transport calculations. However, the acceleration effect still has a bottleneck due to the fact that only the pin-cell homogenized flux can be used to update the fine-mesh flux, such as the flat-source-region flux for MOC calculation. In this paper, a Fine Mesh Finite Difference (FMFD) method based on the Flat Source Region (FSR) level is proposed and implemented in a two-dimensional transport code using Method Of Characteristics (MOC). FMFD could avoid the mapping error from pin-cell-wise mesh to fine FSR mesh. However, the solution of the FMFD system is time-consuming since the scale of the FMFD system is much larger than that of the pin-cell-based CMFD system. Therefore, Two-Level (TL) acceleration scheme is performed where the pin-cell-based CMFD is used to accelerate the solution of FMFD system. Both of these two acceleration methods are based on the Generalized Equivalence Theory (GET). Fourier analysis and numerical results show that FMFD with TL is more efficiency than conventional CMFD.

An accurate and efficient scheme involving unsteady friction for transient pipe flow

Abstract A robust prediction system should monitor all possible hydraulic transients, which is significant for the appropriate and safe operation of pipe systems. A second-order finite volume method (FVM) Godunov-type scheme (GTS) considering unsteady friction factors is introduced to simulate hydraulic transients, which was rarely involved in previous work. One explicit-solution source item approach developed in this work is crucial for the proposed GTS to easily incorporate various forms of the existing unsteady friction models, including original convolution-based models (Zielke model and Vardy–Brown model), simplified convolution-based model (Trikha–Vardy–Brown (TVB) model), and Brunone instantaneous acceleration-based model. Results achieved by the proposed models are compared with experimental data as well as predictions by the classic Method of Characteristics (MOC). Results show that the MOC scheme may produce severe numerical attenuation in the case of a low Courant number. The proposed second-order GTS unsteady friction models are accurate, efficient, and stable even for Courant numbers less than one and sparse grid, and only need much less grid number and computation time to reach the same numerical accuracy. The TVB convolution-based model and Brunone model in the second-order GTS are suggested for further applications in hydraulic transients due to their high accuracy and efficiency.

An Overview of the Numerical Approaches to Water Hammer Modelling: The Ongoing Quest for Practical and Accurate Numerical Approaches

Here, recent developments in the key numerical approaches to water hammer modelling are summarized and critiqued. This paper summarizes one-dimensional modelling using the finite difference method (FDM), the method of characteristics (MOC), and especially the more recent finite volume method (FVM). The discussion is briefly extended to two-dimensional modelling, as well as to computational fluid dynamics (CFD) approaches. Finite volume methods are of particular note, since they approximate the governing partial differential equations (PDEs) in a volume integral form, thus intrinsically conserving mass and momentum fluxes. Accuracy in transient modelling is particularly important in certain (typically more nuanced) applications, including fault (leakage and blockage) detection. The FVM, first advanced using Godunov’s scheme, is preferred in cases where wave celerity evolves over time (e.g., due to the release of air) or due to spatial changes (e.g., due to changes in wall thickness). Both numerical and experimental studies demonstrate that the first-order Godunov’s scheme compares favourably with the MOC in terms of accuracy and computational speed; with further advances in the FVM schemes, it progressively achieves faster and more accurate codes. The current range of numerical methods is discussed and illustrated, including highlighting both their limitations and their advantages.

MOC-CFD coupled model of load rejection in hydropower station

A modelling study investigates the consequences of transient flow conditions due to a turbine load rejection. The case study considers a large hydropower station with a long penstock. A three-dimensional (3D) Computational Fluid Dynamics (CFD) model is used to represent the spiral casing, guide vanes, runner, and draft tube. A one-dimensional (1D) Method of Characteristics (MOC) solver simulates water hammer in the penstock. The two models are coupled, to simulate a full load rejection. The results are compared with reference to field measurements and a pure 1D solver, combining the penstock and a turbine model based on machine and conveyance characteristics. A comparison of the high level data (head, flow, torque and rotational speed) reveals the two models reproduce the field data reasonably well. The exception being rotational speed toward the zero torque region, where both models underestimate speed. The model predicts high cycle pressure fluctuations on the turbine blade, which would produce serious mechanical loading. The source of the fluctuations is determined to be unstable vortices within the runner.

The numerical development of MOC for analyzing the inclined pipelines using the experimental network of Babol Noshirvani University as a case study

Abstract Despite the applications of the Method of Characteristics (MOC) for analyzing the unsteady flows, using this method in networks with variable elevations still has many challenges. In this paper, by developing modified correlations as a computer code, the possibility of analyzing inclined pipelines has been evaluated. For validation and calibration, the results of MOC were compared with the results of EPANET software as well as experimental data. To extract experimental data, the water network of Babol Noshirvani University of Technology (NIT) with a constant head of 7 m three loops, and four inclined branches were employed. While evaluating the capabilities of the developed computer code, the results show that for all pipes, as the number of pressure fluctuations in a specific period increases, the intensity of the pressure fluctuations decreases, and the damping speed increases as well. Moreover, in inclined pipes, unlike noninclined pipes, the intensity of pressure fluctuations will increase as the elevation increases and the cross-sectional distance from the transient event increases as well. The evaluation of the effect of space steps on the accuracy of the solution to the MOC shows that in the study network, considering 20 segments for each pipe, the fastest response time with an error of less than 1% is obtained.

Critical Buckling Generation of TCA Benchmark by the B1 Theory-Augmented Monte Carlo Calculation and Estimation of Uncertainties

The Korea Atomic Energy Research Institute (KAERI) has developed the DeCART2D 2-dimensional (2D) method of characteristics (MOC) transport code and the MASTER nodal diffusion code and has established its own two-step procedure. For design code licensing, KAERI prepared a critical experiment on the verification and validation (V&V) of the DeCART2D code. DeCART2D is able to perform the MOC calculation only for 2D nuclear fuel systems, such as the fuel assembly. Therefore, critical buckling in the vertical direction is essential for comparison between the results of experiments and DeCART2D. In this study, the B1 theory-augmented Monte Carlo (MC) method was adopted for the generation of critical buckling. To examine critical buckling using the B1 theory-augmented MC method, TCA critical experiment benchmark problems were considered. Based on the TCA benchmark results, it was confirmed that the DeCART2D code with the newly-generated critical buckling predicts the criticality very well. In addition, the critical buckling generated by the B1 theory-augmented MC method was bound to uncertainties. Therefore, utilizing basic equations (e.g., SNU S/U formulation) linking input uncertainties to output uncertainties, a new formulation to estimate the uncertainties of the newly generated critical buckling was derived. This was then used to compute the uncertainties of the critical buckling for a TCA critical experiment, under the assumption that nuclear cross-section data have uncertainties.

An Analytical Solution for Unsteady Laminar Flow in Tubes with a Tapered Wall Thickness

Transient fluid flows through tubes are critical in such topics as water hammer, ram pumps and pipeline dynamics. While analytical solutions exist in the literature for simple geometries such as tapered and non-tapered tube diameters, one area that is lacking is the case where the wave speed changes along the length. An example of this is a flexible pipe with a tapered wall thickness. In order to calculate the transient pressure response of such a system, this previously required a computationally expensive gridded method of characteristics (MOC) solution. This paper describes an analytical solution to the dynamic laminar flow of liquid in a tube where the wave speed varies along its length. This frequency-domain solution includes frequency-dependent friction effects. A comparison to a method of characteristics (MOC) solution is used to verify the solution. The paper also discusses some numerical issues and provides an approximate method that can be used for high-frequency calculations where limited numerical precision can cause errors. Finally, a preliminary comparison of the computational performance is presented, in which the new method is an order of magnitude faster to calculate than an MOC solution.

Investigation on the Controlling Factors of Pressure Wave Propagation Behavior Induced by Pulsating Hydraulic Fracturing

Summary Pulsating hydraulic fracturing (PHF) is a promising fracturing technology for unconventional reservoirs because it could improve the hydraulic fracturing efficiency through inducing the fatigue failure of reservoir rocks. Understanding of the pressure wave propagation behavior in wellbores and fractures plays an important role in PHF optimization. In this paper, a transient flow model (TFM) was used to describe the physical process of pressure wave propagation induced by PHF, and this model was solved by the method of characteristics (MOC). Combination of the TFM and MOC was validated with experimental data. The impacts of controlling factors on the pressure wave propagation behavior were fully discussed, and these factors include the frequency of input loading, an injection mode, an injection position, and friction. More than 10,000 sets of pressure wave propagation behaviors in different scenarios were simulated, and their differences were illustrated. In addition, the generation mechanisms of different pressure wave propagation behaviors were explained by the Fourier transform theory and the vibration theory. The important finding is that there is resonance phenomenon in the propagation of the pressure wave, and the resonance frequencies are almost equal to the natural frequencies of a fluid column. As a consequence of resonance phenomenon, the amplitudes of bottomhole pressure (BHP) and fracture tip pressure will increase sharply when the input loading frequency is close to the resonance frequency and less than 5 Hz; otherwise, the resonance phenomenon will disappear. Furthermore, an injection mode can alter the resonance frequency and the amplitude and frequency of the induced pressure wave. In addition, a friction effect can significantly decrease both the resonance frequency and the resonance amplitude. These findings indicate that the optimized input loading frequency should be close to the natural frequency of a fracturing fluid in a wellbore to enhance its BHP.

Field Measurements and Numerical Modeling of Hydraulic Transients in HDPE Pipeline with PRV Interaction

Pressure reducing valves (PRVs) and high-density polyethylene (HDPE) pipes are commonly used in urban water supply systems (UWSS). To study the joint effect of PRV and viscoelasticity on transient wave propagation, extensive experiments have been conducted in a field-scale reservoir-PRV-viscoelastic pipeline system covering a wide range of internal pressure heads (∼10 to 60 m) and air temperatures (12°C–33°C). In addition, a one-dimensional method of characteristics (MOC) based model that incorporates a PRV model and the generalized Kelvin-Voigt (K-V) model for pipe viscoelasticity is developed and validated against field data for the first time. The simulated transient pressures are in good agreement with field measurements. The K-V parameters exhibit a clustered distribution and the mean value for each element can provide a satisfactory simulation. The PRV in hydraulic transients can be interpreted as a quasi-dead-end with a self-adjusting opening, and causes additional positive pressure wave reflections that are gradually damped due to viscoelasticity. The dynamic changes in PRV opening, head loss, pressure wave reflection, and transmission induced by an incident pressure surge are predicted. Due to the joint action of PRV and pipeline viscoelasticity, the pressure oscillations in an intact pipe settle to a value that is considerably higher than the initial PRV set pressure. In a leaking pipe, the downstream pressure reverts to the original set pressure within a few wave cycles. Leaks can be detected by wave reflections in the time domain signals. The existence of leaks is also found to be associated with the the amplification and damping of the frequency response function (FRF) of the system at certain resonance peaks. This study provides new insights into wave propagation in HDPE pipeline with PRV interaction and an original data set for validation of leakage detection methods.