Method Of Characteristics Method Research Articles

Multi-fidelity modeling and analysis of a pressurized vessel-pipe-safety valve system based on MOC and surrogate modeling methods

A pressurized vessel-pipe-safety valve (PVPSV) combination is a commonly used configuration in nuclear power plants, and a good numerical model is essential for the system design, sizing and performance optimization. However, owing to the large-scale and cross-scale features, it is still a challenge to build a system level numerical model with both high accuracy and efficiency. To overcome this, a novel system level modeling method which can synthesize the advantages of various models is proposed in this paper. For system modeling, the analytical approach, the method of characteristics (MOC) and the surrogate model approach are respectively adopted to predict the dynamics of the pressure vessel, the connecting pipe and the safety valve, and different models are connected through data interfaces. With this system model, dynamic simulations were carried out and both the stable and the unstable system responses were obtained. For the model verification purpose, the simulation results were compared with those obtained from experiments and full CFD simulations. A good agreement and a better efficiency were obtained, verifying the ability of the model and the feasibility of the modeling method proposed in this paper.

Computation of the uncollided-flux moments with advanced MOC and QP methods

The growing need to solve radiation protection and shielding problems in reasonable computational times has led to an increasing interest in the use of deterministic approaches. Codes of this type commonly use the discrete-ordinates method. In cases of propagation of point sources in low scattering media, which are typical of shielding studies, the approximation leads to a set of numerical artefacts known as ray effect. One of the most commonly used techniques to mitigate this effect is the calculation of the uncollided flux, which is the most affected flux component, with a fine transport operator. In this paper we propose four different approaches: the method of characteristics (MOC), the trajectory-splitting MOC (TS-MOC), the Monte Carlo MOC (MC-MOC) and the method based on quadrature points (QP). They have been implemented inside the Cartesian solver IDT. This paper contains the formal description of the four numerical methods and their verification over a set of benchmark problems. An analysis of the accuracy and the computation performances of the methods is also given. Finally, a comparison of the four methods on the Kobayashi benchmark problem is proposed.

DEMONSTRATION OF A LINEAR PROLONGATION CMFD METHOD ON MOC

Coarse Mesh Finite Difference (CMFD) method is a very effective method to accelerate the iterations for neutron transport calculation. But it can degrade and even fail when the optical thickness of the mesh becomes large. Therefore several methods, including partial current-based CMFD (pCMFD) and optimally diffusive CMFD (odCMFD), have been proposed to stabilize the conventional CMFD method. Recently, a category of “higherorder” prolongation CMFD (hpCMFD) methods was proposed to use both the local and neighboring coarse mesh fluxes to update the fine cell flux, which can solve the fine cell scalar flux discontinuity problem between the fine cells at the bounary of the coarse mesh. One of the hpCMFD methods, refered as lpCMFD, was proposed to use a linear prolongation to update the fine cell scalar fluxes. Method of Characteristics (MOC) is a very popular method to solve neutron transport equations. In this paper, lpCMFD is applied on the MOC code MPACT for a variety of fine meshes. A track-based centroids calculation method is introduced to find the centroids coordinates for random shapes of fine cells. And the numerical results of a 2D C5G7 problem are provided to demonstrate the stability and efficiency of lpCMFD method on MOC. It shows that lpCMFD can stabilize the CMFD iterations in MOC method effectively and lpCMFD method performs better than odCMFD on reducing the outer MOC iterations.

Effect of Gas/Oil Capillary Pressure on Minimum Miscibility Pressure for Tight Reservoirs

SummaryShale and tight reservoir rocks have pore throats on the order of nanometers, and, subsequently, a large capillary pressure. When the permeability is ultralow (k < 200 nd), as in many shale reservoirs, diffusion might dominate over advection, so that the gas injection might no longer be controlled by the multicontact minimum miscibility pressure (MMP). For gasfloods in tight reservoirs, where k > 200 nd and capillary pressure is still large, however, advection likely dominates over diffusive transport, so that the MMP once again becomes important. This paper focuses on the latter case to demonstrate that the capillary pressure, which has an impact on the fluid pressure/volume/temperature (PVT) behavior, can also alter the MMP.The results show that the calculation of the MMP for reservoirs with nanopores is affected by the gas/oil capillary pressure, owing to alteration of the key tie lines in the displacement; however, the change in the MMP is not significant. The MMP is calculated using three methods: the method of characteristics (MOC); multiple mixing cells; and slimtube simulations. The MOC method relies on solving hyperbolic equations, so the gas/oil capillary pressure is assumed to be constant along all tie lines (saturation variations are not accounted for). Thus, the MOC method is not accurate away from the MMP but becomes accurate as the MMP is approached when one of the key tie lines first intersects a critical point (where the capillary pressure then becomes zero, making saturation variations immaterial there). Even though the capillary pressure is zero for this key tie line, its phase compositions (and, hence, the MMP) are impacted by the alteration of all other key tie lines in the composition space by the gas/oil capillary pressure. The reason for the change in the MMP is illustrated graphically for quaternary systems, in which the MMP values from the three methods agree well. The 1D simulations (typically slimtube simulations) show an agreement with these calculations as well. We also demonstrate the impact of capillary pressure on CO2-MMP for real reservoir fluids. The effect of large gas/oil capillary pressure on the characteristics of immiscible displacements, which occur at pressures well below the MMP, is discussed.

Least squares linear source approach in method of characteristics

Method of characteristics (MOC) has been widely used in reactor physics calculation due to its high computational precision and great ability to deal with complex geometry. The linear source approximation is proposed to reduce the number of the meshes to improve the efficiency of the MOC method. In this research, a new least squares linear source MOC method is proposed and implemented. The numerical results show that the new least squares linear source MOC code is cable to deal with tiny mesh, such as gap in the PWR cell and the comparison among diamond difference (DD) method and several linear source MOC methods is performed. Numerical results indicate that the least squares linear source MOC method achieves desire precision when using much less mesh and the new method is stable with the number of the mesh increasing.

A modified method for modelling, optimization and control of an anti-solvent crystallization process

Crystallization is a complex process of heat and mass transfer, its product quality is affected by many factors, thus it is a challenge to control this process. Modelling and optimization of the manipulated variables is a feasible way to improve the product quality of crystallization. In this paper, a modified numerical scheme is developed to solve the process model of anti-solvent batch crystallization. The proposed numerical scheme can handle growth, nucleation, agglomeration and breakage kinetics by means of the Method of Characteristics (MOC), which provides an accurate and efficient result of the crystal size distribution (CSD). Then Non-linear Model Predictive Control (NMPC) and non-linear Moving Horizon Estimation (MHE) are presented to provide a temperature profile for optimizing the CSD. The result of case studies indicates that the calculated results of the MOC modified process model were in agreement with the experimental results, which indicated that the MOC method could deal with the specific crystallization process efficiently and accurately. The NMPC-MHE optimized temperature control strategy was tested using the verification experiment of anti-solvent crystallization of β-Artemether, the validated result indicated that the method developed had high feasibility and accuracy. The computational time for the MHE and NMPC formulations with 90 control and sampling steps is well within the sampling interval, which allows for real time process control.

A new approach to three-dimensional neutron transport solution based on the method of characteristics and linear axial approximation

A new approach based on the method of characteristics (MOC) is proposed to solve the neutron transport equation. A new three-dimensional (3D) spatial discretization is applied to avoid the instability issue of the transverse leakage iteration of the traditional 2D/1D approach. In this new approach, the axial and radial variables are discretized in two different ways: the linear expansion is performed in the axial direction, then, the 3D solution of the angular flux is transformed to be the planar solution of 2D angular expansion moments, which are solved by the planar MOC sweeping. Based on the boundary and interface continuity conditions, the 2D expansion moment solution is equivalently transformed to be the solution of the axially averaged angular flux. Using the piecewise averaged angular flux at the top and bottom surfaces of 3D meshes, the planes are coupled to give the 3D angular flux distribution. The 3D CMFD linear system is established from the surface net current of every 3D pin-mesh to accelerate the convergence of power iteration. The STREAM code is extended to be capable of handling 3D problems based on the new approach. Several benchmarks are tested to verify its feasibility and accuracy, including the 3D homogeneous benchmarks and heterogeneous benchmarks. The computational sensitivity is discussed. The results show good accuracy in all tests. With the CMFD acceleration, the convergence is stable. In addition, a pin-cell problem with void gap is calculated. This shows the advantage compared to the traditional 2D/1D MOC methods.

Geometry pretreatment method for the 3D modular characteristic method based on AutoCAD

The method of characteristics (MOC) is a competitive and widely implemented method for the 2D and 2D/1D transport calculations. However, it does not play significant of a role for the 3D transport calculations as it does in the 2D cases because of three reasons: the difficulty of pretreatment of 3D complicated geometries, tremendous amount of ray tracing data, and a comparably long running time. The focus of this work is on the geometry pretreatment based on the AutoCAD software for the 3D modular characteristic method, which was previously proposed to reduce the ray tracing data. The proposed geometry pretreatment consists of four steps: (1) the problem to be solved is modeled within AutoCAD, (2) the modeled geometry is then automatically dissected into modules, and this process is referred to as modularization, (3) during the modularization, all typical modules (unique modules) and their arrangement will be calculated, (4) characteristic segment information is generated for all typical modules. Finally the output information is used by the 3D modular characteristic code to perform the transport calculations. Numerical results demonstrate the faithful representation capability of the proposed geometry pretreatment method, which enables the MOC method have the real capability of dealing with the 3D complicated problems.

The Powerful Method of Characteristics Module in Advanced Neutronics Lattice Code KYLIN-2

Due to powerful geometry treatment capability, method of characteristics (MOC) currently becomes one of the best method to solve neutron transport equation. In MOC method, boundary condition treatment, complex geometry representation, and advanced acceleration method are the key techniques to develop a powerful MOC code to solve complex problem. In this paper, we developed a powerful MOC module, which can treat different boundary conditions with two methods. For problems with special border shapes and boundary condition, such as rectangle, 1/8 of square, hexagon, 1/3 of hexagon, 1/6 of hexagon problems with reflection, rotation, and translation boundary condition, the MOC module adopts periodic tracking method, with which rays can return to start point after a certain distance. For problems with general border shapes, the MOC module uses ray prolongation method, which can treat arbitrary border shapes and boundary conditions. Meanwhile, graphic user interface based on computer aided design (CAD) software is developed to generate the geometry description file, in which geometry is represented by “lines and arcs” method. With the graphic user interface, the geometry and mesh can be described and modified correctly and fast. In order to accelerate the MOC transport calculation, the generalized coarse mesh finite difference (GCMFD) is used, which can use irregular coarse mesh diffusion method to accelerate the transport equation. The MOC module was incorporated into advanced neutronics lattice code KYLIN-2, which was developed by Nuclear Power Institute of China (NPIC) and used to simulate the assembly of current pressurized water reactor (PWR) and advanced reactors, to solve the transport equation with multigroup energy structure in cross sections database. The numerical results show that the KYLIN-2 code can be used to calculate 2D neutron transport problems in reactor accurately and fast. In future, the KYLIN-2 code will be released and gradually become the main neutron transport lattice code in NPIC.

The Random Ray Method for neutral particle transport

A new approach to solving partial differential equations (PDEs) based on the method of characteristics (MOC) is presented. The Random Ray Method (TRRM) uses a stochastic rather than deterministic discretization of characteristic tracks to integrate the phase space of a problem. TRRM is potentially applicable in a number of transport simulation fields where long characteristic methods are used, such as neutron transport and gamma ray transport in reactor physics as well as radiative transfer in astrophysics. In this study, TRRM is developed and then tested on a series of exemplar reactor physics benchmark problems. The results show extreme improvements in memory efficiency compared to deterministic MOC methods, while also reducing algorithmic complexity, allowing for a sparser computational grid to be used while maintaining accuracy.

GENESIS: A Three-Dimensional Heterogeneous Transport Solver Based on the Legendre Polynomial Expansion of Angular Flux Method

A heterogeneous transport solver in three-dimensional (3-D) geometry, GENESIS, is developed incorporating recent developments in the method of characteristics (MOC) in 3-D geometry. The Legendre Polynomial Expansion of Angular Flux (LEAF) method is implemented in the GENESIS code, in which neutron transport is calculated in two-dimensional (2-D) characteristics planes rather than in one-dimensional characteristics lines adopted in the conventional approach of 3-D MOC. Unlike the planar MOC method that combines 2-D MOC calculations through axial leakages, the GENESIS code explicitly considers angular and spatial dependence of outgoing and incoming angular fluxes between axial planes. Thus, the GENESIS code eliminates a crucial approximation used in the planar MOC method: No approximation is used for axial leakage treatment. The GENESIS code can handle flexible shapes of objects in rectangular or hexagonal geometry. A two-level, multigroup generalized coarse mesh rebalance acceleration method is adopted for efficient convergence of neutron transport calculation. Performance of the GENESIS code is verified through various benchmark calculations. The calculation results indicate the fidelity of the GENESIS code based on the LEAF method.

A time-dependent neutron transport method of characteristics formulation with time derivative propagation

A new time-dependent Method of Characteristics (MOC) formulation for nuclear reactor kinetics was developed utilizing angular flux time-derivative propagation. This method avoids the requirement of storing the angular flux at previous points in time to represent a discretized time derivative; instead, an equation for the angular flux time derivative along 1D spatial characteristics is derived and solved concurrently with the 1D transport characteristic equation. This approach allows the angular flux time derivative to be recast principally in terms of the neutron source time derivatives, which are approximated to high-order accuracy using the backward differentiation formula (BDF). This approach, called Source Derivative Propagation (SDP), drastically reduces the memory requirements of time-dependent MOC relative to methods that require storing the angular flux. An SDP method was developed for 2D and 3D applications and implemented in the computer code DeCART in 2D. DeCART was used to model two reactor transient benchmarks: a modified TWIGL problem and a C5G7 transient. The SDP method accurately and efficiently replicated the solution of the conventional time-dependent MOC method using two orders of magnitude less memory.

Analytical Solutions for Gas Displacements With Bifurcating Phase Behavior

Summary Minimum miscibility pressure (MMP) is one of the most important parameters in the design of a successful gasflooding process. The most-reliable methods to calculate the MMP are based on slimtube experiments, 1D slimtube simulations, mixing-cell calculations, and the analytical methods known as the method of characteristics (MOC). The calculation of MMP by use of MOC is the fastest method because it relies solely on finding the key tie lines in the displacement path. The MOC method for MMP estimation in its current form assumes that the composition path is a series of shocks from one key tie line to the next. For some oils, however, these key tie lines do not control miscibility, and the MMP calculated by use of the key-tie line approach can be significantly in error. The error can be as high as 5,000 psia for heavier oils or CO2 displacements at low temperature in which three-phase hydrocarbon regions can exist (L1–L2–V). At higher pressures, the two- or three-phase region can split (or bifurcate) into two separate two-phase regions (L1–L2 and L1–V regions). Thus, for the MMP calculation from MOC to be correct, we must calculate the entire composition path for this complex phase behavior, instead of relying on the shock assumption from one key tie line to the next. In this paper, the MOC-composition route is developed completely for the bifurcating phase-behavior displacement for pure CO2 injection by use of a simplified pseudoternary system that is analogous to the complex phase behavior observed for several real displacements with CO2. We develop the MOC analytical solutions by honoring all constraints required for a unique solution—velocity, mass balance, entropy, and solution continuity. The results show that a combination of shocks and rarefaction waves exists along the nontie-line path, unlike previous MOC solutions reported to date. We show that by considering the entire composition path, not just the key tie lines, the calculated MMP agrees with the mixing-cell method. We also show that, in this complex ternary displacement, the displacement mechanism has features of a both condensing and vaporizing (C/V) drive, which was thought to be possible only for gasfloods with four or more components. For pure CO2 injection, the solution also becomes discontinuous for oils that lie on the tie line envelope curve. Finally, we show that shock paths within the two-phase region are generally curved in composition space and that there is no MMP for some oil compositions considered in the displacements by CO2. Recovery can be large even though the MMP is not reached.

Limitations of Current Method-of-Characteristics (MOC) Methods Using Shock-Jump Approximations To Predict MMPs for Complex Gas/Oil Displacements

Summary An accurate minimum miscibility pressure (MMP) is one of the key factors in miscible-gasflood design. There is a variety of experimental and analytical methods to determine the MMP, but the most-reliable methods are slimtube experiments, 1D slimtube simulations, mixing-cell models, and the fast key-tie-line approach using the method of characteristics (MOC). Direct comparisons of all these methods generally agree well, but there are cases in which they do not. No explanation has yet been given for these anomalies, although the MMP is critically important to recovery. The focus of this paper is to explain when current MOC results assuming that shocks exist from one key tie line to the next may not be reliable, and how to identify when this is the case. We demonstrate, using fluid characterizations from Middle East oils, that the MMPs using this MOC method can be more than 6,500 psia greater than those calculated using a recently developed mixing-cell method. The observed differences in the MMP increase substantially as the API gravity of the oil decreases, likely the result of the onset of L1-L2-V behavior. We show that the key tie lines determined using this MOC method do not control miscibility for such cases. We explain the reasons for these differences using simplified pseudoternary models and show how to determine when an error exists. We also offer a way to correct the MMP predictions using the MOC for these complex gas/oil displacements without solving for the complete compositional path.

Multiple-Mixing-Cell Method for MMP Calculations

SummaryThe minimum miscibility pressure (MMP) is a key parameter governing the displacement efficiency of gasfloods. There are several methods to determine the MMP, but the most accurate methods are slim-tube experiments, analytical methods, and numerical-simulation/cell-to-cell methods. Slim-tube experiments are important to perform because they use actual crude oil, but they are costly and time consuming. Analytical methods that use the method of characteristics (MOC) are very fast and help to understand the structure of gasfloods. MOC, however, relies on finding the unique and correct set of key tie lines in the displacements, which can be difficult. Slim-tube simulation methods and their simplified cell-to-cell derivatives require tedious fluid and rock inputs, and their MMP estimates can be clouded by dispersion.This paper presents a simple and accurate multiple-mixing-cell method for MMP calculations that corrects for dispersion, and is faster and less cumbersome than 1D simulation methods. Unlike previous mixing-cell methods, our cell-to-cell mixing model uses a variable number of cells, and is independent of gas/oil ratio, volume of the cells, excess oil volumes, and the amount of gas injected. The new method only relies on robust P/T flash calculations using any cubic equation-of-state (EOS). The calculations begin with only two cells and perform additional cell-to-cell contacts between resulting equilibrium-phase compositions based on equilibrium gas moving ahead of the equilibrium liquid phase. We show for a variety of oil and gas compositions that all key tie lines can be found to the desired accuracy, and that they are nearly identical to those found using analytical MOC methods. Our approach, however, is more accurate and robust than those from MOC because we do not make approximations regarding shocks along nontie-line paths, and the unique set of key tie lines converges automatically.The MMP using our mixing-cell method can be calculated in minutes using an Excel spreadsheet and is estimated from a novel bisection method of the minimum tie-line lengths observed in the cells at four or five pressures. Our multiple-mixing-cell method can calculate either the MMP or the minimum miscibility for enrichment (MME) independent of the number of components in the gas or oil. Our approach further supports the notion that the MMP is independent of fractional flow because we obtain the same key tie lines independent of how much fluid is moved from one cell to another.

Diffusion coefficients for LMFBR cells calculated with MOC and Monte Carlo methods

The present work discusses the calculation of the diffusion coefficient of a lattice of hexagonal cells, with both “sodium present” and “sodium absent” conditions. Calculations are performed in the framework of lattice theory (also known as fundamental mode approximation). Unlike the classical approaches, our heterogeneous leakage model allows the calculation of diffusion coefficients under all conditions, even if planar voids are present in the lattice. Equations resulting from this model are solved using the method of characteristics (MOC). Independent confirmation of the MOC result comes from Monte Carlo calculations, in which the diffusion coefficient is obtained without any of the assumptions of lattice theory. It is shown by comparison to the Monte Carlo results that the MOC solution yields correct values of the diffusion coefficient under all conditions, even in cases where the classic calculation of the diffusion coefficient fails. This work is a first step in the development of a robust method to calculate the diffusion coefficient of lattice cells. Adoption into production codes will require more development and validation of the method.

Godunov’s Method for Simulatinons of Fluid-Structure Interaction in Piping Systems

Constant coefficient one-dimensional linear hyperbolic systems of partial differential equations (PDEs) are often used for description of fluid-structure interaction (FSI) phenomena during transient conditions in piping systems. In the past, these systems of equations have been numerically solved with method of characteristics (MOCs). The MOC method is actually the most efficient and accurate method for description of the single-phase transient in the cold liquid where the constant coefficient mathematical model describes phenomenon with sufficient accuracy. In energy production systems where hot pressurized liquid is used for heat transfer between the heat source and the steam generator, more complex and nonlinear mathematical models are needed to describe transient flow and these models cannot be solved with MOC method because the models are not constant. In addition, the MOC method can be used for pipes having discontinuities like elbows, geometrical changes, material properties changes, etc., but only with some extra numerical modeling. An interesting alternative is explicit characteristic upwind numerical method, known as Godunov’s method that is frequently used for nonlinear systems or systems where properties change with position. In the present study, applicability of the Godunov’s method for the FSI analyses is tested with eight first order PDEs mathematical model. The conventional linear mathematical model is improved with convective term that makes the system nonlinear and additional terms that enable simulations of the FSI in arbitrarily shaped piping systems located in a plane. Two PDEs describe pressure waves in the single-phase fluid and six PDEs describe axial, lateral, and rotational stress waves in the pipe. The applied system of equations has stiff source terms. This numerical problem is solved introducing implicit iterations. The proposed model is verified with a rod impact experiment that is carried out on single-elbow pipe hanging on wires. Godunov’s method is found as a very promising numerical method for simulations of the FSI problems.

A hybrid method for the numerical prediction of enthalpy transport in fluid flow

A hybrid method (HM) is proposed for the prediction of convective enthalpy transport and the boiling boundary location. The momentum equation is solved with the SIMPLE procedure, while the energy equation is solved with the Method of Characteristics (MOC) and with the application of the Lagrange interpolation polynomial (LIP). The MOC method is applied on an equidistant grid. The initial values at the starting points of characteristic paths are calculated with the LIP. The accuracy of the method is demonstrated on tests of the transient boiling boundary prediction and the well known problem of a propagating discontinuity. The dependence of the hybrid method on the spatial integration step size and the degree of the LIP is analyzed. The LIP of the third degree is recommended, by which practically “exact” solutions with acceptable number of nodes are obtained.