Optimization of sensor deployment sequences for hazardous gas leakage monitoring and source term estimation

Nowadays, chemical safety has attracted considerable attention, and chemical gas leakage monitoring and source term estimation(STE) have become hot spots. However, few studies have focused on sensor layouts in scenarios with multiple potential leakage sources and wind conditions, and studies on the risk information(RI) detection and prioritization order of sensors have not been performed. In this work, the monitoring area of a chemical factory is divided into multiple rectangles with a uniform mesh. The RI value of each grid node is calculated on the basis of the occurrence probability and normalized concentrations of each leakage scenario. A high RI value indicates that a sensor at a grid node has a high chance of detecting gas concentrations in different leakage scenarios. This situation is beneficial for leakage monitoring and STE. The methods of similarity redundancy detection and the maximization of sensor RI detection are applied to determine the sequence of sensor locations. This study reveals that the RI detection of the optimal sensor layout with eight sensors exceeds that of the typical layout with 12 sensors. In addition, STE with the optimized placement sequence of the sensor layout is numerically simulated. The statistical results of each scenario with various numbers of sensors reveal that STE is affected by sensor number and scenarios(leakage locations and winds). In most scenarios, appropriate STE results can be retained under the optimal sensor layout even with four sensors. Eight or more sensors are advised to improve the performance of STE in all scenarios. Moreover, the reliability of the STE results in each scenario can be known in advance with a specific number of sensors. Such information thus provides a reference for emergency rescue.

Accident source term and radiological consequences of a small modular reactor

Considering the growing global demand for energy and the need for countries to achieve climate goals,there is an increasing global interest in small modular reactors(SMRs)and their applications.Accident source term and radiological consequence evaluations of SMRs are key components of nuclear and radiation safety reviews,which affect the site,exclusion area(EAB),and low population zone outer boundaries.Based on the design characteristics of the SMR and accident analysis results,a theoretical model of a whole-core fuel cladding damage accident was constructed to study the radioactivity released into the environment and its consequences.The accident source term and radiation dose calculation models were established to analyze the released amounts of radionuclides and the total effective dose affecting individuals at the site boundary.The results showed that the amount of radionuclides released into the environment after a whole-core fuel cladding damage accident reached 10^(14) Bq,among which the release amount of ^(133)Xe was the largest.The total effective dose at the site boundary 30 days after the accident was 8.65 mSv.The highest total effective dose affecting individuals occurred to the east-north-east.The results of the accident source term and radiological consequence provide technical support for site boundary dose assessments and reviews of SMRs.

An improved four-dimensional variation source term inversion model with observation error regularization

Aiming at the Four-Dimensional Variation source term inversion algorithm proposed earlier,the observation error regularization factor is introduced to improve the prediction accuracy of the diffusion model,and an improved Four-Dimensional Variation source term inversion algorithm with observation error regularization(OER-4DVAR STI model)is formed.Firstly,by constructing the inversion process and basic model of OER-4DVAR STI model,its basic principle and logical structure are studied.Secondly,the observation error regularization factor estimation method based on Bayesian optimization is proposed,and the error factor is separated and optimized by two parameters:error statistical time and deviation degree.Finally,the scientific,feasible and advanced nature of the OER-4DVAR STI model are verified by numerical simulation and tracer test data.The experimental results show that OER-4DVAR STI model can better reverse calculate the hazard source term information under the conditions of high atmospheric stability and flat underlying surface.Compared with the previous inversion algorithm,the source intensity estimation accuracy of OER-4DVAR STI model is improved by about 46.97%,and the source location estimation accuracy is improved by about 26.72%.

MULTIPLICITY OF SOLUTIONS AND SOURCE TERMS IN A FOURTH ORDER NONLINEAR ELLIPTIC EQUATION

The authors investigatc relations between multiplicity of solutions and sourceterms of the fourth order nonlinear elliptic boundary value problem under Dirichlet boundary condition △2u＋c△u = bu+＋f inΩ, wherc Ω is a bounded open set in Rn with smoothbonndary and the nonlinearity bu+ crosses eigenvalues of △2 ＋c△. They investigate therelatiolls when the source term is constant and when it is generated by two eigenfuntions.

Exact Solution of Fractional Diffusion Model with Source Term used in Study of Concentration of Fission Product in Uranium Dioxide Particle

The exact solution of fractional diffusion model with a location-independent source term used in the study of the concentration of fission product in spherical uranium dioxide （U02） particle is built. The adsorption effect of the fission product on the surface of the U02 particle and the delayed decay effect are also considered. The solution is given in terms of Mittag-Leffler function with finite Hankel integral transformation and Laplace transformation. At last, the reduced forms of the solution under some special physical conditions, which is used in nuclear engineering, are obtained and corresponding remarks are given to provide significant exact results to the concentration analysis of nuclear fission products in nuclear reactor.

Least-squares finite-element method for shallow-water equations with source terms

Numerical solution of shallow-water equations （SWE） has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of source terms in momentum equations, such as the bottom slope and friction of bed, compounds the difficulties further. In this paper, a least-squares finite-element method for the space discretization and θ-method for the time integration is developed for the 2D non-conservative SWE including the source terms. Advantages of the method include： the source terms can be approximated easily with interpolation functions, no upwind scheme is needed, as well as the resulting system equations is symmetric and positive-definite, therefore, can be solved efficiently with the conjugate gradient method. The method is applied to steady and unsteady flows, subcritical and transcritical flow over a bump, 1D and 2D circular dam-break, wave past a circular cylinder, as well as wave past a hump. Computed results show good C-property, conservation property and compare well with exact solutions and other numerical results for flows with weak and mild gradient changes, but lead to inaccurate predictions for flows with strong gradient changes and discontinuities.

TWO REGULARIZATION METHODS FOR IDENTIFYING THE SOURCE TERM PROBLEM ON THE TIME-FRACTIONAL DIFFUSION EQUATION WITH A HYPER-BESSEL OPERATOR

In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional stability.Then,we give the optimal error bound for this inverse problem.Next,we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem,and give corresponding error estimates under different regularization parameter selection rules.Finally,we verify the effectiveness of the method through numerical examples.

Approximation of Derivative for a Singularly Perturbed Second-Order ODE of Robin Type with Discontinuous Convection Coefficient and Source Term

In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving numerical method is proposed. An e-uniform global error estimate for the numerical solution and also to the numerical derivative are established. Numerical results are presented, which are in agreement with the theoretical predictions.

Well-posedness for A Plate Equation with Nonlocal Source term

In this paper,we investigate the initial boundary value problem for a plate equation with nonlocal source term.The local,global existence and exponential decay result are established under certain conditions.Moreover,we also prove the blow-up in finite time and the lifespan of solution under certain conditions.

Influence of active and passive equipment for advanced pressurized water reactor on thermal hydraulic and source term behavior in severe accidents

Extensive studies have been carried out on the behavior of core degradation and fission products of common pressurized water reactors(PWRs).However,few of them have investigated the relationship between thermal hydraulic and fission product behavior in advanced passive PWRs.Due to the impact of thermal hydraulic be-haviors in different accident sequences on the release and transportation of fission products,an integrated severe accident analysis(ISAA)code with highly coupled thermal hydraulic and source term calculations is required to simultaneously analyze thermal hydraulic and source term behavior.For advanced passive PWRs,important safety systems that may affect the behavior of the core and fission products should be considered.It is therefore necessary to simulate the thermal hydraulic and fission product behavior of advanced passive PWRs.In this study,the ISAA code is adopted to simulate the occurrence of a hypothetical double ended cold leg LBLOCA of HPR1000 in three scenarios of equipment failure.The results show that the high-temperature fuel rods and cladding ma-terials exhibit delayed failure at the lower position of the active core,whereas earlier failure at higher position during the reflooding.Active and passive equipment affects fuel temperature,the oxidation conditions of the fuel,the interaction of fission products and structural materials,and the state of the fuel,thereby affecting the release of fission products in the fuel.HPR1000 only relies on passive equipment to relieve the core degradation in severe accidents,realize the in-vessel retention of melt,and eliminate the ex-vessel release possibility of fission product.It is hoped that the results can provide references for HPR1000 to formulate the severe accident management guidelines(SAMG).

Research on inversion method for complex source-term distributions based on deep neural networks

This study proposes a source distribution inversion convolutional neural network (SDICNN), which is deep neural network model for the inversion of complex source distributions, to solve inversion problems involving fixed-source distributions. A function is developed to obtain the distribution information of complex source terms from radiation parameters at individual sampling points in space. The SDICNN comprises two components:a fully connected network and a convolutional neural network. The fully connected network mainly extracts the parameter measurement information from the sampling points,whereas the convolutional neural network mainly completes the fine inversion of the source-term distribution. Finally, the SDICNN obtains a high-resolution source-term distribution image. In this study, the proposed source-term inversion method is evaluated based on typical geometric scenarios. The results show that, unlike the conventional fully connected neural network, the SDICNN model can extract the two-dimensional distribution features of the source terms, and its inversion results are better. In addition, the effects of the shielding mechanism and number of sampling points on the inversion process are examined. In summary, the result of this study can facilitate the accurate assessment of dose distributions in nuclear facilities.

High Order Accurate Direct Arbitrary-Lagrangian-Eulerian ADER-MOOD Finite Volume Schemes for Non-Conservative Hyperbolic Systems with Stiff Source Terms

In this paper we present a 2D/3D high order accurate finite volume scheme in the context of direct Arbitrary-Lagrangian-Eulerian algorithms for general hyperbolic systems of partial differential equations with non-conservative products and stiff source terms.This scheme is constructed with a single stencil polynomial reconstruction operator,a one-step space-time ADER integration which is suitably designed for dealing even with stiff sources,a nodal solver with relaxation to determine the mesh motion,a path-conservative integration technique for the treatment of non-conservative products and an a posteriori stabilization procedure derived from the so-called Multidimensional Optimal Order Detection(MOOD)paradigm.In this work we consider the seven equation Baer-Nunziato model of compressible multi-phase flows as a representative model involving non-conservative products as well as relaxation source terms which are allowed to become stiff.The new scheme is validated against a set of test cases on 2D/3D unstructured moving meshes on parallel machines and the high order of accuracy achieved by the method is demonstrated by performing a numerical convergence study.Classical Riemann problems and explosion problems with exact solutions are simulated in 2D and 3D.The overall numerical code is also profiled to provide an estimate of the computational cost required by each component of the whole algorithm.

A New Approach of High OrderWell-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms.In our earlier work[31–33],we designed high order well-balanced schemes to a class of hyperbolic systems with separable source terms.In this paper,we present a different approach to the same purpose:designing high order well-balanced finite volume weighted essentially non-oscillatory(WENO)schemes and RungeKutta discontinuous Galerkin(RKDG)finite element methods.We make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly,if a small modification on either the initial condition or the flux is provided.The computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG method.The same idea can be applied to the finite volume WENO schemes.We will first describe the algorithms and prove the well balanced property for the shallow water equations,and then show that the result can be generalized to a class of other balance laws.We perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions,the non-oscillatory property for general solutions with discontinuities,and the genuine high order accuracy in smooth regions.

POINTWISE AND UPWIND DISCRETIZATIONS OF SOURCE TERMS IN OPEN-CHANNEL FLOOD ROUTING

Upwind algorithms are becoming progressively popular for river flood routing due to their capability of resolving trans-critical flow regimes. For consistency, these algorithms suggest natural upwind discretization of the source term, which may be essential for natural channels with irregular geometry. Yet applications of these upwind algorithms to natural river flows are rare, and in such applications the traditional and simpler pointwise, rather than upwind discretization of the source term is used. Within the framework of a first-order upwind algorithm, this paper presents a comparison of upwind and pointwise discretizations of the source term. Numerical simulations were carried out for a selected irregular channel comprising a pool-riffle sequence Jn the River Lune, England with observed data. It is Shown that the impact of pointwise discretization, compared to the upwind, is appreciable mainly in flow zones with the Froude number closer to or larger than unity. The discrepancy due to pointwise and upwind discretizations of the source term is negligible in flow depth and hence in water surface elevation, but well manifested in mean velocity and derived flow quantities. Also the occurrence of flow reversal and equalisation over the pool-riffle sequence in response to increasing discharges is demonstrated.

THAI experimental research on hydrogen risk and source term related safety systems

In the defense-in-depth concept employed for the safety of nuclear installations,maintaining integrity of containment as the last barrier is of high importance to limit the release of radioactivity to the environment in case of a severe accident.The active and passive safety systems implemented in containments of light water reactors(LWRs)are designed to limit the consequences of such accidents.Assessing the performance and reliability of such systems under accident conditions is critical to the safety of nuclear installations.In the aftermath of the Fukushima accident,there has been focus on re-examining the existing safety systems to demonstrate their capabilities for a broader range of boundary conditions comprising both the early as well as the late phases of an accident.In addition to the performance testing of safety systems,their interaction with containment atmosphere needs detailed investigations to evaluate the effects of operation of safety systems on H2 risk and fission product(FP)behavior in containment,which may ultimately have an impact on the source term to the environment.In this context,an extensive containment safety related experimental research has been conducted in a thermalhydraulics,hydrogen,aerosols,and iodine test facility(THAI,60 m3,single vessel)/(THAI+,80 m3,two interconnected vessels).Related to the subject of this paper,experimental investigations covered performance testing of various safety and mitigation systems,i.e.,containment spray,passive autocatalytic recombiner(PAR),pressure suppression pool(water pools),and effects of their operation on H2 risk and in-containment FP behavior.The experimental results have provided a better phenomenological understanding and database for validation and further improvement of a safety analysis tool based on computation fluid dynamic(CFD)and lumped parameter(LP)modeling approach.This paper summarizes the main insights obtained from the aforesaid THAI experimental research covering safety systems installed in containments of LWRs.The relevance of experimental outcomes for reactor safety purpose is also discussed.

Estimation of a source term in a quasi steady twodimensional heat transfer problem： application to an electron beam welding

In previous work, we have analyzed the feasibility of the estimation for a source term S（x, y, z） in a transversal section, The present study is concerned with a twodimensional inverse phase change problem. The goal is the estimation of the dissipated heat flux in the liquid zone （reconstruction of a source term in the energy equation） from experimentally measured temperatures in the solid zone. This work has an application in the electron beam welding of steels of thickness about 8cm. The direct thermometallurgical problem is treated in a quasi steady two-dimensional longitudinal section （x, y）. The beam displacement is normally in the y direction. But in the quasi steady simulation, the beam is steady in the study section. The sample is divided in the axial direction z in few sections. At each section, a source term is defined with a part of the beam and creates a vaporized zone and a fused zone. The goal of this work is the rebuilding of the complete source term with the estimations at each section. In this paper, we analyze the feasibility of the estimation. For this work, we use only the simulated measurements without noise.

Forest Canopy Flow Analysis Using Turbulence Model with Source/Sink Terms

A computational fluid dynamics( CFD) model was presented to simulate wind flow over a forest canopy for analyzing the wind flow within and above forest canopies. Unlike previous studies on the canopy flow,the effect of canopy contour on the canopy was considered to develop the simulation method into a more general but complex case of wind flow over a forest canopy,using cedrus deodara and cinnamomum camphora. The desire of this work is mainly motivated to provide a rational way for predicting the wind flow within and above vegetation canopies. The model of canopy is not incorporated in the geometrical model,and it uses a porous domain combined with k-ε two-equation turbulence model with source / sink terms. The objectives of this paper are to analyze the contour of pressure and velocity and compare the simulation results with other works and field measurements. Results are encouraging,as the model profiles of mean velocity( u) qualitatively agree well with other works compared with and quantitatively have similar explanations as several authors. In conclusion, it is demonstrated that the adoption turbulence model with source / sink terms for forest canopies is proved to be a physically accurate and numerically robust method. The model and method are recommended for future use in simulating turbulent flows in forest canopies.

Efficient Finite Difference WENO Scheme for Hyperbolic Systems withNon-conservativeProducts

Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.

Two-dimensional shallow water equations with porosity and their numerical scheme on unstructured grids

In this study, porosity was introduced into two-dimensional shallow water equations to reflect the effects of obstructions, leading to the modification of the expressions for the flux and source terms. An extra porosity source term appears in the momentum equation. The numerical model of the shallow water equations with porosity is presented with the finite volume method on unstructured grids and the modified Roe-type approximate Riemann solver. The source terms of the bed slope and porosity are both decomposed in the characteristic direction so that the numerical scheme can exactly satisfy the conservative property. The present model was tested with a dam break with discontinuous porosity and a flash flood in the Toce River Valley. The results show that the model can simulate the influence of obstructions, and the numerical scheme can maintain the flux balance at the interface with high efficiency and resolution.