Three-dimensional Diffusion Equation Research Articles

A Generalized Selectively Relaxed Matrix Splitting Preconditioning Strategy for Three-Dimensional Flux-Limited Multi-Group Radiation Diffusion Equations

A Generalized Selectively Relaxed Matrix Splitting Preconditioning Strategy for Three-Dimensional Flux-Limited Multi-Group Radiation Diffusion Equations

Solutions of a three-dimensional multi-term fractional anomalous solute transport model for contamination in groundwater

The study presents a meshless computational approach for simulating the three-dimensional multi-term time-fractional mobile-immobile diffusion equation in the Caputo sense. The methodology combines a stable Crank-Nicolson time-integration scheme with the definition of the Caputo derivative to discretize the problem in the temporal direction. The spatial function derivative is approximated using the inverse multiquadric radial basis function. The solution is approximated on a set of scattered or uniform nodes, resulting in a sparse and well-conditioned coefficient matrix. The study highlights the advantages of meshless method, particularly their simplicity of implementation in higher dimensions. To validate the accuracy and efficacy of the proposed method, we performed numerical simulations and compared them with analytical solutions for various test problems. These simulations were carried out on computational domains of both rectangular and non-rectangular shapes. The research highlights the potential of meshless techniques in solving complex diffusion problems and its successful applications in groundwater contamination and other relevant fields.

New Stability Results of the Modified Craig-Sneyd Scheme in a Multidimensional Diffusion Equation with Mixed Derivative Terms

The time-dependent multidimensional diffusion equations with mixed derivative terms have been widely used in mathematics. Due to the mixed derivative terms, it is difficult to solve this pattern of multidimensional diffusion equations. The Modified Craig-Sneyd scheme (MCS), which has been applied to value the power derivatives, is one of the most promising methods for solving such equations. Therefore, it is of great importance to study its stability. The unconditional stability problem of MCS for a multidimensional diffusion equation with mixed derivative terms is investigated by taking into account the sizes. A new sufficient condition, necessary condition, and equivalence condition on the parameter of the MCS scheme for unconditional stability are proposed in case of the two and three-dimensional diffusion equations.

Steady-state and transient simulations of the MTR research reactor using the high-fidelity N-TH coupling method

The tremendous advancement in computer technology makes it feasible to perform high-fidelity and multi-physics numerical simulations of reactors, allowing for a more accurate and comprehensive description of the phenomena that occur within nuclear reactors. The application of the high-fidelity coupled neutronic and thermal-hydraulic codes to execute the safety analysis and assessment of the fresh highly enriched uranium core of the IAEA-10 MW MTR research reactor is presented in this research. A three-dimensional coupled neutronic and thermal-hydraulic code is realized using the open-source C++ library OpenFOAM. The coupled code used in this study utilizes an internal coupling approach, where the identical program and grid system are utilized to solve both the neutron physical and thermal-hydraulic fields. The neutron physical field in the reactor is solved by the three-dimensional multi-group time-dependent neutron diffusion equations, and the CFD methodology is applied in solving the thermal-hydraulic governing equations. The global and local steady-state calculation results of MTR are obtained, which are in accordance with the results issued from the IAEA TECDOC and other previous relevant studies. The unprotected loss of flow accident transient condition is considered to conduct the simulation and analysis of the MTR research reactor. The evolutions of temporal and spatial distributions of important parameters of the MTR research reactor are acquired under the ULOFA transient condition, revealing the responses of the reactor core during the transient accident without SCRAM. The results obtained show that the high-fidelity N-TH coupling code can be utilized to analyze the safety of the MTR research reactor.

Diffusion of light in turbid media with internal reflections

We relate the the Kubelka-Munk equations for the description of the intensity transfer of light in turbid media to a one-dimensional diffusion equation, which is obtained by averaging the three-dimensional diffusion equation over the lateral directions. This enables us to identify uniquely the Kubelka-Munk parameters and derive expressions for diffuse reflection and transmission coefficients including the effect of internal reflections. Without internal reflections we recover the Kubelka-Munk formulae for these coefficients. We show that the Kubelka-Munk equations are the proper radiative-transfer equations for the one-dimensional diffusion problem.

Errors Incurred in Local Convective Heat Transfer Coefficients Obtained through Transient One-Dimensional Semi-Infinite Conduction Modeling: A Computational Heat Transfer Study

In typical turbulent flow problems, detailed heat transfer coefficient (h) maps obtained through short-duration experiments are based on inverse heat transfer methods that take the wall temperatures measured via liquid crystals or infrared thermography as input, and an error minimization routine is adopted to determine the best value of h that satisfies the wall temperature temporal evolution under a certain change in fluid temperature. A common practice involves modeling the solid as a one-dimensional semi-infinite medium by selecting the solid material that has low thermal conductivity and low thermal diffusivity. However, in certain flow scenarios, the neglection of the lateral heat diffusion may lead to significant errors in the deduced h values. It is imperative to understand the reasons behind large errors that may be incurred by using the 1D heat conduction assumption in order to accurately determine high-resolution h maps for better heat exchanger designs in a wide range of thermal management applications. This paper presents a computational heat transfer study on different jet impingement scenarios to demonstrate the errors incurred in the determination of h when calculated under the assumption of one-dimensional (1-d) heat conduction into a solid. To this end, three different cases are studied: (a) single jet, (b) array jet (theoretical distribution), (c) array jet (experimental distribution), along with three different mainstream temperature evolution profiles representing step change, moderately fast transient and slow transient nature of flow driving the heat transfer in the solid. A known distribution of heat transfer coefficient (“true h”) for each of the three cases is considered, and three-dimensional transient heat diffusion equations were solved to populate temperatures of each node in the solid at every time step. It is found that stagnation zones’ h1d calculations were lower than the “true h” while the low heat transfer zones exhibited significantly higher h1d compared to the “true h”. For the array jet (experimental distribution) case, it was observed that errors can be as high as 10% in certain low heat transfer zones. Different data reduction procedures, configurations, and conditions explored in this study indicate that a suitable balance can be achieved if shorter time durations in transient experiments are used as a reference for tracking in h1d calculations to keep the deviations from the “true h” low.

Analytical Solutions Formulated in the Time Domain for Three-Dimensional Heat Diffusion Equation

Two different strategies are provided to generate solutions to the three-dimensional heat diffusion equation. The first strategy is inspired by the well-known one-dimensional heat polynomial, which consists of an infinite set of polynomials, which are solutions to the one-dimensional heat diffusion equation. The second strategy is based on an exponential type function. None of the solutions presented here can be obtained by the method of separation of variables. The mathematical developments proving that, indeed, the particular solutions generated with both strategies satisfy the three-dimensional heat diffusion equation are presented. The analytical solutions are validated by generating the corresponding numerical solutions with the method of finite differences. When comparing both analytical and numerical solutions, it is found that they are identical. In addition, as part of the results, it is found that there are exponential solutions that reproduce the behavior of polynomial solutions. Finally, an example of the use of heat polynomials in engineering applications is provided.

A comparison between nonlinear and constant thermal properties approaches to estimate the temperature in LASER welding simulation

The nonlinear thermophysical properties significantly affect the temperature field and the appearance of the weld bead in the LASER Beam Welding (LBW). Then, it is vital to have a well-defined numerical model for analyzing the thermal behavior of the welded material. Nonetheless, many papers still address the welding simulation using constant thermal properties. In this way, this paper proposes a three-dimensional thermal analysis of an unsteady LBW aiming to compare the difference between the constant and nonlinear thermophysical properties approaches. It applied the Finite Volume Method (FVM) to solve the nonlinear three-dimensional heat diffusion equation with an enthalpy function to model the phase change using a fully implicit scheme. In traditional models, these considerations promote a significant increase in computational time for the convergence of the method. Thus, CUDA-C in-house parallel routines were implemented and executed in a Graphics Processing Unit (GPU) to solve this problem. Lab-controlled experiments validated the proposed methodology. The results highlighted the importance of using the nonlinear approach. Furthermore, a detailed study demonstrated the difficulty of knowing precisely the placement of thermocouples, given the high-temperature gradient in the welding processes. The proposed methodology demonstrated to be a faster, cheaper, and efficient way to simulate the LBW.

The Use of Imitation Models at Developing and Introducing Information-Control Systems

Imitation models for computing the environmental water pollution level depending on the intensity of pollution sources created by the author over the years are presented. For this purpose, an additive model of a non-stationary random process is considered. For the modeling of its components, models that consider only dilution and self-purification processes are proposed for waste water and three-dimensional turbulent diffusion equations for river waters, and multidimensional Gaussian Markov series are proposed for modeling the random component. The purpose, the capabilities and the peculiarities of such imitation models are discussed taking into account the peculiarities of the water objects. The modular principle of creating imitation models is proposed to facilitate their development and use.

A Three-Dimensional Anisotropic Diffusion Equation-Based Video Recognition Model for Classroom Concentration Evaluation in English Language Teaching

In this paper, a three-dimensional anisotropic diffusion equation is used to conduct an in-depth study and analysis of students’ concentration in video recognition in English teaching classrooms. A multifeature fusion face live detection method based on diffusion model extracts Diffusion Kernel (DK) features and depth features from diffusion-processed face images, respectively. DK features provide a nonlinear description of the correlation between successive face images and express face image sequences in the temporal dimension; depth features are extracted by a pretrained depth neural network model that can express the complex nonlinear mapping relationships of images and reflect the more abstract implicit information inside face images. To improve the effectiveness of the face image features, the extracted DK features and depth features are fused using a multicore learning method to obtain the best combination and the corresponding weights. The two features complement each other, and the fused features are more discriminative, which provides a strong basis for the live determination of face images. Experiments show that the method has excellent performance and can effectively discriminate the live nature of faces in images and resist forged face attacks. Based on the above face detection and expression recognition algorithms, the classroom concentration analysis system based on expression recognition is designed to achieve real-time acquisition and processing of classroom images, complete student classroom attendance records using face detection and face recognition methods, and analyze students’ concentration from the face integrity and facial expression of students facing the blackboard by combining face detection and expression recognition to visualize and display students’ classroom data for teachers, students, and parents with more data support and help.

Preliminary assessments of variations in the form-function-based maximum pin power in CANDU analyses

In the current CANDU full-core physics analysis, the maximum pin power within a bundle is determined by multiplying the bundle power obtained from solving the three-dimensional neutron diffusion equation with a ratio of the highest pin power to the average pin power, obtained from burnup-dependent infinite lattice calculations. Since the geometry of the fuel channel changes during the life of the reactor, the ratio of the highest to the average pin power changes as well. The objective of the current paper is to quantify such variations. Two sources of variation are considered: (i) pressure tube diametral creep (PTDC); and (ii) the placement of the fuel bundle inside the fuel channel in the lattice model. The results from the study indicate that the maximum pin power could vary up to 1.2%, depending on the amount of the PTDC, the fuel burnup, and the fact that a concentric or non-concentric lattice model is used. Therefore, it is recommended that the lattice model used for deriving the form function be as consistent as possible to the core condition being analysed, especially with respect to the PTDC value.

Effects of surface wettability on pool boiling process: Dynamic and thermal behaviors of dry spots and relevant critical heat flux triggering mechanism

In this paper, we investigate experimentally and numerically the influences of surface wettability on the boiling process from nucleate boiling to critical heat flux (CHF). The boiling surfaces with four different wettabilities (apparent contact angles) are examined. Using a simple optic setup, sequential phase distribution images on the boiling surfaces were acquired, and we analyzed the dynamics of microlayer, bulk liquid region, and dry spots with the processed images. The variation of surface wettability significantly affected the liquid and vapor phase behaviors on the boiling surface, such as the growth and shrinkage of the microlayer and dry spot, wetting velocity and area portion of the dry spot. As a result, the CHF increased with the surface wettability, and we found the CHF is triggered from a local dry spot where the liquid wets the spot no longer and is violently scattered by strong evaporation at the triple contact line. Our numerical simulation results, in which the three-dimensional transient heat diffusion equation was solved for the boiling substrate using the phase distribution images as boundary conditions, indicated that certain dry spots (critical dry spot) can be highly overheated under high heat flux condition, and they lead significant heat transfer degradation and, eventually, the CHF. With the numerical results and observation of the phase distribution images at the CHF conditions, we postulated the allowable wetting temperature and formulated it as a function of surface wettability through considering the balance between vapor recoil and surface tension forces. The wetting temperature increased with surface wettability, and it was confirmed by the numerical results. Finally, we proposed a qualitative relation to elucidate the influence of surface wettability on the CHF based on the heat conduction via the vapor on a critical dry spot and heat flux partitioning concept.

Efficient second-order ADI difference schemes for three-dimensional Riesz space-fractional diffusion equations

In this paper, a three-dimensional time-dependent Riesz space-fractional diffusion equation is considered, and an alternating direction implicit (ADI) difference scheme is proposed, in which a weighted and shifted Grünwald difference scheme (see Tian et al. (2015) [33]) is utilized for the discretizations of space-fractional derivatives, and a fractional-Douglas-Gunn type ADI method is utilized for the discretization of time derivative. The method is proved to be unconditionally stable and convergent with second-order accuracy both in time and space with respect to a weighted discrete energy norm. Efficient implementation of the method is carefully discussed, and then based on fast matrix-vector multiplications, a fast conjugate gradient (FCG) solver for the resulting symmetric positive definite linear algebraic system is developed. Numerical experiments support the theoretical analysis and show strong effectiveness and efficiency of the method for large-scale modeling and simulations. Finally, a linearized ADI scheme based on second-order extrapolation method is developed and tested for the nonlinear Riesz space-fractional diffusion equation.

Accuracy and speed of splitting methods for three-dimensional space–time fractional diffusion equation with [formula omitted]-Caputo derivatives

The paper deals with finite-difference schemes for a three-dimensional diffusion equation with ψ-Caputo derivative with respect to the time and space variables. Theoretical and experimental estimates of accuracy and performance of implicit and splitting schemes are given. Finite-difference schemes are combined with algorithms that accelerate computations on the base of fixed memory principle and expansion of integral operator kernel into series. Computational experiments are conducted on a test problem that has an analytical solution for the case of the Caputo–Katugampola derivative. In the experiments we focus on the issue of interdependence between accuracy and speed of calculations. Based on the obtained estimates, we present an algorithm for automatic selection of optimal computational scheme.

Simultaneous simulation of changes in the NaCl concentrations of Japanese radish and an aqueous solution during cooking

When cooking foods in a broth, the NaCl concentration in the ingredients increases owing to diffusion, whereas the amount of NaCl in the broth decreases. The broth's concentration is affected by evaporation. This study's purpose was to simultaneously simulate the NaCl concentrations in the ingredients and broth, which change from moment to moment during cooking. The NaCl concentration in the samples was predicted via a three-dimensional diffusion equation, and the amount of NaCl diffusing in the samples was determined. The NaCl concentration in the aqueous solution was predicted using a calculated amount of NaCl in the samples from the solution and the experimental value of evaporation. The predicted values of the NaCl concentrations in the samples and aqueous solution were in good agreement with the experimental values. Therefore, herein, the simultaneous simulation of changes in the NaCl concentration in the samples and aqueous solution during cooking was possible.

Variational nodal method for three-dimensional multigroup neutron diffusion equation based on arbitrary triangular prism

With the development of nuclear technology, new concepts of reactor core have been continuously proposed, and new types of geometric assembly and reactor core configuration would be adopted to achieve better characteristics. Traditional nodal method based on rectangle or hexagonal nodal is no longer applicable for complex geometry modeling. Given that triangles have the advantage of constructing arbitrarily complex geometry, in this study, variational nodal method of three-dimensional multigroup neutron diffusion theory based on arbitrary triangular prism was developed. ANSYS code is used to divide calculation region into arbitrary triangle mesh, which is then transformed into regular triangle mesh by coordinate transformation, and a set of orthogonal polynomial basis function is built. Variables such as nodal neutron scalar flux and neutron current on nodal surface are expanded based on those polynomial basis functions. The response relationship between neutron partial neutron current and neutron flux can be obtained by variational principle, and traditional power method is employed to solve those response matrix system. Based on above, TriVNM code was developed, and 4 types of typical benchmarks including rectangle, hexagonal assembly reactor and reactor with curved boundary were employed to verify TriVNM. Both power distributions and keff calculated by TriVNM code agree well with reference results, and the maximum relative error of all the benchmarks’ power doesn’t exceed 1% and error of keff doesn’t exceed 50 pcm. TriVNM code achieves the same or higher level of calculation accuracy compared with other diffusion codes. Mesh sensitivity analysis shows that for large mesh size TriVNM code still has good accuracy.

ASSESSMENT OF GEOECOLOGICAL CONSEQUENCES OF SUBDUCTION PROCESSES

t. The aim of the work is to develop a numerical-analytical method for assessing the geoecological consequences of volcanic activity accompanying the subduction process. The boundary problem was formulated, including the three-dimensional transport and diffusion equation and boundary conditions at the bottom and surface of the water. For research, a block structure with quasi-homogeneous layers is introduced. In each block, a differential factorization method is implemented and integral representations of solutions are obtained. Calculations for a model problem are carried out, conclusions are formulated about the features of the behavior of the heavy and light fractions of volcanic ejections depending on the speed of currents, intensity and duration of the eruption.

VERIFICATION OF THE DIFFUSION AND TRANSPORT SOLVERS WITHIN DIF3D FOR 3D HEXAGONAL GEOMETRIES

The DIF3D code (DIFfusion 3D) has been a workhorse of fast reactor analysis work at Argonne National Laboratory for over 40 years. DIF3D was primarily built in the late 1970s as a three-dimensional multigroup diffusion equation solver operating on semi-structured grid geometries. In the mid-1990s, transport capabilities needed for high-leakage reactor configurations were added to DIF3D with the variational anisotropic nodal transport approach. Recent reactor design activities at Argonne are requiring that a thorough verification of the Argonne Reactor Computation (ARC) codes be performed. With DIF3D being central to the entire ARC system, the verification efforts are focused on the 3D Cartesian, 3D triangular, and 3D hexagonal core geometry options of DIF3D. Validation activities, while needed for the ongoing design activities at Argonne, are handled at a project-specific level. This paper summarizes the verification work so far on the forward and adjoint forms of the fixed source, inhomogeneous fixed source, and k-eigenvalue steady state transport and diffusion equations as implemented specifically for 3D triangular and hexagonal geometries in DIF3D. Since analytic solutions of the neutron diffusion and transport equations are either limited in scope or not possible, this verification required multiple tiers of problems unique to each solver and geometry type, each testing features independent and complementary arguments for why this separate testing of functionalities is acceptable. This separate testing was also supplemented with a high-level integral check of each the diffusion and transport capabilities and applicable geometries.

Modified nonlinear function specification and temperature moving sensor to estimate the heat supply in a welding process

The estimation of the imposed heat rate in welding processes has always been an obstacle to improve the thermal efficiency of these processes. Thus, the use of inverse problem techniques is an alternative procedure to estimate the heat rate. In this study, the problem is highly nonlinear; therefore, the heat flux provided by the welding process is estimated by the iterative Function Specification method. This method was modified to estimate the heat supply based on the heat rate sensitivity coefficient, which represents the influence of the welding power in the temperatures of 304 stainless steel plates. In order to avoid the problem of low thermal sensitivity due to the movement of the welding source on the upper face of the plate, this methodology is based on the concept of temperature moving sensor. The software COMSOL was used to solve the three-dimensional heat diffusion equation with enthalpy function to model the phase change problem. 10 type K thermocouples, equally spaced, were attached by capacitive discharge on the opposite heated surface to measure the temperature. The efficiency of the process was calculated with the estimated net heat rate which ranged from 63% to 87%. The average efficiency obtained was 75% for this welding process. A relation between the energy rate given by the power supply and the estimated efficiency was also observed. The thermal efficiency of the arc decreases as the power increases. Thus, an analysis using the Taguchi method was performed in order to better understand the influence of the welding parameters on the efficiency of the process. It was concluded that with a significance level of 10%, the welding current and arc length have a strong correlation with thermal efficiency. These results are discussed and compared with results obtained experimentally by other researchers. Besides, an uncertainty analysis in the estimated heat rate is also presented.

New finite element neutron kinetics coupled code system FENNECS/ATHLET for safety assessment of (very) Small and Micro Reactors

Cores of (very) small and medium size reactor–(v)SMR – are often characterized by irregular geometries. Monte Carlo methods are not yet mature enough for transient applications. To perform coupled transient safety assessments of (v)SMRs, the Finite ElemeNt NEutroniCS (FENNECS) code is being developed at GRS. It solves the time-dependent and steady-state three-dimensional few-group diffusion equation in Galerkin finite element representation using upright triangular prisms with linear basis functions as spatial elements. FENNECS is also coupled with the GRS thermal-hydraulic system code ATHLET. For the special meshing, an external meshing tool is being developed as a Python software module. This paper considers the Heat Pipe Micro Reactor core. For the generation of cross sections to be used in FENNECS, Monte Carlo Serpent models are developed for two configurations: all absorber faces of control drums are turned out of the core (All Rods Out – ARO – configuration) and all absorber faces are turned in (All Rods In – ARI – configuration). The Serpent multiplication factors are in good agreement with the benchmark. In the FENNECS model, control drums are approximated by a 12-edges polygon or by a 24-edges polygon. The FENNECS multiplication factors agree well with Serpent reference results for both configurations. This demonstrates the applicability of FENNECS to (v)SMRs.