One-dimensional Transport Equation Research Articles

Analytical solution of a non-homogeneous boundary-value problem for the transport equation in an earth to air energy exchanger by initial base analysis method

In this paper, a one-dimensional time-dependent transport equation with inhomogeneous Dirichlet boundary conditions, often encountered in the study of an Earth to Air Energy Exchanger, is solved by a practical analytical method. The initial condition function is not randomly set to verify the transport equation at the initial time, but determined as a first approximation to the solution of the transport equation (at t=0). As in the homotopy analysis method, the solution of the transport equation is expressed by the set of a basis function constructed with the obtained initial condition function. The components of the solution in this base, are time’s functions deduced from the boundary conditions of the problem. Suitable endings are noticeable between the proposed model and experimental measurements.

A new seasonal frozen soil water-thermal coupled migration model and its numerical simulation.

In this paper, a mathematical model based on spherical differential unit cell is proposed as a model for studying seasonal freeze-thaw soil space infinitesimal differential unit cell. From this model, the basic equations of permafrost moisture and heat flow motion are directly derived, then the linked equations form the permafrost water-heat coupled transport model. On this basis, the one-dimensional seasonal permafrost water-heat transport equation is derived. The model reduces the original spatial three-variable coordinate system (parallel hexahedron) into a coupled equation with a single spherical radius (R) as the independent variable, so the iterations of the numerical simulation algorithm is greatly reduced and the complexity is decreased. Finally, the model is used to simulate the seasonal freeze-thaw soil in the ShiHeZi region of Xinjiang, China. The principle of the simulation is to collect the soil temperature and humidity values of the region in layers and fixed-points using a homemade freeze-thaw soil sensor, after that we solve it by numerical calculation using MATLAB. The analysis results show that the maximum relative error of the model we proposed is 4.36, the minimum error is 0.98, and the average error is 2.515. The numerical simulation results are basically consistent with the measured data, then the proposed model is consistent with the matching states of permafrost moisture content and soil temperature in the region at different times. In addition, the experiments also demonstrate the reliability and accuracy of the model.

An a posteriori strategy for adaptive schemes in time for one-dimensional advection-diffusion transport equations

Stability condition is a more restrictive constraint that leads to unnecessary small-time steps with respect to the accuracy and results in computational time wastage. We propose a node by node adaptive time scheme to relax the stability constraint enabling a larger global time step for all the nodes. A nonlinear procedure for optimising both the schemes in time and space is proposed in view of increasing the numerical efficiency and reducing the computational time. The method is based on a four-parameter family of schemes we shall tune in function of the physical data (velocity, diffusion), the characteristic size in time and space, and the local regularity of the function leading to a nonlinear procedure. The a posteriori strategy we adopt consists in, given the solution at time tn, computing a candidate solution with the highest accurate schemes in time and space for all the nodes. Then, for the nodes that present some instabilities, both the schemes in time and space are modified and adapted in order to preserve the stability with a large time step. The updated solution is computed with node-dependent schemes both in time and space. For the sake of simplicity, only convection-diffusion problems are addressed as a prototype with a two-parameters five-points finite difference method for the spatial discretisation together with an explicit time two-parameters four-stages Runge–Kutta method. We prove that we manage to obtain an optimal time-step algorithm that produces accurate numerical approximations exempt of non-physical oscillations.

A divide-and-conquer algorithm for seismic data approximation by the Laguerre series

An algorithm of the Laguerre transform for approximating functions on large intervals is proposed. The idea of the considered approach is that the calculation of improper integrals of rapidly oscillating functions is replaced by a solution of an initial boundary value problem for the one-dimensional transport equation. It allows one to successfully avoid the problems associated with the stable implementation of the Laguerre transform. A divide-and-conquer algorithm based on shift operations made it possible to significantly reduce the computational cost of the proposed method. Numerical experiments have shown that the methods are economical in the number of operations, stable, and have satisfactory accuracy for seismic data approximation.

Taylor dispersion of elongated rods

Particles transported in fluid flows, such as cells, polymers, or nanorods, are rarely spherical. In this study, we numerically and theoretically investigate the dispersion of an initially localized patch of passive elongated Brownian particles constrained to one degree of rotational freedom in a two-dimensional Poiseuille flow, demonstrating that elongated particles exhibit an enhanced longitudinal dispersion. In a shear flow, the rods translate due to advection and diffusion and rotate due to rotational diffusion and their classical Jeffery's orbit. The magnitude of the enhanced dispersion depends on the particle's aspect ratio and the relative importance of its shear-induced rotational advection and rotational diffusivity. When rotational diffusion dominates, we recover the classical Taylor dispersion result for the longitudinal spreading rate using an orientationally averaged translational diffusivity for the rods. However, in the high-shear limit, the rods tend to align with the flow and ultimately disperse more due to their anisotropic diffusivities. Results from our Monte Carlo simulations of the particle dispersion are captured remarkably well by a simple theory inspired by Taylor's original work. For long times and large Peclet numbers, an effective one-dimensional transport equation is derived with integral expressions for the particles' longitudinal transport speed and dispersion coefficient. The enhanced dispersion coefficient can be collapsed along a single curve for particles of high aspect ratio, representing a simple correction factor that extends Taylor's original prediction to elongated particles.

Solving one-dimensional advection diffusion transport equation by using CDV wavelet basis

Solving one-dimensional advection diffusion transport equation by using CDV wavelet basis

Generating the Laguerre expansion coefficients by solving a one-dimensional transport equation

A new approach for calculating Laguerre expansion coefficients is proposed. In solving applied problems, initial data may be specified as time series with a constant discretization step. In this case, the use of quadratures of high-order accuracy is limited due to their instability. In order to overcome these difficulties, this paper considers an approach in which algorithms are proposed to calculate integral Laguerre transform by solving a one-dimensional transport equation. In contrast to the direct calculation of improper integrals of rapidly oscillating functions, these procedures make it possible to calculate the expansion coefficients of a Laguerre series expansion with better stability, higher accuracy, and less computational burden. The numerical experiments have shown that the methods are economical in terms of operation count, stable, and have reasonable accuracy in practical calculations.

Characteristics of quasi-ballistic heat conduction in a multiple materials system based on the solution of the Boltzmann transport equation

Phonons will exhibit quasi-ballistic heat transport when the mean free path is longer than the dimension of the medium through which the phonons travel. Such a transport regime can yield information about the phonon mean free path. The primary focus of this study is on numerically solving the one-dimensional transient Boltzmann transport equation to investigate how the phonon mean free path can be determined in accordance with the characteristics of quasi-ballistic heat conduction in a multiple materials system. The Boltzmann transport equation is solved for heat conduction within an aluminum film with a double-layered structure based on the discrete ordinates method. The metal film is deposited onto a silicon substrate. Frequency-dependent boundary conditions are used to define the difference in phonon dispersion between the metal film and the substrate. The frequency-dependent Boltzmann transport equation is used to determine how the thermal conductivity changes with the modulation frequency when phonons travel quasi-ballistically. The distribution of the phonon mean free path is discussed, and comparisons are made between theoretical results at different modulation frequencies. The results indicated that the effective thermal conductivity of the multiple materials system does not depend upon the modulation frequency. Phonons with mean free paths longer than the penetration depth do not contribute to the effective thermal conductivity. The effect of quasi-ballistic heat conduction is noticeable in semiconductor alloys even at room temperature.

Simulation for non-homogeneous transport equation by Nyström method

In this work we solve numerically the one-dimensional transport equation with semi-reflective boundary conditions and non-homogeneous domain. The proposed methodology consists of applying the Nyström method in order to discretize the integral formulation of this problem which is an equation involving weakly singular integral operators. For this purpose, analytical and computational techniques were applied to deal with the singularities. The Nyström method is an integral method which approximates the integral operator by a numerical quadrature and turns the integral equation into a finite dimensional linear system. This formulation allows us to use any function to describe both scattering cross section and total cross section. The algorithm is implemented in C language with the use of routines of GNU scientific library and computational techniques for code optimization. The scalar flux was calculated for two numerical quadrature, namely Gauss-Legendre quadrature and Boole's rule. The numerical results were determined for transport problem with homogeneous and non-homogeneous domains. In order to validate the proposed method-ology, our numerical results were compared with those from the literature and presented with several correct significant digits.

ON TWO MODELS RELATED TO THE FLOODING AND DEPOSITION OF THE DON DELTA

The results of modeling changes in the water surface level in the eastern part of the Taganrog Bay and the main Don branches in its delta area are presented. A numerical study of the process of saltwater inflow from the Taganrog Bay to the Don delta has been carried out. The hydrodynamics in the Taganrog Bay, as well as the saltwater transport process, are specified using the corresponding two-layer models. A system of Don arms is presented in the form of a graph, the edges of which correspond to open channels, and the vertices correspond to branching points and end nodes. The flow in the main Don branches is described by the Saint-Venant equation. It is assumed that there is no distributed lateral inflow, and the channel cross-section has a parabolic profile. Saltwater inflow into the arms is described by a one-dimensional transport equation. Boundary conditions are specified for each sleeve. At the branching nodes, conditions are set for the equality of the water levels, as well as the equality of the inflowing and outflowing discharges. The description of the algorithm of the process of flooding/drainage of the Don delta area is given. Considering the values of the depths at the nodes of the flat grid, the cells located in water or on land are determined. A logical array characterizing the type of cells (“water”, “land”) sets the configuration of the entire computational domain. Comparison of the calculation results with the observed values of salinity and water level is carried out.

Luenberger Observers for Infinite-Dimensional Systems, Back and Forth Nudging, and Application to a Crystallization Process

This paper deals with the observer design problem for time-varying linear infinite-dimensional systems. We address both the problem of online estimation of the state of the system from the output via an asymptotic observer, and the problem of offline estimation of the initial state via a back and forth nudging algorithm. In both contexts, we show under a weak detectability assumption that a Luenberger-like observer may reconstruct the so-called observable subspace of the system. However, since no exact observability hypothesis is required, only a weak convergence of the observer holds in general. Additional conditions on the system are required to show the strong convergence. We provide an application of our results to a batch crystallization process modeled by a one-dimensional transport equation with periodic boundary conditions, in which we try to estimate the crystal size distribution from the chord length distribution.

Ghost solutions with centered schemes for one-dimensional transport equations with Neumann boundary conditions

Ghost solutions with centered schemes for one-dimensional transport equations with Neumann boundary conditions

Uniform adiabatic limit of Benney type systems

In this paper, we show that solutions of the cubic nonlinear Schrödinger equation are asymptotic limit of solutions to the Benney system. Due to the special characteristic of the one-dimensional transport equation, same result is obtained for solutions of the one-dimensional Zakharov and 1d-Zakharov–Rubenchik systems. Convergence is reached in the topology \(L^2({\mathbb R})\times L^2({\mathbb R})\) and with an approximation in the energy space \(H^1({\mathbb R})\times L^2({\mathbb R})\). In the case of the Zakharov system, this is achieved without the condition \(\partial _t n(x,0) \in \dot{H}^{-1}({\mathbb R})\) for the wave component, improving previous results.

Strongly Out-of-Equilibrium Simulations for Electron Boltzmann Transport Equation Using Modal Discontinuous Galerkin Approach

In this study, we present simulations of strongly out-of-equilibrium conditions for the one-dimensional electron Boltzmann transport equation (BTE) in semiconductor devices. An explicit modal discontinuous Galerkin method is employed to solve the electron BTE along with the simplest collisional model–relaxation time approximation. The electron BTE system is discretized in momentum and time, and applied to the description of the dynamics of non-linear electron transport under a strong static electric field. A third-order explicit SSP-RK scheme is adopted for the temporal discretization of the resulting semi-discrete ODE. For the steady-state electron BTE, the analytical solutions are derived at low and high electric fields which are used to validate the numerical approach. The computed solution show a good agreement with derived analytic solution in wide range of tested parameters and regimes. An extensive range of numerical simulations has also been performed to investigate the impact of exploited flow parameters on the electron BTE solutions. Ultimately, these results predict that the modal DG approach is particularly effective in treating the strongly out-of-equilibrium regimes encountered like in ultrafast transport dynamics.

Separate-effect experiments and modeling for two-phase flow under geometric restrictions

While significant efforts have been made to investigate two-phase flows in vertical upward channels, limited work has been performed to study the effects stemming from geometric configurations on two-phase flow transport, such as flow restrictions and change in flow orientation. This presents significant shortcomings in the application of predictive models to practical systems, including nuclear power reactors. It has been well observed that geometric restrictions (e.g., elbows, spacer-grid, and inlet configurations) and change in flow orientation (e.g., upward-to-horizontal, horizontal-to-downward) can cause significant effects on the hydrodynamics of two-phase flow. In view of these, high-resolution separate-effect experiments have been performed to establish a two-phase flow database that is accurate, extensive and good enough for multi-phase CFD codes development, to develop predictive models accounting for geometric effects, and to improve the performance of nuclear reactor system analysis codes. Recent studies on the effects of the vertical-upward and vertical-downward 90-degree elbows, spacer grid, inlet configurations, and change in flow orientation are summarized. Some of the major two-phase flow phenomena investigated in the present study include flow regime transition, pressure loss, two-phase flow parameters, flow development, and interfacial drag. To demonstrate the functionality of the newly developed models, the one-dimensional steady-state one-group interfacial area transport equation is employed to predict two-phase flow transport under geometric effects.

Symmetries of the [formula omitted] approximation to the Boltzmann neutron transport equation

This work explores conditional translation and scaling symmetries of a P3 approximation to the one-dimensional (1D) Boltzmann neutron transport equation with arbitrary material properties. The principal outcomes of this analysis are twofold: 1) the identification of symmetries using Lie group theory is exhaustive, and thus provides definitive insight into scenarios where the differential equation(s) under consideration are expected to hold across phase space position or dimensional scale; and 2) invariance under transformations may be leveraged to construct changes of variables under which reduced-order structures or exact solutions with special properties may be derived. In support of these goals, this work includes the application of Lie group theory to the aforementioned neutron transport model. A variety of conditional translation and scaling symmetries are obtained (e.g., depending on the functional form of various macroscopic cross section functions), and an example scaleinvariant numerical solution of the 1D spherical P3 equations is provided.

Semi-analytical solution for one-dimensional unsteady sediment transport model in open channel with concentration-dependent settling velocity

This paper aims to solve the one-dimensional unsteady suspended sediment transport equation in open channels through a semi-analytical approach. The presence of a large amount of particles in flow change the settling velocity of a particle, and this phenomenon, commonly known as hindered settling effect, must be taken into account to deal with high concentrated flows. Inclusion of this mechanism makes the governing equation nonlinear, and together with this nonlinear governing equation, a generalized bottom boundary condition is taken in terms of deposition velocity and equilibrium bottom concentration. An explicit series solution is presented using the method of lines based homotopy analysis method, and the convergence of the series solution is gained through a convergence control parameter. The solution is validated by comparing it with the existing solution as well as a numerical one. Apart from that, the solution has also been validated under a steady-state condition with available experimental data. Results are interpreted both graphically and physically. It is found that the hindered settling effect is dominant in the main flow region only, for sediment free inlet for all types of turbulent diffusion coefficients. On the other hand, in the case of uniform sediment concentration at the inlet, hindered settling affects the concentration in the top portion of the channel too for linear and parabolic profiles of turbulent diffusion coefficients.

Numerical estimation of the oxygen impurity transport in the Aditya tokamak

The present study illustrates an approach to solving the one-dimensional impurity transport equation in tokamak plasma, without the loss of generality, using a semi-implicit numerical method. The radial profiles of number density distribution of different ionization states of oxygen (Z = 1 to Z = 8) have been estimated using the ionization and recombination rate coefficient data from the Atomic Data and Analysis Structure database. The rate coefficients are determined based on the electron temperature and number density distribution in the Aditya tokamak. The radial number density profile of the O4+ ion, obtained using the semi-implicit impurity transport equation, has been applied further in determining the radial emissivity profiles of the characteristic 650.024 nm transition of the Be-like O4+ ion. The simulated O4+ emissivity is then matched with respect to the experimental emissivity values along inboard (high toroidal magnetic field) and outboard (low toroidal magnetic field) regions of the Aditya tokamak. The comparison estimates the nature of the radial diffusivity profile of oxygen ions, which in turn demonstrates the nature of oxygen transport in Aditya plasma. Large values of oxygen ion diffusivity (> neo-classical values) are observed toward the plasma edge in both regions of the Aditya tokamak. These diffusivity values point toward a fluctuation induced transport in the edge region of Aditya plasma. The simulated results are benchmarked with the STRAHL code. The semi-implicit form of the radial impurity transport equation can be extended to deduce the distributions of other light impurities in tokamak plasma.

Hybrid FEM and FDM Approach for Monolithic Catalytic Converters with Pore Diffusion

A transient, three-dimensional model is developed for monolith catalytic converters with washcoat pore diffusion. Due to the unique configuration of monolith converters, the flow within the channel can be considered fully developed laminar flow, and one-dimensional transport equations can be formulated for the flow and species. On the other hand, a three-dimensional equation is necessary for the monolith solid temperature to capture the effects of significant external heat loss or flow maldistribution. The proposed model employs a three-dimensional finite-element method (FEM) for the solid temperature equation and a one-dimensional finite-difference method (FDM) for the flow and species equations. A recently developed asymptotic approach (Bissett, Emiss. Control Sci. Technol. 1(1), 3–16, 2015, Bissett, Emiss. Control Sci. Technol. 5(1), 45–54, 2019) is utilized to solve the washcoat pore diffusion and surface coverages equations. The three-dimensional FEM meshes are chosen so the nodes also serve for the one-dimensional transport equations. All equations are solved fully coupled, without lagging or reliance upon external tools. The developed solver can efficiently simulate general cross-section geometries (circle, oval, rectangle, etc.) for a catalytic monolith under nonuniform inlet distribution and detailed washcoat pore diffusion. The solver is verified by comparing the simulation results to the one-dimensional simulation results for a simple geometry under uniform inlet conditions. The model can be applied to flow through general monolith catalytic reactors with single or dual washcoat layers and a wide range of kinetics.

Newly Developed Modeling Application for Quench Characteristic Analysis of YBCO Pancake Coil

We demonstrate a newly developed three-dimensional (3-D) heat conduction analysis model for a yttrium barium copper oxide (YBCO) pancake coil by including the turn-to-turn and layer-to-layer heat conduction in the one-dimensional heat transport equation. The YBCO tape is represented by two thermal components, which are in thermal contact through a thermal resistance. The turn-to-turn and layer-to-layer thermal contacts are defined with thermal conduction coefficients of the insulation. For validating the accuracy of the simulation model, a quench propagation analysis of a racetrack pancake coil is performed, and the simulation results are compared with the experimental results presented in a published paper. The comparison shows that the simulation model can give a reliable forecast of the quench propagation characteristics. When considering the reduction of the thermal conduction efficiency induced due to temperature variation near the hotspot, the simulation results are in high consistency with the experimental results. A 3-D surface map of the temperature profile is displayed to show the impact of the thermal conductivity coefficient of the insulation. The simulation model is proven to be a powerful tool to study the quench propagation characteristics of the YBCO coil.