One-dimensional Transport Equation Research Articles

High-Resolution Sensor System to Investigate Solute Transport in Saturated Sediments

Highlights Innovative sensor system was developed to detect solute transport at very high time resolutions. The sensors measure electrical conductivity as voltage drops at a frequency of up to 10 Hz, allowing for detailed solute breakthrough curves (BTCs). One-dimensional solute transport equations were modeled numerically using the finite differences technique, and the results show good agreement between the modeled BTCs and those obtained in the laboratory for all materials. Abstract. Hyporheic exchange flow, the movement of river water into streambed sediments and its subsequent return to the surface water after suffering biogeochemical transformations, impacts a stream’s water quality and ecology. These processes, which are the basis for the self-purification capacity of river systems, depend on the quantity and travel time of the flow within the sediments. Different tracer-based approaches have been deployed for studying flow in saturated sediments, but few of these experimental methodologies are able to attain the high temporal resolutions that are needed to adequately capture the passage of a solute plume in a porous medium. To allow for a better description of such flows, we designed and fabricated an innovative, low-cost sensor system able to detect solute transport at multiple locations simultaneously at very high time resolutions. The sensors measure electrical conductivity as voltage drops at a frequency of up to 10 Hz, allowing for detailed solute breakthrough curves (BTCs). The fabricated sensors were tested in a laboratory experimental setup designed to examine solute transport under conditions that reproduce 1D flow in saturated sediments. The sensor system was able to successfully detect electrical conductivity (expressed as voltage drop) in real-time, at high temporal resolution, and at multiple locations. Solute BTCs collected with the system were consistent and highly repeatable under the same conditions. One-dimensional solute transport equations were modeled numerically using the finite differences technique, and the results showed good agreement between the measured and modeled BTCs for all materials. The proposed system is inexpensive compared to conventional EC probes, easy to operate, and sensitive to low solute concentrations. Keywords: Hyporheic exchange flow, Saturated sediments, Sensor, Solute transport.

A NUMERICAL STUDY ON THE REDUCTION OF GREENHOUSE GASEOUS COMPONENT (CO2) DUE TO THE ADDITION OF H2 IN THE FUEL STREAM OF THE COUNTERFLOW CH4/AIR DIFFUSION FLAME

In this study, a series of 1-D and steady-state numerical simulations have been performed for the prediction of the effect of the addition of H2 on the characteristics of a non-sooting counterflow CH4/Air diffusion flame using detailed chemical reaction model, which is composed of 325 elementary chemical reactions and 53 chemical species. Under the steady-state assumption, a set of one-dimensional transport equations of mass, momentum, species, and energy along with the equation of state has been solved numerically at the atmospheric conditions over the counterflow configuration by exploiting an efficient numerical code, OPPDIF (a Fortran Program for Computing Opposed-Flow Diffusion Flames). The grid adaption technique has been used to achieve better convergence as well as to ensure the maximum accuracy of the simulated results. It is found that the flame temperature is increased due to the addition of H2 with CH4, which is injected into the fuel stream. The elevation in the temperature is caused by the augmentation of the integrated heat release rate of the elementary reactions supported by the active radicals (H, O, and OH), which are generated by the higher reactivity of H2. Besides, it is found that the mole fractions of H2O are increased as the percentage of H2 in the loading fuel (CH4) is increased and also, it is identified that the chain propagating reaction, OH + H2 => H2O + H is dominating one which produces highest amount of H2O. Furthermore, it is noticed that the indirect greenhouse gas or precursor, CO is reduced when H2 is added to CH4. Consequently, the mole fraction of the principle greenhouse gas, CO2 is decreased significantly when the fuel, CH4 percentage is modified by the higher percentage of H2. The sensitivity analysis of elementary reactions reveals the fact that the chemical reaction: OH + CO => H + CO2 is a dominating reaction in producing a lower amount of CO2 when the volume fraction of H2 is increased in the fuel (CH4) stream. In the presence of 75 % H2 in CH4, the pressure-dependent reaction, O + CO (+M) => CO2 (+M) appears as another chemical route that also generates greenhouse gas, CO2 but its contribution is negligibly small.

An analytical model for wearable thermoelectric generators harvesting body heat: An opportunistic approach

Wearable thermoelectric generators (WTEGs) have the potential to transform body heat into electrical energy, thereby establishing a sustainable power reservoir for wearable electronics and offering a promising alternative to traditional chemical batteries. This paper presents an opportunistic analytical model that investigates the energy conversion performance of a human skin-WTEG system. The model incorporates a one-dimensional bioheat transport equation to account for metabolic heat generation within skin tissue and heat exchange resulting from the blood perfusion, while a comprehensive three-dimensional analysis of the WTEG captures convective heat dissipation occurring between the WTEG and ambient surroundings. Analytical solutions for the temperature profile and power density of WTEGs are obtained, along with an approximate closed-form expression for the matched load resistance. The results demonstrate the significant impact of blood perfusion and convective heat dissipation on the WTEG performance. The optimal fill factor and height of thermoelectric couples for maximum power output are also determined. This model will assist in predicting and optimizing the performance of WTEG devices in practical applications.

Three-Dimensional Numerical Transfer Model Using a Monotonized Z-Scheme

The aim of this work is to investigate a three-dimensional second-order monotone finite-difference scheme for the transport equation. The investigation is conducted for model three-dimensional transport equations of an incompressible medium. The properties of the three-dimensional extension of the Z-scheme with nonlinear correction are studied in this work. This study is an extension of the author’s previous works, where a nonlinear correction of one-dimensional transport equations was constructed. The proposed scheme uses “skew differences” containing values from different time layers instead of fluxes for the correction. The monotonicity of the obtained nonlinear scheme is numerically studied for a family of limiter functions for both smooth and non-smooth continuous solutions. The constructed scheme is absolutely stable but loses the monotonicity property when the Courant step is exceeded. The distinctive feature of the proposed finite-difference scheme is the minimalism of its template. The constructed numerical scheme is designed for models of plasma instabilities of various scales in the low-latitude ionospheric plasma of the Earth. One of the real problems that arise in solving such equations is the numerical modeling of strongly non-stationary medium- and small-scale processes in the low-latitude ionosphere of the Earth under conditions of the occurrence of the Rayleigh–Taylor instability and other types of instabilities, leading to the phenomenon of F‑scattering. Due to the fact that transport processes in the ionospheric plasma are controlled by the magnetic field, it is assumed that the plasma is incompressible in the direction perpendicular to the magnetic field.

Effect of inlet gas and liquid velocity profiles on one-group interfacial area transport equation in a vertical large rectangular channel

Bubbly flows appear in many heat and mass transfer equipment. The interfacial area concentration is a critical parameter in determining the performance of the heat and mass transfer systems. The one-dimensional one-group interfacial area transport equation (IATE) was developed based on area-averaged bubbly flow data collected under uniform inlet flow boundary conditions. The applicability of the IATE to the flow systems with non-uniform inlet flow boundary conditions was not comprehensively examined. This study investigated the effect of inlet liquid and gas velocity profiles on measured interfacial area concentration changes along the flow direction. The data used in this study were collected for vertical upward air-water two-phase flow in a rectangular channel with a gap length of 10 mm and width of 200 mm. The test conditions were the superficial gas velocity ranging from 0.0870 to 0.288 m/s and the superficial liquid velocity ranging from 0.515 to 2.53 m/s. The interfacial area concentrations with uniform and non-uniform inlet flow boundary conditions measured in a large vertical rectangular channel were compared under the same area-averaged flow rate conditions. Then, the predictive capability of the IATE to the bubbly flow test data with uniform and non-uniform inlet flow boundary conditions was assessed. Some non-uniform inlet flow boundary conditions, including channel-center peaking of liquid flow (CPF), channel-center peaking of gas flow (CPG), single-side-wall peaking of liquid flow (SPF), and single-side-wall peaking of gas flow (SPG), significantly affected the interfacial area concentration changes along the axial location at relatively high superficial gas or liquid velocity conditions. The IATE's predictive capability deteriorated for several non-uniform inlet flow boundary conditions, including double-side-wall peaking of gas flow (DPG) and single-side-wall peaking of liquid flow (SPF), specifically at relatively high superficial liquid velocity conditions. The maximum percentage error of the interfacial area concentration prediction by the IATE for these non-uniform inlet flow boundary conditions reached 75%.

A semi-analytical solution for one-dimensional pollutant transport equation in different types of river networks

This study proposes a semi-analytical solution to a one-dimensional advection-dispersion equation considering reaction and source terms to predict and describe pollutant transport in various river networks using the Laplace transform approach. Moreover, the proposed semi-analytical solution is expanded for variable coefficients of the transport equation by increasing the number of middle nodes and associated intervals. The presented method solves the mass transport equation by considering advection and dispersion phenomena for one branch of the river network. The final mass balance equation is thus obtained in different types of river networks for each node. Regarding the connection matrix, the matrices of the characteristics of flow, pollutant, and network geometry are first determined as the model inputs. Then, by considering the diffusion and final mass conservation (Laplace transform of mass balance equation) equations for each node in the arbitrary river network, the mass balance and diffusion matrices are calculated in terms of the Laplace variable (s). Thus, a complex system of nonlinear algebraic equations is produced in terms of s, which can be solved using inverse Laplace algorithms to determine the concentration value at every node. The present study introduces four practical scenarios of pollutant transport in river networks in order to evaluate the proposed semi-analytical solution; the first is a branch-type river network with constant coefficients. The second case is a loop-type river network with variable coefficients. The third case simultaneously considers a point pollutant source and a distributed pollutant source in a branch-type river network. The fourth case is a branch-type river network under the dispersion-dominant phenomenon with a point source. Results show that the proposed semi-analytical solution is adequately capable of modeling pollutant transport and indicating its critical features in complex states of various river networks. Moreover, exploring the literature shows that no analytical solution has been introduced for river networks before undertaking the present study.

Modeling of turbulent diffusion terms for one-dimensional interfacial area transport equation in vertical round channels

The two-fluid model is comprised of the mass, momentum, and energy equations for liquid and gas phases separately. The terms of interfacial transfer govern mass, momentum, and energy transfer across the gas-liquid interface. The interfacial area concentration is critical to formulating the interfacial transfer terms. The one-dimensional Interfacial Area Transport Equation (IATE) has excellent potential in dynamically predicting the interfacial area concentration. The sink and source terms modeling for the interfacial area concentration is vital in the successful IATE development. The current IATE in bubbly flows considers the sink terms caused by bubble coalescence owing to random bubble collision and wake entrainment. The source term is caused by bubble breakup owing to turbulent impact. The turbulent diffusion term has not been considered in the current practice of one-dimensional IATE. This study developed the equation to predict the turbulent diffusion terms in the one-dimensional IATE for bubbly flow in vertical round channels. The predictive capability of the developed equation was validated with the collected database. The non-dimensionalized turbulent diffusion terms predicted by the developed equation agreed with those directly calculated by the experimental data. The developed turbulent diffusion term was also implemented in the one-dimensional IATE. The interfacial area concentrations predicted by the one-dimensional IATE with the turbulent diffusion term had a satisfactory agreement with the experimental results by the statistical parameters of the mean relative deviation of 3.76% and the mean absolute relative deviation of 10.1%.

BITS: an efficient transport solver based on a collocation method with B-spline basis

A B-spline Interpolation Transport Solver (BITS) based on a collocation method is developed. It solves transport equations as a generalized interpolation problem, taking the first-order accuracy in time and the second-order accuracy in space along with a predictor–corrector or under-relaxation iteration method. Numerical tests show that BITS can solve one-dimensional transport equations for tokamak plasma more accurately without additional computation cost, compared to the finite difference method transport solver which is widely used in existing tokamak transport codes.

О влиянии диффузии на точность измерения расхода жидкости методом лагранжевых меток

By solving a one-dimensional scalar transport equation with diffusion, it is shown that the diffusion effect can lead to an overestimation of the velocity determined by the Lagrangian particle tracking method, which consists in measuring the time of passage of a flow tracer between two sensors located in the channel. To obtain model signals, the transport equation was solved numerically for different initial scalar distributions, after which the scalar values at two spatial points were recorded. It is shown that in liquid metals characterized by high electrical and thermal conductivity, the effect can be no-ticeable. The limiting Peclet number, starting from which diffusion effects do not introduce error into measurements, was determined. It is shown that for high-frequency signals the diffusion effect will not manifest itself due to the rapid attenuation of perturbations. The paper provides estimates of Peclet numbers with turbulent transfer coefficients taken into account. It is shown that in developed turbulent flows the diffusion effect can appear due to a significant contribution of small-scale turbulence.

Improving the one-dimensional interfacial area transport equation using a physics-informed machine learning method

The interfacial area concentration modeling has been greatly improved with the development of the interfacial area transport equation (IATE). The one-dimensional IATE, which is obtained through area-averaging the three-dimensional form of the model, has been adopted into many thermal hydraulic analysis codes. Towards improving the performance of one dimensional interfacial area transport equation, along with many of the existing studies on developing sophisticated source and sink constitutive models describing the liquid-bubble and bubble-bubble interactions, additional efforts should be made on improving the accuracies of the closure models supporting the calculation of these constitutive models. In this study, a physics-informed machine learning method was proposed that couples the machine learning and the IATE model for improving the performance of the existing IATE models. The machine learning model serves to predict the errors of the closure models that calculate the key parameters related to the interfacial area transports. The machine learning model is guided by the physical models, and its training is through an IATE model-based loss optimization. To effectively train the machine learning model, a data augmentation method was developed based on the existing IAC database. The newly proposed method that couples the trained machine learning model and the IATE model gives a 12% of improvement compared with the state-of-art IATE model. This new method and the approach on training the model provide good reference on how to effectively couple the machine learning and physical model for an improved modeling of the indirectly observable physical phenomena.

Numerical Caseology by Lagrange Interpolation for the 1D Neutron Transport Equation in a Slab

Here, we are concerned with a new, highly precise, numerical solution to the one-dimensional neutron transport equation based on Case’s analytical, singular eigenfunction expansion (SEE). While a considerable number of numerical solutions currently exist, understandably, because of its complexity even in one dimension, there are only a few truly analytical solutions to the neutron transport equation. In 1960, Case introduced a consistent theory of the SEE for a variety of idealized transport problems and forever changed the landscape of analytical transport theory. Several numerical methods, including the Fn method, were based on the theory. What is presented is yet another, called the Lagrange order N method (LNM) featuring the simplicity and precision of the Fn method, but for a more convenient and natural Lagrangian polynomial basis.

Modeling effective thermal conductivity enhanced by surface waves using the Boltzmann transport equation

The thermal management of semiconductors at the device level has become a crucial issue owing to the high integration density and miniaturization of microelectronic systems. Because surface phonon polaritons (SPhPs) exhibit long propagation lengths, they are expected to contribute significantly to the heat dissipation in microelectronic systems. This study aims to numerically estimate the heat transfer due to SPhPs in a thin SiO2 film. The one-dimensional Boltzmann transport equation (BTE) is solved using the estimated propagation length based on the SPhP dispersion curves. The temperature profiles and heat fluxes are predicted and demonstrate the size effect of the film on the effective in-plane thermal conductivity of the SiO2 film. The results indicate that the temperature distribution was constant regardless of the film length and thickness because the propagation length was much longer than the film length. In addition, the heat flux increased with decreasing film thickness owing to the depth-averaged energy transfer. The effective thermal conductivities predicted using the BTE differed by ~ 16.5% from the values obtained from the analytical expression. The numerical results of this study can provide valuable data when studying the thermal behavior of SPhPs.

Numerical estimates of extreme material responses

Numerical models have to be used in most applications to characterize approximately the random fields describing the spatial fluctuation of material properties and estimate extremes and other functionals of materials responses. These models, referred to as finite dimensional (FD) models, are deterministic functions of the spatial coordinate and finite sets of random variables. Conditions are established under which the distributions of extremes and other functionals of material responses can be estimated from samples of material responses to FD material models. An one-dimensional stochastic transport equation is used to illustrate the construction and FD models and assess the accuracy of the extreme material response based on these models.

Existence analysis and numerical approximation for a second-order model of traffic with orderliness marker

We propose a toy model for self-organized road traffic taking into account the state of orderliness in drivers’ behavior. The model is reminiscent of the wide family of generalized second-order models (GSOM) of road traffic. It can also be seen as a phase-transition model. The orderliness marker is evolved along vehicles’ trajectories and it influences the fundamental diagram of the traffic flow. The coupling we have in mind is non-local, leading to a kind of “weak decoupling” of the resulting [Formula: see text] system; this makes the mathematical analysis similar to the analysis of the classical Keyfitz–Kranzer system. Taking advantage of the theory of weak and renormalized solutions of one-dimensional transport equations [Panov, 2008], which we further develop on this occasion in the Appendix, we prove the existence of admissible solutions defined via a mixture of the Kruzhkov and the Panov approaches; note that this approach to admissibility does not rely upon the classical hyperbolic structure for [Formula: see text] systems. First, approximate solutions are obtained via a splitting strategy; compactification effects proper to the notion of solution we rely upon are carefully exploited, under general assumptions on the data. Second, we also address fully discrete approximation of the system, constructing a BV-stable Finite Volume numerical scheme and proving its convergence under the no-vacuum assumption and for data of bounded variation. As a byproduct of our approach, an original treatment of local GSOM-like models in the BV setting is briefly discussed, in relation to discontinuous-flux LWR models.

Stable Difference Schemes with Interpolation for Delayed One-Dimensional Transport Equation

In this article, we consider the one-dimensional transport equation with delay and advanced arguments. A maximum principle is proven for the problem considered. As an application of the maximum principle, the stability of the solution is established. It is also proven that the solution’s discontinuity propagates. Finite difference methods with linear interpolation that are conditionally stable and unconditionally stable are presented. This paper presents applications of unconditionally stable numerical methods to symmetric delay arguments and differential equations with variable delays. As a consequence, the matrices of the difference schemes are asymmetric. An illustration of the unconditional stable method is provided with numerical examples. Solution graphs are drawn for all the problems.

Advances in the theory of existence and numerical simulations for the one-dimensional transport equation

The steady-state transport equation with internal sources and semi-reflective boundaries in a participative medium in slab geometry was analytically and numerically solved in this work. The existence theory in Hölder spaces when the sources belong to the space of continuous functions were established, generalizing results of previous works. Also, the Nyström method combines with singularity removal techniques was applied to produce an algorithm for calculating the scalar flux and criticality. The analytical estimates and the numerical methodology were obtained by looking at integral formulation as a Fredholm equation of the second type. Tabulated numerical results show the efficiency of the proposed methodology.

An analytical nodal method for solving the SN transport equation in 2-D Cartesian geometry

A novel Analytical Nodal Discrete Ordinates (ANDO) method for the solution of the discrete ordinates (SN) neutron transport equation in Cartesian geometry with isotropic scattering and constant neutron source is presented. A nodal method approximates the multi-dimensional transport equation as a system of coupled one-dimensional transport equations along each coordinate axis through transverse integration. The resulting transverse-integrated equations are then solved separately using a one-dimensional solution technique. The one-dimensional solution utilized in the new ANDO method is based on a recent closed-form analytical solution in slab geometry (WANG and BYAMBAAKHUU, 2017). The novel solution approach allows for a compact closed-form solution on each computational cell. Results for constant and linear transverse leakage approximations are presented. The ANDO method possesses a number of favorable properties, demonstrated with numerical results, such as high accuracy, rapid convergence, asymptotic preserving, positivity preserving, near linear computational complexity, and local-hp adaptivity. The ANDO method can also be easily extended to higher-order transverse leakage approximations, to three-dimensional Cartesian geometry, and to multi-group.

On a 1D model with nonlocal interactions and mass concentrations: an analytical-numerical approach* *Abreu was partially supported by CNPq 306385/2019-8, BR and Ferreira was partially supported by FAPESP and CNPq, BR.

We study global well-posedness and finite time blow-up of solutions for a nonlinear one-dimensional transport equation with nonlocal velocity , ν > 0, and measure initial data. Such model arises in fluid mechanics in vortex-sheet problems and its nonlocal feature comes from the presence of a singular integral operator (Hilbert transform ) in the velocity field. In the viscous case ν > 0, we analytically obtain an explicit condition on the size of the initial data for the global well-posedness in the framework of pseudomeasure spaces. In fact, we can give the condition depending on the initial-mass and analyze how the flow evolves from singular measures. Also, we numerically study blow-up of concentration type and global diffusion-smooth behavior of solutions. We obtain numerics that indicate the threshold value 2π for the initial-data mass that decides between blow-up or global smoothness of solutions. Such value is the same obtained for regular initial-data and by means of entropy methods. Thus, it seems to be intrinsic to the nonlocal PDE and independent of a particular framework, approach and initial-data regularity. The inviscid case ν = 0 is remarkable: simulations for model , evidence that the solution presents blow-up of concentration type with mass-preserving, while an attenuation effect is observed for the model with opposite sign in the nonlinearity , for any nontrivial (positive) measure as initial data.

Improving the One-Dimensional Interfacial Area Transport Equation Using a Physics-Guided Machine Learning Method

Improving the One-Dimensional Interfacial Area Transport Equation Using a Physics-Guided Machine Learning Method

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.