Flux Equations Research Articles

On the Maxwell-Stefan diffusion limit for a reactive mixture of polyatomic gases in non-isothermal setting

In this article we deduce a mathematical model of Maxwell-Stefan type for a reactive mixture of polyatomic gases with a continuous structure of internal energy. The equations of the model are derived in the diffusive limit of a kinetic system of Boltzmann equations for the considered mixture, in the general non-isothermal setting. The asymptotic analysis of the kinetic system is performed under a reactive-diffusive scaling for which mechanical collisions are dominant with respect to chemical reactions. The resulting system couples the Maxwell-Stefan equations for the diffusive fluxes with the evolution equations for the number densities of the chemical species and the evolution equation for the temperature of the mixture. The production terms due to the chemical reaction and the Maxwell-Stefan diffusion coefficients are moreover obtained in terms of general collision kernels and parameters of the kinetic model.

Key Uncertainties in the Recent Air‐Sea Flux of CO2

AbstractThe contemporary air‐sea flux of CO2 is investigated by the use of an air‐sea flux equation, with particular attention to the uncertainties in global values and their origin with respect to that equation. In particular, uncertainties deriving from the transfer velocity and from sparse upper ocean sampling are investigated. Eight formulations of air‐sea gas transfer velocity are used to evaluate the combined standard uncertainty resulting from several sources of error. Depending on expert opinion, a standard uncertainty in transfer velocity of either ~5% or ~10% can be argued and that will contribute a proportional error in air‐sea flux. The limited sampling of upper ocean fCO2 is readily apparent in the Surface Ocean CO2 Atlas databases. The effect of sparse sampling on the calculated fluxes was investigated by a bootstrap method, that is, treating each ship cruise to an oceanic region as a random episode and creating 10 synthetic data sets by randomly selecting episodes with replacement. Convincing values of global net air‐sea flux can only be achieved using upper ocean data collected over several decades but referenced to a standard year. The global annual referenced values are robust to sparse sampling, but seasonal and regional values exhibit more sampling uncertainty. Additional uncertainties are related to thermal and haline effects and to aspects of air‐sea gas exchange not captured by standard models. An estimate of global net CO2 exchange referenced to 2010 of −3.0 ± 0.6 Pg C/year is proposed, where the uncertainty derives primarily from uncertainty in the transfer velocity.

Flux Equations for Osmotically Moderated Sorption–Diffusion Transport in Rigid Microporous Membranes

The development of non-aqueous reverse osmosis separations via the use of rigid microporous membrane materials suggests that the flux equations originally developed for swellable polymeric membrane...

The Representation of Hydrological Dynamical Systems Using Extended Petri Nets (EPN)

AbstractThis work presents a new graphical system to represent hydrological dynamical models and their interactions. We propose an extended version of the Petri Nets mathematical modeling language, the Extended Petri Nets (EPN), which allows for an immediate translation from the graphics of the model to its mathematical representation in a clear way. We introduce the principal objects of the EPN representation (i.e., places, transitions, arcs, controllers, and splitters) and their use in hydrological systems. We show how to cast hydrological models in EPN and how to complete their mathematical description using a dictionary for the symbols and an expression table for the flux equations. Thanks to the compositional property of EPN, we show how it is possible to represent either a single hydrological response unit or a complex catchment where multiple systems of equations are solved simultaneously. Finally, EPN can be used to describe complex Earth system models that include feedback between the water, energy, and carbon budgets. The representation of hydrological dynamical systems with EPN provides a clear visualization of the relations and feedback between subsystems, which can be studied with techniques introduced in nonlinear systems theory and control theory.

Multi-layer flows of immiscible fractional Maxwell fluids with generalized thermal flux

Unsteady laminar flows and heat transfer of n-immiscible fractional Maxwell fluids in a channel are investigated under influence of time-dependent pressure gradient. The isothermal channel walls have translational motions in their planes with time-dependent velocities. Governing equations of the mathematical model are based on the generalized constitutive equations for shear stress and thermal flux described by the time-fractional Caputo derivative. Analytical and semi-analytical solutions for velocity, shear stress, and temperature fields are obtained by using finite sine-Fourier and Laplace transforms. In the case of semi-analytical solutions, the inverse Laplace transforms are obtained numerically by employing the Talbots algorithms. Using the software Mathcad, numerical calculations have carried out and results are presented in graphical illustrations in order to analyze the memory effects on the fluid temperature and motion. It is found that in fluids with thermal memory the heat transfer is slower compared with the ordinary fluid, while the fractional velocity parameters act as braking/accelerating factors of the fluids.

Fitting thermal conductivity and optimizing thermoelectric efficiency in [formula omitted] nanowires

We consider a thermoelectric energy generator constituted by a Si∕Ge nanowire of length L. The dependence on composition and temperature of its thermal conductivity is analyzed in view of three series of experimental data obtained at the constant temperatures T=300K, T=400K, and T=500K. The best-fit curve is determined by a nonlinear regression method (NLRM). Then, under the hypothesis of nonlinear constitutive equation for the heat flux, we investigate the thermoelectric efficiency of the system as function of the composition of the nanowire and of the difference of temperature applied to its ends. For each temperature we calculate the value of the composition which realizes the optimal efficiency of the thermoelectric energy conversion. The corresponding value of the thermal conductivity is determined as well.

Excitations of the Myers-Perry black holes

We demonstrate separability of the dynamical equations for all p-form fluxes in the Myers-Perry-(A)dS geometry, extending the earlier results for electromagnetic field. In the physically important cases of p = (1–4), we explicitly write the ODEs governing the dynamics of separable solutions.

New sensor measures biological activity in soil at field scale in real time

New sensor measures biological activity in soil at field scale in real time

Time-dependent integral transport in one-dimensional infinite media using dimensionless variables and the reduced collision formulation

A new method for calculating time-dependent integral neutron transport single collision fluxes has been developed for one-dimensional infinite media problems using dimensionless variables and the reduced collision formulation. By transforming the integral scalar flux equation into dimensionless variables, one may immediately integrate over the time domain and reduce the problem complexity to integrations strictly over the spatial domain, significantly reducing the computational work required to obtain a solution. The results have been compared to published benchmark solutions for validation. Further, reformulating the scalar flux as a Green’s function has allowed for the determination of an analytical expression for the relative error of any collision level scalar flux to the summed scalar flux. This provides the absolute number of collisions required for convergence at any given mean free time, using previous work in one-dimensional Cartesian transport and derived work herein for the one-dimensional spherical transport.

Evolution mechanisms and fundamental equations of social water cycle fluxes

Abstract The rise of socio-hydrology, addressing the interactions between human and water systems, is regarded as an innovative perspective to researches achieving the sustainable use of water resources. Revealing the social water fluxes, in terms of magnitude, structure, and variations under changing environment, could advance the understanding of water cycling under the dual driving forces: natural and anthropogenic. This study attempts to formulate the fundamental equations of the social water cycle by focusing on the evolution mechanisms of social water cycle fluxes. The endogenously dynamic characteristics of social water cycling are portrayed, i.e., the gradual change mechanism and the catastrophe mechanism, therefore dividing the evolution processes into four stages. Then, social water cycle flux reaches its peak and completes the first stage of evolution. The evolution process is an S-shaped curve process. After the peak, it enters the next evolutionary stage, where the pattern varies with the intensities of the gradual change mechanism and the catastrophe mechanism. The coordination relationships of these two mechanisms and the fluctuating characteristics in each stage are studied as well. Case studies are investigated in 39 countries globally to verify the fitting of the fundamental equations and evolution mechanisms.

Electromagnetic Analysis and Performance Investigation of a Flux-Switching Permanent Magnet Machine

This paper aims to present a general and effective analytical approach to calculate the air gap flux density and the back electromotive force (EMF) of a flux-switching permanent magnet (FSPM) machine. The proposed analytical expression of the air gap flux density is based on an improved air gap permeance function considering the geometries of slotted stator core pieces and magnets between stator teeth as well as the salient rotor poles. The back EMF equation is accurately derived using the proposed air gap flux density equation expressed in terms of practical machine dimensions and thus it provides the key design factors as well as details of the back EMF production mechanism. To validate the proposed analytical expressions, they are applied to the case study of a 12-slot 10-pole FSPM machine, and the finite element analysis results confirm the analytical predictions. Besides, for the proposed analytical model, the effects of the machine’s geometries on back EMF characteristics are investigated. The investigation shows that the ratio of rotor slot opening to slot pitch has a significant effect on the back EMF, and its optimal value is suggested. The proposed equations also provide a mean to choose the slot and pole combinations to obtain a higher power density.

Sensorless Control of Surface-Mounted Permanent Magnet Synchronous Motor Drives Using Nonlinear Optimization

This paper proposes a novel nonlinear optimization-based sensorless control scheme for surface-mounted permanent magnet synchronous motors (SPMSM) over wide speed range operations. Two algebraic constraints derived from the voltage equations and the flux equations of the SPMSM system are fundamental to the establishment of the proposed scheme. The position estimation algorithm generates the rotor position estimate via the stator flux estimates obtained by minimizing an equality constrained scalar function. Specifically, the flux information is extracted both from the voltage-based flux model and the current-based flux model, which contributes to its robustness against the inherent weakness of the pure integrator to dc offsets. Meanwhile, a nonlinear observer, based on the standard gradient search, is constructed for speed estimation. The proposed method is unique because the position estimation and the speed estimation are achieved independently; thus, the complexity associated with the speed dependence in position estimation is eliminated. Further, it is simple to implement. The feasibility and effectiveness of the proposed position algorithm and the speed observer have been verified by experiments under different operating conditions.

Estimating a baseline of soil CO2 flux at CO2 geological storage sites.

This study aims to determine a baseline for natural soil carbon dioxide (CO2) flux at the surface based on long-term field measurements, with the ultimate purpose to detect the gas leakage at CO2 geological storage sites. CO2 surface monitoring is a tool that measures the safety and effectiveness of CO2 capture and storage (CCS), a technology which is believed to be a reliable approach to mitigate the CO2 emission. However, the fluctuations of naturally occurring CO2 in soil layers complicate the leakage detection as the soil connects both the underground layers and the atmosphere. In this regard, this study not only investigates the natural surface CO2 flux behavior but also develops an equation to estimate the surface CO2 flux with respect to the soil moisture content and temperature. To meet this end, two values within the CO2 flux equation were defined and calculated based on the field measurements; a, representing a water saturation-dependent value, and b, representing the temperature sensitivity (independent of the water saturation). The results show a good agreement between estimated and measured data. Upon which, the maximum baseline for surface CO2 flux was derived and used as a threshold to detect the potential CO2 leakage in the candidate field (INAS, Japan).

A Spatially Resolved and Quantitative Model of Early Atherosclerosis.

Atherosclerosis is a major burden for all societies, and there is a great need for a deeper understanding of involved key inflammatory, immunological and biomechanical processes. A decisive step for the prevention and medical treatment of atherosclerosis is to predict what conditions determine whether early atherosclerotic plaques continue to grow, stagnate or become regressive. The driving biological and mechanobiological mechanisms that determine the stability of plaques are yet not fully understood. We develop a spatially resolved and quantitative mathematical model of key contributors of early atherosclerosis. The stability of atherosclerotic model plaques is assessed to identify and classify progression-prone and progression-resistant atherosclerotic regions based on measurable or computable in vivo inputs, such as blood cholesterol concentrations and wall shear stresses. The model combines Darcy's law for the transmural flow through vessels walls, the Kedem-Katchalsky equations for endothelial fluxes of lipoproteins, a quantitative model of early plaque formation from a recent publication and a novel submodel for macrophage recruitment. The parameterization and analysis of the model suggest that the advective flux of lipoproteins through the endothelium is decisive, while the influence of the advective transport within the artery wall is negligible. Further, regions in arteries with an approximate wall shear stress exposure below 20% of the average exposure and their surroundings are potential regions where progression-prone atherosclerotic plaques develop.

Light scattering in a spatially-correlated particle field: Role of the radial distribution function

Radiative transfer through particle-laden media such as clouds can be impacted by variations in particle spatial distributions. Due to mixing and inertial effects of droplets suspended in the almost always turbulent atmosphere, cloud particles are often spatially-correlated. The correlations result in clusters and voids within the droplet field that, even when smaller than the photon mean free path, can lead to deviations from the exponential extinction law. Prior work has numerically investigated these departures from exponential attenuation in absorptive media; this work extends those results for a scattering medium. The problem is explored with a Monte Carlo Ray Tracing (MCRT) program capable of tracking light attenuation through both perfectly random (uncorrelated) and spatially correlated collections of scatterers and/or absorbers. The MCRT program is favorably compared to two-stream flux equations, and numerical exploration of the pure-absorption case is used to determine the sampling statistics necessary to characterize radiative transmission within the numerical simulation. Light transmission through fields of spatially-correlated, non-absorbing, scattering particles is explored. Particles are distributed following a Matérn Point Process, which allows cluster strength and size, as well as the usual variables of particle scattering cross section and number density to be varied. The results show that the degree of non-exponential attenuation is determined by the magnitude and shape of the radial distribution function, which describes correlations in discrete (non-continuous) particle distributions. Parametric studies revealed that the number of clusters and cluster radius, factors in the Matérn radial distribution function, impact direct, diffuse and backward radiative transfer. The Matérn RDF is shown to be consistent with a previous “cloudlet” approach, providing a bridge between the analytical cloudlet model and continuous correlation function approaches.

Solutions to the 3D Transport Equation and 1D Diffusion Equation for Passive Tracers in the Atmospheric Boundary Layer and Their Applications

Abstract A solution to the 3D transport equation for passive tracers in the atmospheric boundary layer (ABL), formulated in terms of Green’s function (GF), is derived to show the connection between the concentration and surface fluxes of passive tracers through GF. Analytical solutions to the 1D vertical diffusion equation are derived to reveal the nonlinear dependence of the concentration and flux on the diffusivity, time, and height, and are employed to examine the impact of the diffusivity on the diurnal variations of CO2 in the ABL. The properties of transport operator H and their implications in inverse modeling are discussed. It is found that H has a significant contribution to the rectifier effect in the diurnal variations of CO2. Since H is the integral of GF in time, the narrow distribution of GF in time justifies the reduction of the size of H in inverse modeling. The exponential decay of GF with height suggests that the estimated surface fluxes in inverse modeling are more sensitive to the observations in the lower ABL. The solutions and first mean value theorem are employed to discuss the uncertainties associated with the concentration–mean surface flux equation used to link the concentrations and mean surface flux. Both analytical and numerical results show that the equation can introduce big errors, particularly when surface flux is sign indefinite. Numerical results show that the conclusions about the evolution properties of passive tracers based on the analytical solutions also hold in the cases with a more complicated diffusion coefficient and time-varying ABL height.

Computational Study of MHD Nanofluid Flow Possessing Micro-Rotational Inertia over a Curved Surface with Variable Thermophysical Properties

This work presents a numerical investigation of viscous nanofluid flow over a curved stretching surface. Single-walled carbon nanotubes were taken as a solid constituent of the nanofluids. Dynamic viscosity was assumed to be an inverse function of fluid temperature. The problem is modeled with the help of a generalized theory of Eringen Micropolar fluid in a curvilinear coordinates system. The governing systems of non-linear partial differential equations consist of mass flux equation, linear momentum equations, angular momentum equation, and energy equation. The transformed ordinary differential equations for linear and angular momentum along with energy were solved numerically with the help of the Keller box method. Numerical and graphical results were obtained to analyze the flow characteristic. It is perceived that by keeping the dynamic viscosity temperature dependent, the velocity of the fluid away from the surface rose in magnitude with the values of the magnetic parameter, while the couple stress coefficient decreased with rising values of the magnetic parameter.

Investigation of two-phase change flow in mechanical seal with complex solid surfaces

Phase change flows between seal faces play an important role in the performance of mechanical seals. Cavitation and boiling phenomena are observed to occur between mechanical seal faces. Most studies on phase change flows in mechanical seals focus on only one phenomenon. However, in different mechanical seal structures, for example, sector-groove seals, cavitation and boiling may appear simultaneously when the temperature and speed are changed. This study contributes to ongoing investigations in the field by proposing a phase change flow model considering both cavitation and boiling in mechanical seals. First, the governing equations and main features of the proposed model are discussed based on the mass conservation law, which is particularly useful in cases where both cavitation and boiling occur. Then, numerical schemes were constructed by employing the liquid volume fraction and analyzed by considering two different forms of fluid flux equations. Case studies of two models, namely, the wavy face seal and hydrostatic seal, were conducted and separately analyzed, and the numerical results demonstrated the feasibility of the proposed model. Then, numerical simulation and experiment of a sector-groove seal were conducted to demonstrate the advantage of the proposed model. The proposed model is especially effective for analyzing the two-phase flow field bounded by complex solid surfaces.

Dissipative properties of relativistic two-dimensional gases

The constitutive equations for the heat flux and the Navier tensor are established for a high temperature dilute gas in two spatial dimensions. The Chapman–Enskog procedure to first order in the gradients is applied in order to obtain the dissipative energy and momentum fluxes from the relativistic Boltzmann equation. The solution for such equation is written in terms of three sets of orthogonal polynomials which are explicitly obtained for this calculation. As in the three dimensional scenario, the heat flux is shown to be driven by the density, or pressure, gradient additionally to the usual temperature gradient given by Fourier’s law. For the stress (Navier) tensor one finds, also in accordance with the three dimensional case, a non-vanishing bulk viscosity for the ideal monoatomic relativistic two dimensional gas. All transport coefficients are calculated analytically for the case of a hard disk gas and both the non-relativistic and ultrarelativistic limits for the constitutive equations are addressed.

Chromium removal and water recycling from electroplating wastewater through direct osmosis: Modeling and optimization by response surface methodology

Background: Considering the carcinogenic effects of heavy metals, such as chromium, it is essential to remove these elements from water and wastewater. Direct osmosis is a new membrane technology, which can be a proper alternative to conventional chromium removal processes. Methods: The wastewater samples were collected from an electroplating unit, located in Alborz industrial city, Qazvin, Iran. Magnesium chloride was used as the draw solution, and a semipermeable membrane (Aquaporin) was used in this study. The experiments were designed, using response surface methodology (RSM) and central composite design (CCD) with draw solution concentration (0.5- 1.5 M), feed solution concentration (4-12 mg/L), and experiment time (30-90 minutes) as variable factors. The chromium concentration and water flux were also measured, based on atomic absorption spectrophotometry and water flux equation, respectively. Results: Direct osmosis was highly efficient in chromium removal and water recycling. Water flux and chromium removal efficiency were 15.6 LMH and 85.58%, respectively, under optimal conditions (draw solution = 1.27 mol/L, feed solution = 4 mg/L, and experiment time = 90 min). In terms of validity, the results predicted by the quadratic polynomial model were in good agreement with the responses reported in the laboratory. Conclusion: In direct osmosis, the use of magnesium chloride as the draw solution resulted in the acceptable chromium removal from electroplating wastewater. Using this method, chromium concentration in wastewater reduced to a level lower than the discharge standards, established by Iran’s Department of Environment.