Flux Equations Research Articles

A model for time-dependent grain boundary diffusion of ions and electrons through a film or scale, with an application to alumina

A model for ionic and electronic grain boundary transport through thin films, scales or membranes with columnar grain structure is introduced. The grain structure is idealized as a lattice of identical hexagonal cells – a honeycomb pattern. Reactions with the environment constitute the boundary conditions and drive the transport between the surfaces. Time-dependent simulations solving the Poisson equation self-consistently with the Nernst-Planck flux equations for the mobile species are performed. In the resulting Poisson-Nernst-Planck system of equations, the electrostatic potential is obtained from the Poisson equation in its integral form by summation. The model is used to interpret alumina membrane oxygen permeation experiments, in which different oxygen gas pressures are applied at opposite membrane surfaces and the resulting flux of oxygen molecules through the membrane is measured. Simulation results involving four mobile species, charged aluminum and oxygen vacancies, electrons, and holes, provide a complete description of the measurements and insight into the microscopic processes underpinning the oxygen permeation of the membrane. Most notably, the hypothesized transition between p-type and n-type ionic conductivity of the alumina grain boundaries as a function of the applied oxygen gas pressure is observed in the simulations. The range of validity of a simple analytic model for the oxygen permeation rate, similar to the Wagner theory of metal oxidation, is quantified by comparison to the numeric simulations. The three-dimensional model we develop here is readily adaptable to problems such as transport in a solid state electrode, or corrosion scale growth.

An analytical approach for solving a nodal formulation of two-dimensional fixed-source neutron transport problems with linearly anisotropic scattering

In this work, the formulation of the Analytical Discrete Ordinates (ADO) method for two-dimensional fixed-source neutron transport problems is extended to problems with anisotropic scattering. The methodology is developed from the discrete ordinates approximation of the two-dimensional transport, whose equations are integrated transversally within homogenized regions of the domain, yielding one-dimensional equations for the average angular fluxes. Such one-dimensional equations are solved by the ADO method, after considering approximations for the transverse leakage terms, and their solutions are written in terms of eigenvalues and eigenfunctions. The relevant feature of the ADO method for the isotropic scattering cases, which reduced the order of the eigenvalue problems to half of the number of discrete directions, is also preserved for the present anisotropic cases. Explicit spatial variables expressions are derived for the average angular fluxes in regions of interest defined in the domain. The full domain solution is defined by the coupling of local solutions in different regions, without the use of sweeping procedures. As usual in nodal schemes, auxiliary equations are necessary and, in this case, the unknown region-edge angular fluxes are approximated by constants embedded in the source term. Numerical results for region-average scalar fluxes are obtained and compared with the ones available in the literature to illustrate the feasibility of keeping the computational efficiency already verified in treating problems with isotropic scattering, with the use of the ADO method.

Quantifying differences in thermal dissipation probe calibrations for Eucalyptus globulus species and E. nitens × globulus hybrid

Calibration of sap flux density equations for Eucalytus is required when using sapflow thermal dissipation probes to avoid large underestimations of transpiration and water use. Eucalyptus plantations are expanding in response to global wood demand but are raising concerns about their impacts on water supplies. Sustainable plantation management in areas with water conflicts will require accurate assessments of tree and stand water use. Thermal dissipation probes have been used to estimate tree water use, but recent work suggests that species-specific calibrations may be required to obtain accurate results. In this study, we quantified sap flux density (SFD) in 2-year-old Eucalyptus globulus Labill (Eg) and E. nitens × globulus (Eng) species using the thermal dissipation method developed by Granier. For each species we compared the original Granier equation with species-specific calibrations using whole tree potometers over a 36-h period. Our results showed that on average, Granier’s original equation significantly underestimated SFD in both species, and when scaled to the stand level, tree transpiration (Ec) was significantly lower compared to onsite calibrations. The Granier method also underestimated nocturnal transpiration for both genotypes. Measured calibration coefficients were similar and not statistically different between Eg and Eng. These results highlight the importance of species-specific calibrations using thermal dissipation probes for Eucalyptus species to improve stand water use estimates and inferences about ecological impacts.

PIV-based estimation of unsteady loads on a flat plate at high angle of attack using momentum equation approaches

This work presents, compares and discusses results obtained with two indirect methods for the calculation of aerodynamic forces and pitching moment from 2D Particle Image Velocimetry (PIV) measurements. Both methodologies are based on the formulations of the momentum balance: the integral Navier–Stokes equations and the “flux equation” proposed by Noca et al. (J Fluids Struct 13(5):551–578, 1999), which has been extended to the computation of moments. The indirect methods are applied to spatio-temporal data for different separated flows around a plate with a $$16\mathrm {:}1$$ chord-to-thickness ratio. Experimental data are obtained in a water channel for both a plate undergoing a large amplitude imposed pitching motion and a static plate at high angle of attack. In addition to PIV data, direct measurements of aerodynamic loads are carried out to assess the quality of the indirect calculations. It is found that indirect methods are able to compute the mean and the temporal evolution of the loads for two-dimensional flows with a reasonable accuracy. Nonetheless, both methodologies are noise sensitive, and the parameters impacting the computation should thus be chosen carefully. It is also shown that results can be improved through the use of dynamic mode decomposition (DMD) as a pre-processing step.

A Nernst-Planck analysis on the contributions of the ionic transport in permeable ion-exchange membranes to the open circuit voltage and the membrane resistance in reverse electrodialysis stacks

The contributions of the ionic transport through a cation-exchange membrane, with perfectly stirred bathing solutions, on the open circuit voltage and the membrane ohmic resistance under reverse electrodialysis conditions, have been theoretically investigated. The Nernst-Planck flux equations, together with the Donnan equilibrium relations and the electrical neutrality condition, are numerically solved by using the network simulation method. On the one hand, the voltage-current characteristic, the polarization curves and the contributions of the ion-exchange membrane to the open circuit voltage and the membrane resistance are studied in a single system with NaCl solutions. The results are compared to those analytically obtained by using linear variations of ionic concentrations inside the membrane. On the other hand, a similar study is done for a membrane immersed in solutions constituted by a mixture of NaCl and MgCl2. The uphill transport of the divalent cation against its concentration gradient is considered in the interpretation of the results. The numerical results are compared to those analytically obtained by considering the hypothesis of total co-ion exclusion inside the membrane.

Investigation of MHD effects and heat transfer for the upper-convected Maxwell (UCM-M) micropolar fluid with Joule heating and thermal radiation using a hyperbolic heat flux equation

In this article, we investigate MHD flow of micropolar upper-convected Maxwell (UCM) fluid with the application of a recently proposed hyperbolic heat flux equation thus allowing explaining the time relaxation characteristics for heat flux. The aim is to incorporate microstructural effects in the upper-convected Maxwell fluid with Joule heating and thermal radiation. In this respect, microrotation of the fluid particles is taken into account and as a result the angular-momentum balance equation is obtained. The resulting governing flow equations are transformed into their respective system of ODEs by using suitable similarity transformations. As a consequence two non-dimensional parameters arise thus accounting for microstructural effects in the UCM fluid. Effects of these microstructural parameters on macroscopic velocity, microrotations and temperature profiles are examined. Moreover, different other physical parameters for instance, Magnetic parameter, Eckert number, Prandtl number, fluid and thermal relaxation time affecting the velocities and temperature profiles are also studied and discussed in case of both classical and micropolar fluid flow. The accuracy of the numerical solution obtained by the shooting method is shown by comparing the skin friction coefficient values in the case of classical fluid flow with existing results in the literature.

Specific features of the flow structure in a reactive type turbine stage

The results of experimental studies of the gas dynamics for a reactive type turbine stage are presented. The objective of the studies is the measurement of the 3D flow fields in reference cross sections, experimental determination of the stage characteristics, and analysis of the flow structure for detecting the sources of kinetic energy losses. The integral characteristics of the studied stage are obtained by averaging the results of traversing the 3D flow over the area of the reference cross sections before and behind the stage. The averaging is performed using the conservation equations for mass, total energy flux, angular momentum with respect to the axis z of the turbine, entropy flow, and the radial projection of the momentum flux equation. The flow parameter distributions along the channel height behind the stage are obtained in the same way. More thorough analysis of the flow structure is performed after interpolation of the experimentally measured point parameter values and 3D flow velocities behind the stage. The obtained continuous velocity distributions in the absolute and relative coordinate systems are presented in the form of vector fields. The coordinates of the centers and the vectors of secondary vortices are determined using the results of point measurements of velocity vectors in the cross section behind the turbine stage and their subsequent interpolation. The approach to analysis of experimental data on aerodynamics of the turbine stage applied in this study allows one to find the detailed space structure of the working medium flow, including secondary coherent vortices at the root and peripheral regions of the air-gas part of the stage. The measured 3D flow parameter fields and their interpolation, on the one hand, point to possible sources of increased power losses, and, on the other hand, may serve as the basis for detailed testing of CFD models of the flow using both integral and local characteristics. The comparison of the numerical and experimental results, as regards local characteristics, using statistical methods yields the quantitative estimate of their agreement.

The mechanism of heat transfer enhancement using longitudinal vortex generators in a laminar channel flow with uniform wall temperature

Heat transfer enhancements are used extensively to improve the performance of heat exchangers and to reduce their running cost. Longitudinal vortices can enhance heat transfer greatly. To enrich the mechanism of heat transfer enhancement produced by longitudinal vortices, this paper addresses the mechanisms of heat transfer enhancement enforced by longitudinal vortices generated by vortex generators (VGs) from another view point. The view point is based on the convective transport equation of heat flux, a description of convective heat transfer process in the process parameter of heat flux q. Based on this view point, the heat transfer enhancement mechanism is explained by the contributions of the combinations of flux with velocity gradient and velocity with flux gradient to the heat transfer. To find these contributions the heat transfer in a channel with and without VGs is studied numerically, and the differences of these contributions under conditions with and without VGs are compared in the frame of the convective transport equation of heat flux. The results show that longitudinal vortices greatly enhance the local convection contribution terms that promote the local transport of heat flux in span direction, which enforces the transports of heat flux along the main flow direction and heat flux along the normal direction by enlargements of the values of the corresponding convection contribution terms. The later results in the increase of the heat flux per unit temperature difference on the top and the bottom surfaces, and hence the heat transfer enhancement is enforced by longitudinal vortices. The local terms intensify the local transport of heat flux in span direction, but do not do so in an average measure in the presently studied case.

Analytical Study and Experimental Validation of Copper II Sulfate and Potassium Ferri/Ferrocyanide Thermocells Using Onsager Flux Equations

Thermocells convert heat energy directly into electrical energy through charge-transfer reactions at the electrode–electrolyte interface. To perform an analytical study on the behavior of thermocells, the Onsager flux relationship was applied to thermocells, which used aqueous copper II sulfate and aqueous potassium ferri/ferrocyanide as the electrolyte. The transport coefficient matrices were calculated for each electrolyte and applied to several simulations, which were subsequently validated through experimental testing and comparison to previous literature results. The simulation is shown to correctly predict the short circuit current, maximum power output, and power conversion efficiency. Validation demonstrates that the simulation model developed, using the Onsager flux equations, works for thermocells with different electrode materials (platinum, copper, charcoal, acetylene black, and carbon nanotube), electrode spacing, and temperature differentials. The power dependence of the thermocell on concentration and electrode spacing, with respect to the Seebeck coefficient, maximum power output, and relative efficiency, is also shown.

Governing equations of transient soil water flow and soil water flux in multi-dimensional fractional anisotropic media and fractional time

Abstract. In this study dimensionally consistent governing equations of continuity and motion for transient soil water flow and soil water flux in fractional time and in fractional multiple space dimensions in anisotropic media are developed. Due to the anisotropy in the hydraulic conductivities of natural soils, the soil medium within which the soil water flow occurs is essentially anisotropic. Accordingly, in this study the fractional dimensions in two horizontal and one vertical directions are considered to be different, resulting in multi-fractional multi-dimensional soil space within which the flow takes place. Toward the development of the fractional governing equations, first a dimensionally consistent continuity equation for soil water flow in multi-dimensional fractional soil space and fractional time is developed. It is shown that the fractional soil water flow continuity equation approaches the conventional integer form of the continuity equation as the fractional derivative powers approach integer values. For the motion equation of soil water flow, or the equation of water flux within the soil matrix in multi-dimensional fractional soil space and fractional time, a dimensionally consistent equation is also developed. Again, it is shown that this fractional water flux equation approaches the conventional Darcy equation as the fractional derivative powers approach integer values. From the combination of the fractional continuity and motion equations, the governing equation of transient soil water flow in multi-dimensional fractional soil space and fractional time is obtained. It is shown that this equation approaches the conventional Richards equation as the fractional derivative powers approach integer values. Then by the introduction of the Brooks–Corey constitutive relationships for soil water into the fractional transient soil water flow equation, an explicit form of the equation is obtained in multi-dimensional fractional soil space and fractional time. The governing fractional equation is then specialized to the case of only vertical soil water flow and of only horizontal soil water flow in fractional time–space. It is shown that the developed governing equations, in their fractional time but integer space forms, show behavior consistent with the previous experimental observations concerning the diffusive behavior of soil water flow.

Kinetics of extraction of Ti(IV) from SO42– medium by Cyanex 301 dissolved in kerosene containing 5% heptan-1-ol

ABSTRACTThe kinetics of Ti(IV) extraction by Cyanex 301 (HA) were investigated by measuring initial flux of Ti(IV) transfer (F, kmol/m2s), using a Lewis cell, operated at 3 Hz. The empirical flux equation at 298 K is found to be as follows: F (kmol/m2s) = 10–4.288 [Ti(IV)] (1 + 447 [H+])–1 [HA](o) (1 + 1.18[SO42–])–1. The activation energy, Ea, has been measured to be within 37–60 kJ/mol, depending on experimental parameters and temperature region. The ΔS± value is always highly negative. Analysis of the flux equation has been done, given various parametric conditions, to elucidate the mechanism of extraction. The rate-determining chemical reaction step, in most parametric conditions, appears to be as follows: TiO2+ + A– → TiOA+; and this step occurs via an SN2 mechanism as suggested by high negative ΔS± values. However, in certain cases, the extraction process appears as intermediate controlled as supported by Ea value of less than ~48 kJ/mol.

Flux concentration on tubular receiver of compound parabolic collector by surface areal irradiance method of ray tracing

Ray tracing for solar collectors finds wide application for designing the components. In this research, a solar flux concentration on the tubular receiver of compound parabolic collector (CPC) is determined by a novel approach of surface areal irradiance (SAI). The objective of this method is to determine the solar reflected irradiance at the receiver by incorporating effective areas of reflector and receiver. These areas are determined by 2D graphical ray tracing technique. Multiple ray tracing iterations are performed on various geometric possibilities of receiver–collector combinations by varying receiver diameter and height. The collector dimensions are fixed as per authors previously designed CPC to reduce the number of possibilities of receiver – collector combinations. The results of these iterations are presented in terms of utilization ratio (U), projection ratio (P) etc. based on which concentrated solar flux and heat equations are determined. It is observed that there is saturation point for all receiver diameters at which a receiver – collector combination gives highest utilization ratio and the corresponding height gives the best location for that receiver. SAI method gives a strong methodology for designing solar collectors on the basis of utilization ratios and heat equations.

Comparative Study of Small Electrical Machines With Soft Magnetic Composite Cores

In this paper, various kinds of electrical machines with soft magnetic composite (SMC) cores are compared, based on the qualitative and quantitative comparison methods. In the first part, the performances of five typical electrical machines with SMC cores are qualitatively compared. Simplified power equations for transverse flux, axial flux, and radial flux electrical machines are deduced to show the main difference among them and key design points of each machine. In the second part, the outer rotor claw pole machine (CPM) and outer rotor transverse flux machine (TFM) are comprehensively compared in a quantitative way, based on the three-dimensional finite-element method. It shows that the power capability of the outer rotor CPM is much higher than that of the TFM. On the other hand, the outer rotor CPM has higher cogging torque and no-load losses than the TFM. Furthermore, the four outer rotor radial flux machines are optimized and compared with the outer rotor CPM. The calculated results of the outer rotor TFM are compared with the experiment results, showing that the analysis results match well with the experiment ones. Several useful and interesting conclusions have been obtained for the electrical machines with SMC cores.

Investigation of Heat Fluid Analysis in Ventilated Disc Brake with Respect to Structure Geometry

The brake systems are composed of brake disc, brake pad and caliper and, these three parts play an important role for braking. In this study, heat fluid analysis is conducted for five different ventilated disc models, and two piece brake disc model separated in rotor and housing is used. In this case, each model has a different number of holes and vent shape. The basic heat flux and braking power equations are applied for the heat fluid analysis. The cooling performance with/without the braking operation is also analyzed for given five models where the material properties and boundary conditions are set to be identical. From our analysis results, it is found that the number of disc holes and ventilated pins strongly influences on the cooling performance.

Quantifying streambed advection and conduction heat fluxes

AbstractGroundwater and accompanying heat fluxes are particularly relevant for aquatic habitats as they influence living conditions both within the river and streambed. This study focuses on the theory and the development of new equations to estimate conduction and advection heat fluxes into and out of the bed, correcting some earlier misunderstandings and adding parameterizations that extend our understanding of timing of heat fluxes. The new heat flux equations are illustrated using Catamaran Brook (New Brunswick, Canada) stream/streambed temperature data. We show important relationships between fluxes when the surface water temperature (1) follows a sinusoidal function superimposed on a steady state condition (constant deep streambed temperature) and (2) when sinusoidal variations in stream temperature at two frequencies (annual and diel) are superimposed. When the stream temperature is used as a prescribed boundary condition, the contribution of bed fluid fluxes to stream temperature occurs through the effects of conductive thermal gradients, not through direct contribution/mixing of cold/warm water. Boundary conditions can be modified, however, to account for direct contribution of cold/warm water (e.g., localized upwelling) and consequences for the conduction heat flux. Equations developed allow for prediction of conductive fluxes to the bed during summer driven by diel and annual temperature fluctuations of the stream water and good agreement was observed between analytic solutions and field data. Results from this study provide a better insight into groundwater and heat fluxes which will ultimately result in better stream temperature models and a better management of fisheries resources.

Adaptive Torque Estimation for an IPMSM with Cross-Coupling and Parameter Variations

This paper presents a new adaptive torque estimation algorithm for an interior permanent magnet synchronous motor (IPMSM) with parameter variations and cross-coupling between d- and q-axis dynamics. All cross-coupled, time-varying, or uncertain terms that are not part of the nominal flux equations are included in two equivalent mutual inductances, which are described using the equivalent d- and q-axis back electromotive forces (EMFs). The proposed algorithm estimates the equivalent d- and q-axis back EMFs in a recursive and stability-guaranteed manner, in order to compute the equivalent mutual inductances between the d- and q-axes. Then, it provides a more accurate and adaptive torque equation by adding the correction terms obtained from the computed equivalent mutual inductances. Simulations and experiments demonstrate that torque estimation errors are remarkably reduced by capturing and compensating for the inherent cross-coupling effects and parameter variations adaptively, using the proposed algorithm.

Phonon and electron temperature and non-Fourier heat transport in thin layers

We present a thermodynamic model of heat conductor which allows for different temperatures of phonons and electrons. This model is applied to calculate the steady-state radial temperature profile in a circular thin layer. The compatibility of the obtained temperature profiles with the second law of thermodynamics is investigated in view of the requirement of positive entropy production and of a nonlocal constitutive equation for the entropy flux.

Influence of particle size in silo discharge

Recently Janda et al. [Phys. Rev. Lett. 108 , 248001 (2012)] reported an experimental study where it was measured the velocity and volume fraction fields of 1 mm diameter stainless steel beads in the exit of a two-dimensional silo. In that work, they proposed a new expression to predict the flow of granular media in silos which does not explicitly include the particle size as a parameter. Here, we study if effectively, there is not such influence of the particle size in the flux equations as well as investigate any possible effect in the velocity and volume fraction fields. To this end, we have performed high speed motion measurements of these magnitudes in a two-dimensional silo filled with 4 mm diameter beads of stainless steel, the same material than the previous works. A developed tracking program has been implemented to obtain at the same time both, the velocity and volume fraction. The final objective of this work has been to extend and generalize the theoretical framework of Janda et al. for all sizes of particles. We have found that the obtained functionalities are the same than in the 1 mm case, but the exponents and other fitting parameters are different.

A Study of Charged Cylindrical Gravitational Collapse with Dissipative Fluid

The present works deals with gravitational collapse of cylindrical viscous heat conducting anisotropic fluid following the work of Misner and Sharp. Using Darmois matching conditions, the dynamical equations are derived and the effects of charge and dissipative quantities over the cylindrical collapse are analyzed. Finally, using the Miller-Israel-Steward causal thermodynamic theory, the transport equation for heat flux is derived and its influence on collapsing system has been studied.

Hydrogen Transport to Fracture Sites in Metals and Alloys: Multiphysics Modelling

Generalised continuum model of hydrogen transport to fracture loci is developed for the purposes of analysis of the hydrogenous environment assisted fracture (HEAF). The model combines the notions of the theories of gas flow, surface science, and diffusion and trapping in stressed solids. Derived flux and balance equations describe the species migration across different states (gas, adsorbed specie at the gas-metal interface, interstitial solute in metal bulk) and a variety of corresponding sites of energy minimums along the potential relief for hydrogen in a system. The model accounts for the local kinetics of hydrogen interchange between the closest dissimilar neighbour sites and for the nonlocal interaction of hydrogen trapping in definite positions with the species wandering in their farer surroundings. In particular situations, certain balance equations of the model may degenerate into equilibrium constraints, as well as some terms in the generalised equations may be insignificant. A series of known theories of hydrogen transport in material-environment system can be recovered then as particular limit cases of the generalised model. Presented theory can help clarifying the advantages and limitations of particularised models so that appropriate one may be chosen for the analysis of a particular HEAF case.