Flux Equations Research Articles

The temperature field study on the three-dimensional surface moving heat source model in involute gear form grinding

Form grinding is one of the most important finishing processes for obtaining high surface quality gears. Nevertheless, to avoid thermally induced burning and structural change poses a major challenge for this manufacturing technology. Based on the moving heat source model and the superposition principle of heat sources, a comprehensive analytical thermal model for analyzing the heat transfer mechanism of involute gear form grinding was derived. A three-dimensional distribution model of equivalent grinding heat source considering the non-linear distribution of tangential grinding force, grinding parameters, and heat partition ratio was proposed. The FEM (finite element method) simulation of the grinding temperature field was carried out to predict the grinding temperature field using a nested net heat flux equation. Research results have shown that the grinding temperature increases nonlinearly with the rolling angle increasing and the evident peak grinding zone temperature appearing near the tip of the tooth. Through the comparison with the previous experimental measurement results, it is found that the temperature distribution of analytical calculation and numerical simulation agrees well with the measured results.

A Study on the Relationship Between the Boundary Flux Parameters and Membrane Process Requirements

The boundary flux concept permits to describe the fouling behaviour of membrane systems as a function of the operating time. The method relies on a set of equations that is possible to integrate in time, thus permitting to evaluate the separation process outcome and performances. This study focuses on the relationship between the membrane area requirements and specific parameters of the boundary flux concept on different membrane systems characterized by different waste feed streams and operating conditions. The target of the analysis was to identify which parameters of the boundary flux equations are the most influent one on the membrane area requirements. The knowledge of the relevant parameters can strongly assist membrane process designers to minimize the capex of the developed plants. The study was performed on many different systems, available in literature, and at different operating conditions. In order to avoid the triggering of irreversible fouling, the operation must be performed in sub-boundary operating conditions, and as a consequence, the condition of the permeate flux Jp equal to the boundary flux Jb must be met at the end of operation. Summarizing briefly the findings, it was possible to observe that the membrane area requirements are minimized for higher pure water membrane permeability values (w) for recoveries up to 75% Ymax, but then, in the range up to 90% Ymax, to achieve the same target, it appears to be more important to have higher Jb values. In other words, in those systems characterized by VRF (volume recovery factor) less than 4, high membrane permeabilities appears advantageous for the choice, even in presence of some extent of fouling. This appears not to be the case for those systems targeting higher VRF values: in this case, high Jb values must be achieved. Since the value of Jb depends on many parameters, such as T, Re and other physical-chemical characteristics, the minimization of the membrane area requirements requires the proper design of the membrane process, pre-treatment steps and operating conditions choice.

Numerical study of plane Couette flow: turbulence statistics and the structure of pressure–strain correlations

AbstractThe paper presents the results of direct numerical simulation of turbulent plane Couette flow. The calculations were performed for Reynolds numbers Re =U0H/ν(His the height of the channel, ±U0/2 is the motion velocities of the lower and upper walls, respectively,νis the kinematic viscosity) from 5200 (where viscous effects significantly affect the flow structure) to 80000 (where a logarithmic layer is clearly observed). Estimates of terms of the equation for the balance of turbulent Reynolds stresses are obtained, which indicate the importance of the kinetic energy transport by velocity fluctuations. The vertical transport of the turbulent momentum flux is less important and partly compensated by the transport of pressure fluctuations. It is shown that in the logarithmic layer the normal components of the ‘pressure–strain rate’ correlation tensor are described in the framework of the ‘isotropization of production’ model, and in the central part of the channel they are described by the linear Rotta model [29]. The additive model considering both the interaction of the velocity field fluctuation and the influence of the mean velocity gradient is a good approximation only for the off-diagonal component of the tensor entering the balance equation for the turbulent momentum flux.

Interaction of elementary waves with a weak discontinuity in an isothermal drift-flux model of compressible two-phase flows

In this paper, we study the interaction of elementary waves of the Riemann problem with a weak discontinuity for an isothermal no-slip compressible gas-liquid drift flux equation of two-phase flows. We construct the solution of the Riemann problem in terms of a one parameter family of curves. Using the properties of elementary waves, we prove a necessary and sufficient condition on initial data for which the solution of the Riemann problem consists of a left shock, contact discontinuity, and a right shock. Moreover, we derive the amplitudes of weak discontinuity and discuss the interactions of weak discontinuity with shocks and contact discontinuity. Finally, we carry out some tests to investigate the effect of shock strength and initial data on the jump in shock acceleration and the amplitudes of reflected and transmitted waves.

Critical review: incorporating the arrangement of mitochondria and chloroplasts into models of photosynthesis and carbon isotope discrimination.

The arrangement of mitochondria and chloroplasts, together with the relative resistances of cell wall and chloroplast, determine the path of diffusion out of the leaf for (photo)respired CO2. Traditional photosynthesis models have assumed a tight arrangement of chloroplasts packed together against the cell wall with mitochondria located behind the chloroplasts, deep inside the cytosol. Accordingly, all (photo)respired CO2 must cross the chloroplast before diffusing out of the leaf. Different arrangements have recently been considered, where all or part of the (photo)respired CO2 diffuses through the cytosol without ever entering the chloroplast. Assumptions about the path for the (photo)respiratory flux are particularly relevant for the calculation of mesophyll conductance (gm). If (photo)respired CO2 can diffuse elsewhere besides the chloroplast, apparent gm is no longer a mere physical resistance but a flux-weighted variable sensitive to the ratio of (photo)respiration to net CO2 assimilation. We discuss existing photosynthesis models in conjunction with their treatment of the (photo)respiratory flux and present new equations applicable to the generalized case where (photo)respired CO2 can diffuse both into the chloroplast and through the cytosol. Additionally, we present a new generalized Δ13C model that incorporates this dual diffusion pathway. We assess how assumptions about the fate of (photo)respired CO2 affect the interpretation of photosynthetic data and the challenges it poses for the application of different models.

A study on the kinetics of extraction of Ti(IV) from sulphate medium by Cyanex 302

The kinetics of Ti(IV)-extraction by Cyanex 302 (H2A2) were investigated by measuring initial Ti(IV)-transfer flux, using a constant interfacial area stirring cell, operated at 3 Hz. The empirical flux equation, at 293 K, is: F = 10−4.40 [Ti(IV)] (1 + 233[H+])−1 [H2A2](o)0.5 (1 + 3.20[SO42−])−1 (F in kmol/m2 s and [ ] terms are in kmol/m3). The activation energy, Ea is measured to be (36–58) × 103 kJ/kmol depending on experimental conditions. The enthalpy change on activation, ΔS±, is always highly negative. Analysis of the flux equation has been done, at various concentration regions of H+ and SO42−, to elucidate the mechanism of extraction. The rate-determining chemical reaction step, irrespective of extraction condition, appears as: TiO2+ + A− → TiOA+. This step occurs in the bulk aqueous phase via an SN2 mechanism. The pseudo first order rate constant for the system is 7.5–55 times smaller than the maximum mass transfer coefficient attainable in a Lewis cell. This indicates that the process is chemically controlled or at least mixed controlled.

Dynamics of viscous dissipative collapse with casual approach in string inspired Gauss–Bonnet gravity

In this work, we have studied the effects of dissipative viscous/heat gravitational collapse on the dynamics of collapsing source in 5D Einstein–Gauss–Bonnet theory using full casual approach. For this purpose, the dynamical equations have been formulated by using Misner–Sharp approach in 5D Einstein–Gauss–Bonnet theory of gravity. Using the Müller–Israel–Stewart theory, the dynamical equations have coupled with casual transport equations for the heat flux, the bulk and shear bulk viscosity to determine the effects of heat flux including thermodynamics viscous/heat coupling coefficients. The applications of this work to certain astrophysical situations are discussed.

Model of Consecutive, Steady-State Underflow for Vertical Tailing Silos

A continuous tailing discharge model is proposed for solving the problems of high fluctuation in underflow concentration and low, unstable actual discharge concentration in vertical tailing silos. On the basis of the mass balance equation, a mathematical model for continuous tailing discharge is derived, and a partial differential equation related to the height of the tailing silo in the compression area and tailing slurry concentration is obtained. The effective solid stress equation and the solid flux equation can be obtained from the results of centrifuge tests and intermittent sedimentation tests. The volume concentrations of underflow corresponding to various heights of tailing surfaces are simulated by using fluid mechanics software under dynamic balancing conditions. The results of the simulations are highly similar to those calculated by the derived mathematical model of continuous tailing discharge, and the results of the industrial test are also closely related to the results of the differential equation, thereby verifying the accuracy of the proposed discharge model.

Contaminant transport in a largely-deformed aquitard affected by delayed drainage

This paper employed a one-dimensional large-deformation model in consideration of the coupling of mechanical consolidation and solute transport, to study the transport of contaminants in a largely-deformed aquitard. An analytical solution has been derived to describe the drawdown variation in a largely-deformed aquitard which is subjected to abrupt hydraulic head decline in adjacent confined aquifers. The pore water flux and void ratio variation were obtained on the basis of the analytical solution. The equation for transient contaminant flux was solved by the finite difference method. A hypothetical case study was done to explore the effect of consolidation on the contaminant transport in a largely-deformed aquitard. The transit time of contaminant transport in the aquitard is mainly determined by the hydraulic conductivity, thickness, partitioning coefficient, void ratio and effective diffusion coefficients of aquitard, as well as the drawdown in the adjacent confined aquifer. The impact of delayed drainage on the contaminant transport in the largely-deformed aquitard is mainly controlled by two factors: the transient water flow and the decrease of aquitard thickness, in the process of aquitard consolidation. The former increases the breakthrough time of contaminant transport in the aquitard, and the latter gives rise to an opposite case with its effect decreasing with increasing contaminant partitioning coefficient for soil particles sorption. A larger deformation, which may be induced by a larger thickness, higher specific storativity of aquitard or a larger drawdown of the adjacent confined aquifer, and a lower hydraulic conductivity of aquitard cause a more significant impact of delayed drainage on the contaminant transport in an aquitard.

Nonlinear Diffusion Acceleration of the Least-Squares Transport Equation in Geometries with Voids

In this paper we show the extension of nonlinear diffusion acceleration (NDA) to geometries containing small voids using a weighted-least-squares (WLS) high-order equation. Even though the WLS equation is well defined in voids, the low-order drift-diffusion equation was not defined in materials with a zero cross section. This paper derives the necessary modifications to the NDA algorithm. We show that a small change to the NDA closure term and a nonlocal definition of the diffusion coefficient solve the problems for void regions. These changes do not affect the algorithm for optically thick material regions while making the algorithm well defined in optically thin ones. We use a Fourier analysis to perform an iterative analysis to confirm that the modifications result in a stable and efficient algorithm. Later in the paper, numerical results of our method are presented. We test this formulation with a small, one-dimensional test problem. Additionally, we present results for a modified version of the C5G7 benchmark containing voids as a more complex, reactor-like problem. We compared our results to Texas A&M’s transport code PDT, utilizing a first-order discontinuous formulation as reference and the self-adjoint angular flux equation with void treatment (SAAF), a different second-order form. The results indicate that the NDA WLS performed comparably or slightly worse then the asymmetric SAAF while maintaining a symmetric discretization matrix.

A stabilized well-balanced RBF-FD meshless method for shallow-water equations

In this work, we are concerned with the study and computing of stabilized radial basis function-generated finite difference (RBF-FD) approximations for shallow-water equations. In order to obtain both stable and highly accurate numerical approximations of convection-dominated shallow-water equations, we use stabilized flat Gaussians (RBFSGA-FD) and polyharmonic splines with supplementary polynomials (RBFPHS-FD) as basis functions, combined with modified method of characteristics. These techniques are combined with careful design for the spatial derivative operators in the momentum flux equation, according to a general criterion for the exact preservation of the “lake at rest” solution in general mesh-based and meshless numerical schemes for the strong form of the shallow-water equations with bottom topography. Both structured and unsructured point clouds are employed for evaluating the influence of cloud refinement, size of local supports and maximal permissible degree of the polynomials in RBFPHS-FD.

Modeling and Simulation Studies Analyzing the Pressure-Retarded Osmosis (PRO) and PRO-Hybridized Processes

Pressure-retarded osmosis (PRO) is viewed as a highly promising renewable energy process that generates energy without carbon emissions in the age of the climate change regime. While many experimental studies have contributed to the quest for an efficiency that would make the PRO process commercially viable, computational modeling and simulation studies have played crucial roles in investigating the efficiency of PRO, particularly the concept of hybridizing the PRO process with reverse osmosis (RO). It is crucial for researchers to understand the implications of the simulation and modeling works in order to promote the further development of PRO. To that end, the authors collected many relevant papers and reorganized their important methodologies and results. This review, first of all, presents the mathematical derivation of the fundamental modeling theories regarding PRO including water flux and concentration polarization equations. After that, those theories and thermodynamic theories are then applied to depict the limitations of a stand-alone PRO process and the effectiveness of an RO-PRO hybridized process. Lastly, the review diagnoses the challenges facing PRO-basis processes which are insufficiently resolved by conventional engineering approaches and, in response, presents alternative modeling and simulation approaches as well as novel technologies.

Nonlinear waves in thermoelastic dielectrics

This paper is addressed to the analysis of wave propagation in electroelastic materials. First the balance equations are reviewed and the entropy inequality is established. Next the constitutive equations are considered for a deformable and heat-conducting dielectric. To allow for discontinuity wave propagation, an appropriate objective rate equation of the heat flux is considered. The thermodynamic consistency of the whole set of constitutive equations is established. Next the nonlinear evolution equations so determined are tested in relation to wave propagation properties. Waves are investigated in the form of weak discontinuities and the whole system of equations for the jumps is obtained. As a particular simple case the propagation into an unperturbed region is examined. Both the classical electromagnetic waves and the thermal waves are found to occur. In both cases the mechanical term is found to be induced by the electrical or the thermal wave discontinuity.

Application of the Morozov–Solov’ev Equations to a Toroidal Magnetic Trap

In 1974, A.I. Morozov and L.S. Solov’ev derived a general system of time-independent equations of two-component ideal plasma hydrodynamics [1]. This system is extremely complicated and cumbersome. For the axisymmetric case, the authors succeeded in writing it in a more compact form by introducing three flux functions for the magnetic field, electrons, and ions. The situation resembles the derivation of the Grad–Shafranov equation, when the problem of static plasma equilibrium is reduced to one second-order equation for the magnetic flux. In this case, in order to close the problem, it was required to specify two arbitrary parameter functions. When considering flows instead of static equilibrium, the number of parameter functions increases. In the present work, the Morozov–Solov’ev equations are used to describe steady-state plasma configurations in toroidal magnetic traps.

Impulse source of high energy neutrons emitted by fusion reactions after compression of d--t gas by cumulative detonation waves

We develop the physical model and the system of equations for the impulse neutron source (INS) of high-energy neutrons ($\sim$14 MeV) emitted by fusion reactions during compression of D-T gas by cumulative detonation waves. The system of INS equations includes a system of gas dynamic equations that takes into account the energy transfer by radiation, equations for the radiation flux, the equation of the shock adiabat (the Hugoniot adiabat) for a compressed gas, and the equation for the neutron yield. We perform the INS dynamics simulation for the spherical and cylindrical geometries, and calculate maximum temperatures of D-T plasma, its density and neutron yield in the pulse. The obtained temperature estimates and simulation results show that the thermonuclear fusion temperatures are reached within this approach, and the fusion reactions proceed. Their yield determines the yield of neutrons.

From the simple reacting sphere kinetic model to the reaction-diffusion system of Maxwell–Stefan type

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 the collisional kernels and parameters of the kinetic model.

A Weighted Least-Squares Transport Equation Compatible with Source Iteration and Voids

Second-order forms of the transport equation allow the use of continuous finite elements (CFEMs). This can be desired in multiphysics calculations where other physics require CFEM discretizations. Second-order transport operators are generally self-adjoint, yielding symmetric positive-definite (SPD) matrices, which allow the use of efficient linear algebra solvers with an enormous advantage in memory usage.Least-squares (LS) forms of the transport equation can circumvent the void problems of other second-order forms but are almost always nonconservative. Additionally, the standard LS form is not compatible with discrete ordinates method (SN) iterative solution techniques such as source iteration. A new form of the LS transport equation has recently been developed that is compatible with voids and standard SN iterative solution techniques. Performing nonlinear diffusion acceleration (NDA) using an independently differenced low-order equation enforces conservation for the whole system and makes this equation suitable for reactor physics calculations. In this context, “independent” means that both the transport and low-order solutions converge to the same scalar flux and current as the spatial mesh is refined, but for a given mesh, the solutions are not necessarily equal.In this paper we show that introducing a weight function into this LS equation improves issues with causality and can render our equation equal to the self-adjoint angular flux (SAAF) equation. Causality is a principle of the transport equation that states that information travels only downstream along characteristics. This principle can be violated numerically. We show how to limit the weight function in voids and demonstrate the effect of this limit on accuracy. Using the C5G7 benchmark, we compare our method to the SAAF formulation with a void treatment (SAAFτ) that is not self-adjoint and has a nonsymmetric coefficient matrix. We show that the weighted LS equation with NDA gives acceptable accuracy relative to the SAAFτ equation while maintaining a SPD system matrix.

Simulations of Nonthermal Electrical Discharges in Air Over Solid Insulating Barrier

The propagation of streamer discharges in air in strongly divergent electric fields and their interaction with solid dielectric barrier inserted between the energized electrodes are analyzed by means of computer simulations. The model used is based on a hydrodynamic approximation of the transport of charge carriers in both media (electrons and ions in air, and electrons and holes in solid) coupled with Poisson’s equation for the electric field and accounts for charge injection and accumulation on gas–solid interfaces. The model is solved numerically utilizing customized finite-element formulation employing streamline upwind Petrov–Galerking method for stabilizing solution of drift–diffusion equations for fluxes of charged species. The results of the simulations performed for conditions of earlier reported experimental studies are presented and discussed. It is shown that streamer discharges may develop in different modes depending upon particular conditions and are capable of propagating along solid surfaces in the direction transverse to the applied field. In the latter case, their progress is fully controlled by space and surface charges accumulated within discharge plasma and on gas–solid interfaces. The developed calculation method creates a base for considering more complicated and practically oriented tasks.

Fundamental equations for primary fluid recovery from porous media

Primary fluid recovery from a porous medium is driven by the volumetric expansion of thein situfluid. For production from a petroleum reservoir, primary recovery accounts for more than half of the total amount of recovered hydrocarbon. The primary recovery process is studied here at the pore scale and the macroscopic scale. The pore-scale flow is first analysed using the compressible Navier–Stokes equations and the mathematical theory for low-Mach-number flow developed by Klainerman & Majda (Commun. Pure Appl. Maths, vol. 34 (4), 1981, pp. 481–524; vol. 35 (5), 1982, pp. 629–651). An asymptotic analysis shows that the pore-scale flow is governed by the self-diffusion of the fluid and it exhibits a slip-like mass flow rate, even though the velocity satisfies the no-slip condition on the pore wall. The pore-scale density equation is then upscaled to a macroscopic diffusion equation for the density which possesses a diffusion coefficient proportional to the fluid’s kinematic viscosity. Darcy’s law is shown to be inapplicable to primary fluid recovery and it should be replaced by a new mass flux equation which depends on the porosity but not on the permeability. This is in stark contrast to the classical result and it can have important implications for hydrocarbon recovery as well as other applications.

Balance and constitutive equations in thermoelectroelastic solids

The article addresses the modeling of thermoelectroelastic solids in the quasi-static approximation. First, the balance equations are revisited with the purpose of contrasting two formulations occurring in the literature: one is based on the Maxwell stress tensor and another involves expressions of force, couple, and power determined via considerations on single dipoles. Next a non-stationary scheme of heat conduction is set up with a rate equation based on an objective time derivative. The thermodynamic restrictions are then determined by allowing also for the dependence on gradients of temperature, deformation gradient, and electric field. The thermodynamic analysis shows that the selected rate equation for the heat flux induces some effects on the Cauchy stress tensor. Some particular cases of constitutive equations for the stress are determined.