Flux Equations Research Articles

Seawater electrodialysis with preferential removal of divalent ions

In this work desalination of a ternary salt mixture and artificial North Sea water is studied with a lab scale electrodialysis stack, which was used in a recycling batch mode. During desalination samples were taken and the ionic composition of the dilute stream was determined. The effect of applied current density (10–300A/m2) on this composition was investigated.A clear effect of applied current density was observed. A lower applied current density leads to a more complete reduction in concentration of divalent ions, in an earlier extent of desalination. This influence of the applied current density could be related to the concentration polarization effects that occur in the diffusional boundary layer and are explained with a model based on the Nernst–Planck flux equation. It was found that the lower initial ion concentration of Ca2+, Mg2+, but also of K+, and SO42− compared to respectively Na+ and Cl−, leads to stronger depletion of these ions in the transport layer adjacent the membrane. These boundary layer effects are more pronounced at higher applied current densities, resulting in reduced transport of ions with a low initial concentration. High monovalent over divalent ion ratios can be found at low applied current.

A comparison of equations commonly used to calculate organic carbon content and marine palaeoproductivity from sediment data

Abstract A comparison is made of different methods for calculating total organic carbon (TOC) in deposits and marine primary productivity using sediment data. The calculation methods include both equations derived by pure regression analysis, and equations that include descriptions of processes such as carbon flux and burial efficiency. The results are compared with present-day measured values of total organic carbon and marine primary productivity, using measured sedimentation rate, dry bulk density and water depth. These measurements cover a large range of marine depositional environments, i.e. shallow to deep water, oxic to anoxic conditions, etc. Both TOC and productivity calculations show some consistent over- or under-prediction of computed values compared with the measured values, while some equations lead to unrealistic results. TOC calculations using process descriptions are influenced more by the choice of burial efficiency equation than by the choice of carbon flux equation. Productivity calculations are influenced by the choice of carbon flux equation. Calculations using measured values from a core show that productivity estimates vary by a factor of two.

Parameterizing Soil Emission and Atmospheric Oxidation-Reduction in a Model of the Global Biogeochemical Cycle of Mercury

Using the GEOS-Chem atmosphere-land-ocean coupled mercury model, we studied the significances of two processes, soil emission and atmospheric oxidation-reduction, in the global biogeochemical cycling of mercury and their parametrization to improve model performance. Implementing an empirical equation for soil emission flux (Esoil) including soil mercury concentration, solar radiation, and surface air temperature as parameters enabled the model to reproduce the observed seasonal variations of Esoil, whereas the default setting, which uses only the former two parameters, failed. The modified setting of Esoil also increased the model-simulated atmospheric concentration in the summertime surface layer of the lower- and midlatitudes and improved the model reproducibility for the observations in Japan and U.S. in the same period. Implementing oxidation of atmospheric gaseous elemental mercury (Hg(0)) by ozone with an updated rate constant, as well as the oxidation by bromine atoms (Br) in the default setting, improved the model reproducibility for the dry deposition fluxes observed in Japan. This setting, however, failed to reproduce the observed seasonal variations of atmospheric concentrations in the Arctic sites due to the imbalance between oxidation and reduction, whereas the model with Br as the sole Hg(0) oxidant in the polar atmosphere could capture the variations.

Kinetics of Partitioning of Zn(II) in Zn(II)/SO4 2− - BTMPPA/Kerosene and H2SO4 - Zn-BTMPPA Complex/BTMPPA/Kerosene Systems

Kinetics of Zn(II)-extraction from SO4 2− medium by purified Cyanex 272 (BTMPPA, H2A2) and of Zn(II)-stripping from Zn(II)-BTMPPA complex by H2SO4 solution have been investigated using the single falling drop technique and F-method of data treatment. At 303 K, the empirical extraction and stripping flux equations are: Ff (kmol/m2s) = 10−8.4 [Zn(II)] [H+ ] (int) −1 ([H2A2 ] (o) 0.5 + 1.07[H2A2 ] (o)) (1+0.708 [SO4 2− ])−1 and Fb (kmol/m2s) = 10−5.25 [Zn-BTMPPA complex] (o) [ H+ ] [H2A2 ] (o) −0.5 (1 + 1.5 [HSO4 −]), respectively. In both cases, values of Ea, ΔH±, and ΔS± have been estimated. With the help of flux equations and Ea-values, the mechanisms of extraction and stripping are proposed. The extraction is diffusion controlled in all parametric conditions at high temperature region (htr). At lower temperature region (ltr), it may be under diffusion, intermediate, or chemical control depending on the experimental parameters. The chemical control step is: . High negative ΔS± values suggest that this step occurs via an SN2 mechanism. On the other hand, the stripping is intermediate control at htr and chemical control at ltr. The chemical controlling step is either: ZnA+ → Zn2+ + A−; or, ZnA+ + HSO4 −→ ZnHSO4 + + A−; depending on [HSO4 −] in the medium. The slow reaction step in stripping occurs via an SN2 mechanism. The equilibrium constant derived from the rate study matches well with that obtained from the equilibrium study.

Simulation of evaporation, coupled liquid water, water vapor and heat transport through the soil medium

In this research, the model of SWCT (soil water content and temperature) was developed for simulation of fluxes of soil water (liquid water and water vapor) and heat. The governing equations for water and heat fluxes were solved using numerical finite difference method. Soil water content and temperature were numerically solved using a fully explicit method. The measurement devices of temperature sensors and TDR probes (for measuring water content) were installed at soil depths of 0.05, 0.35, and 0.5m, at the selected site. The novelty of this study was the simulation of actual evaporation from the soil surface using the Penman–Monteith's equation, in which the value of saturated air vapor pressure was substituted by actual soil vapor pressure at the surface layer. This novelty allowed estimation of the actual evaporation rate based on soil water content and temperature and meteorological data. Simulated water contents followed reasonably well the measured values at three soil depths during the simulation period. Soil water content and temperature were numerically simulated using the SWCT model with coupling liquid water, water vapor, and heat transport and provided reasonably good results with the values of RMSE less than 0.017cm3cm−3 and 2.2°C, respectively.The sensitivity analysis of the simulated soil temperatures were accomplished by changing radiation, input data of air temperatures and initial soil temperatures by ±20% and ±10% of measured values and the maximum values of RMSE changed by 65.7%, 140.3% and 190.7%, respectively. The results showed that air temperature, solar radiation, and initial soil temperature must be measured or predicted with high accuracy in order to produce low error values.

A SPACE–TIME CE/SE METHOD FOR SOLVING HYPERBOLIC HEAT CONDUCTION MODEL

This paper is concerned with the numerical approximation of one and two-dimensional models describing heat conduction in solids at low temperature. The system contains a conservation equation for the energy density and balance equations for the heat fluxes along each characteristic direction. A conservation element and solution element method (CE/E) is proposed for solving the given model. The method has already shown its efficiency and accuracy for solving a wide range of engineering problems. This technique is considerably different in both logic and design from the well-established finite volume schemes. The scheme treats space and time in a unified manner where conserved variables and their gradients are considered as independent unknowns. To illustrate the performance of the CE/SE method, several one and two-dimensional numerical case studies are carried out. The results of kinetic flux vector splitting (KFVS) scheme and central scheme are also presented for the comparison and further validation of our numerical results. The computations of this paper demonstrates that CE/SE method is effective in handling such problems.

Kinetic Analysis of Human and Canine P-Glycoprotein-Mediated Drug Transport in MDR1-MDCK Cell Model: Approaches to Reduce False-Negative Substrate Classification

Madin-Darby canine kidney (MDCK) cells transfected with the multidrug resistance 1 (MDR1) gene, MDR1-MDCK, are widely used as an in vitro model to classify compounds as human P-glycoprotein (hPgp) substrates or nonsubstrates. Because MDCK cells express endogenous canine Pgp (cPgp), which is prone to downregulation after transfection with hPgp, this situation could lead to false-negative classification of hPgp substrates. The aim of this study was to investigate factors that influence hPgp substrate classification in MDR1-MDCK model and to seek ways to reduce false classification. Three-compartment models were used to derive flux equations describing the drug transport processes; factors influencing hPgp substrate classification were evaluated by simulations. Pgp functionality was assessed by determining the bidirectional permeability of a series of test compounds. Expressions of hPgp and cPgp were measured by quantitative polymerase chain reaction (qPCR). Kinetic model analysis revealed that the current net flux ratio calculation for hPgp substrate classification is influenced by endogenous cPgp expression as well as hPgp-cPgp expression ratio; the effect was more pronounced in low hPgp-cPgp region and diminished in high ratio region. On the basis of kinetic considerations, this study provides a rational experimental approach and appropriate mathematical corrections to minimize the potential occurrence of false-negative classification of new molecular entities.

Surfactant transport onto a foam lamella

A modelling study of surfactant evolution on a film surface coupled to lamellar film drainage has been carried out. This study simulates the surfactant transport onto a lamella in a foam fractionation column with reflux: such columns offer greater control over enrichment and recovery of the surface active material. Insoluble surfactant is assumed during the simulation as such surfactants potentially derive more benefit from a reflux system. There are two terms involved in the equation for surfactant flux which are the Marangoni flow – from the Plateau border to the centre of the lamella – and the film drainage – which is directed from the centre of the lamella to the Plateau border. The extent of film drainage is bounded using two extremal assumptions of mobile or rigid interfaces. On a mobile interface, the film drainage dominates the Marangoni effect, while on a rigid interface, the Marangoni effect is dominant. The numerical simulation was carried out using a material point method followed by a bookkeeping operation to regrid the film. Analytical solutions for the case of no drainage and for a (quasi) steady state in which Marangoni flows and drainage are balanced were used to verify the numerical simulation. From the simulations, it can be concluded that the film drainage obtained using surfactant with a mobile interface is much faster than that determined using surfactant with a rigid interface, meaning that surfactant tends to be washed out of the film in the mobile case. The desirable condition in a foam fractionation column is however where the Marangoni flow dominates the liquid drainage which can be achieved when using surfactant that gives a rigid interface. The (quasi) steady state solution verifies the simulation result at later time for the case of film with a rigid interface. An asymptotic solution in the case of film with a mobile interface gives a prediction of the surface concentration of surfactant at the centre of the film, but also predicts growing surfactant concentration gradients and growing Marangoni stresses near the film's edge, which may then invalidate model assumptions.

Thermal convection of non-Fourier fluids. Linear stability

The natural convection of non-Fourier fluids is examined. These fluids possess a relaxation time that reflects the delay in the response of the heat flux with respect to the temperature gradient. Non-Fourier fluids span a wide range of applications, including low-temperature liquids, nanofluids, convection in nano-devices, and fluids subjected to fast heat transfer processes. The constitutive equation for heat flux is frame invariant. For small relaxation time, the neutral stability curve comprises a Fourier branch and an oscillatory branch, intersecting at some wavenumber. Both stationary and oscillatory convection become equally probable as the relaxation time increases, suggesting the existence of a bistable mode. With increasing relaxation time, oscillatory convection becomes increasingly the mode of preference, at the expense of both conduction and stationary convection, with increasing roll size. Although the oscillation frequency decreases monotonically with roll size, it exhibits a maximum against the relaxation time. Finally, the parallels between non-Fourier and non-Newtonian effects are emphasized at the formulation and phenomenological levels.

Robust stabilization of the current profile in tokamak plasmas using sliding mode approach in infinite dimension

This paper deals with the robust stabilization of the spatial distribution of tokamak plasmas current profile using a sliding mode feedback control approach. The control design is based on the 1D resistive diffusion equation of the magnetic flux that governs the plasma current profile evolution. The feedback control law is derived in the infinite dimensional setting without spatial discretization. Numerical simulations are provided and the tuning of the controller parameters that would reject uncertain perturbations is discussed. Closed loop simulations performed on realistic test cases using a physics based tokamak integrated simulator confirm the relevance of the proposed control algorithm in view of practical implementation.

Analytic Equilibria of High-Beta Tokamaks with Toroidal and Poloidal Flows and Pressure Anisotropy Associated with Parallel Heat Flux

Analytic solutions for a reduced set of magnetohydrodynamic equilibrium equations for high-beta tokamaks with toroidal and poloidal flow velocities comparable to the poloidal sound velocity as well as with pressure anisotropy associated with parallel heat flux are presented. The equations for the parallel heat flux are closed with a fluid closure model. Equilibrium profiles including anisotropic ion and electron pressures are self-consistently obtained and their properties are analytically examined. The solution indicates a qualitatively different behavior in the region around the poloidal sound velocity from those for adiabatic pressure [A. Ito and N. Nakajima: Plasma Phys. Control. Fusion 51 (2009) 035007] and the Chew–Goldberger–Low (CGL) double adiabatic pressure for ions since the parallel heat flux modifies the characteristics of sound waves.

Formula omitted] stabilization of the current profile in tokamak plasmas via an LMI approach

This paper deals with the H∞ stabilization of the spatial distribution of the current profile of tokamak plasmas using a Linear Matrix Inequality (LMI) approach. The control design is based on the one dimensional resistive diffusion equation of the magnetic flux that governs the plasma current profile evolution. The feedback control law is derived in the infinite dimensional setting without spatial discretization. The proposed distributed control is based on a proportional–integral state feedback taking into account both interior and boundary engineering actuators. Supporting numerical simulations are presented and tuning of the controller parameters attenuating uncertain disturbances is discussed.

Plasma focus ion beam fluence and flux—For various gases

A recent paper derived benchmarks for deuteron beam fluence and flux in a plasma focus (PF) [S. Lee and S. H. Saw, Phys. Plasmas 19, 112703 (2012)]. In the present work we start from first principles, derive the flux equation of the ion beam of any gas; link to the Lee Model code and hence compute the ion beam properties of the PF. The results show that, for a given PF, the fluence, flux, ion number and ion current decrease from the lightest to the heaviest gas except for trend-breaking higher values for Ar fluence and flux. The energy fluence, energy flux, power flow, and damage factors are relatively constant from H2 to N2 but increase for Ne, Ar, Kr and Xe due to radiative cooling and collapse effects. This paper provides much needed benchmark reference values and scaling trends for ion beams of a PF operated in any gas.

Chronoamperometric response of ion-exchange membrane systems

The electrical response of an ion-exchange membrane system to a square shaped electric potential, has been investigated from both experimental and theoretical viewpoints. The chronoamperometric response of a cation-exchange membrane in contact with a NaCl solution was experimentally obtained for different values of the amplitude of the perturbing electric potential. From the theoretical viewpoint, the system under study is considered to be constituted by a cation-exchange membrane and two diffusion boundary layers on both sides of the membrane. The ionic transport processes are prescribed by the Nernst–Planck flux equations, the electrical neutrality condition, and the Donnan equilibrium relations at the membrane/solution interfaces. The theoretical results are numerically obtained by using the network simulation method. Some of our results confirm that the chonoamperometric techniques can be used to measure the thickness of the diffusion boundary layers. Others provide new insights in relation with the time evolution of the short circuit electric current supplied by the salinity gradient due to the concentration polarization in ion-exchange membrane systems, showing the role played by the concentration polarization in the field of the renewable energies.

Numerical simulation of hydrogen–air reacting flows in rectangular channels with catalytic surface reactions

In this work a prediction was numerically modeled for a catalytically stabilized thermal combustion of a lean homogeneous mixture of air and hydrogen. The mixture flows in a narrow rectangular channel lined with a thin coating of platinum catalyst. The solution using an in-house code is based on the steady state partial differential continuity, momentum and energy conservation equations for the mixture and species involved in the reactions. A marching technique is used along the streamwise direction to solve the 2-D plane-symmetric laminar flow of the gas. Two chemical kinetic reaction mechanisms were included; one for the gas phase reactions consisting of 17 elementary reactions; of which 7 are forward–backward reactions while the other mechanism is for the surface reactions—which are the prime mover of the combustion under a lean mixture condition—consisting of 16 elementary reactions. The results were compared with a former congruent experimental work where temperature was measured using thermocouples, while using PLIF laser for measuring water and hydrogen mole fractions. The comparison showed good agreement. More results for the velocities, mole fractions of other species were carried out across the transverse and along the streamwise directions providing a complete picture of overall mechanism—gas and surface—and on the production, consumptions and travel of the different species. The variations of the average OH mole fraction with the streamwise direction showed a sudden increase in the region where the ignition occurred. Also the rate of reactions of the entire surface species were calculated along the streamwise direction and a surface water production flux equation was derived by calculating the law of mass action’s constants from the concentrations of hydrogen, oxygen and the rate of formation of water near the surface.

Imperfect pitchfork bifurcation in asymmetric two-compartment granular gas

The clustering behavior of a mono-disperse granular gas is experimentally studied in an asymmetric two-compartment setup. Unlike the random clustering in either compartment in the case of symmetric configuration when lowering the shaking strength to below a critical value, the directed clustering is observed, which corresponds to an imperfect pitchfork bifurcation. Numerical solutions of the flux equation using a modified simple flux function show qualitative agreements with the experimental results. The potential application of this asymmetric structure is discussed.

Governing equations in non-isothermal diffusion

Generalizations of Fick's law for the diffusion flux are often considered in the literature by analogy with those for the heat flux. The paper reviews the balance equations for a fluid mixture and provides the equations for the diffusion fluxes. As a consequence, the mass densities are shown to satisfy a system of hyperbolic equations. Moreover, for a binary mixture of ideal gases in stationary conditions, Fick's law is recovered. Next, diffusion fluxes are regarded as constitutive functions and a whole set of thermodynamic restrictions are determined which account for diffusion, heat conduction, viscosity and inhomogeneities. Hyperbolic models for diffusion and heat fluxes are established which involve the co-rotational derivative. The driving term of diffusion turns out to be the gradient of chemical potential rescaled by the temperature.

A hybrid molecular-continuum simulation method for incompressible flows in micro/nanofluidic networks

We present a hybrid molecular-continuum simulation method for modelling nano- and micro-flows in network-type systems. In these types of problem, a full molecular dynamics (MD) description of the macroscopic flow behaviour would be computationally intractable, or at least too expensive to be practical for engineering design purposes. Systems that exhibit multiscale traits, such as this, can instead be solved using a hybrid approach that distinguishes the problem into macroscopic and microscopic dynamics, modelled by their respective solvers. The technique presented in this study is an extension and addition to a hybrid method developed by Borg et al. (J Comput Phys 233:400–413, 2013) for high-aspect-ratio channel geometries, known as the internal-flow multiscale method (IMM). Computational savings are obtained by replacing long channels in the network, which are highly scale-separated, by much smaller, but representative, MD simulations, without a substantial loss of accuracy. On the other hand, junction components do not exhibit this length-scale separation, and so must be simulated in their entirety using MD. The current technique combines all network elements (junctions and channels) together in a coupled simulation using continuum conservation laws. For the case of steady, isothermal, incompressible, low-speed flows, we use the conservation of mass and momentum flux equations to derive a set of molecular-continuum constraints. An algorithm is presented here that computes at each iteration the new constraints on the pressure differences to be applied over individual MD micro-elements (channels and junctions), successively moving closer to macroscopic mass and momentum conservation. We show that hybrid simulations of some example network cases converge quickly, in only a few iterations, and compare very well to the corresponding full MD results, which are taken as the most accurate solutions. Major computational savings can be afforded by the IMM-type approximation in the channel components, but for steady-state solutions, even greater savings are possible. This is because the micro-elements are coupled to a steady-state continuum conservation expression, which greatly speeds up the relaxation of individual micro-components to steady conditions as compared to that of a full MD simulation. Unsteady problems with high temporal scale separation can also be simulated, but general transient problems are beyond the capabilities of the current technique.

Monitoring heat losses using Landsat ETM+ thermal infrared data — a case study at Kuju fumarolic area in Japan

To monitor heat losses using Landsat 7 thermal infrared data from 2002 to 2010 within the active fumarolic region of Kuju volcano in Japan, we used the Stefan-Boltzmann equation for radiative heat flux (RHF) estimation. Heat discharge rate (HDR) was calculated by using the relationship coefficient of RHF and HDR, obtained from two previous studies. The highest total RHF was found to be about 57.7 MW in 2002 and the lowest was about 21.1 MW in 2010. We found the highest HDR, of about 384.5 MW, in 2002 and the lowest, of about 140.8 MW, in 2010. The RHF anomalous areas were showing a declining trend during our study period. The relationship between the land surface temperature (LST) above ambient and RHF was, as expected, in a strong correlation for each result during our study period. Overall, our study was able to delineate the declining trend of heat losses that supports a previous study of similar declining trend of HDR using steam maximum diameter method from the active fumarolic region of Kuju volcano.

Splitting schemes in the mixed finite-element method for the solution of heat transfer problems

Some computational algorithms are discussed for the solution of equations of nonstationary heat conduction in terms of the temperature-heat flux vector. In other words, a nonstationary problem is considered in a mixed formulation, and the spatial approximation is performed by the mixed finite-element method. For the vector equation of a grid heat flux, numerical analysis is carried out for several splitting schemes. The main attention is concentrated on the relative accuracy of the proposed schemes.