Hydrodynamic Flux Research Articles

Hydrodynamic energy flux in a many-particle system

In this letter, we demonstrate energy transfers and thermalization in an isolated ensemble of realistic gas particles. We perform a grid-free classical molecular dynamics simulation of two-dimensional Lenard-Jones gas. We start our simulation with a large-scale vortex akin to a hydrodynamic flow and study its non-equilibrium behavior till it attains thermal equilibrium. In the intermediate phases, small wavenumbers (k) exhibit E(k)∝k−3 kinetic energy spectrum, whereas large wavenumbers exhibit E(k)∝k spectrum. Asymptotically, E(k)∝k for the whole range of k, thus indicating thermalization. These results are akin to those of Euler turbulence despite complex collisions and interactions among the particles.

Large-scale sediment and phosphorus transport in the Three Gorges Reservoir based on a new reservoir operation method

Change of hydrodynamic conditions is a key factor inducing sedimentation, water eutrophication and algal blooms in the Three Gorges Reservoir (TGR). How to mitigate sedimentation and phosphorus (P) retention by improving hydrodynamic conditions in the Three Gorges Reservoir area (TGRA) is an urgent issue in the study of sediment and water environment. In this study, a Hydrodynamic-Sediment-Water quality model for the whole TGRA is proposed considering sediment and P inputs from numerous tributaries, and a new reservoir operation method namely the tide-type operation method (TTOM) is used to investigate the large-scale sediment and P transport in the TGR based on the model. Results indicate that the TTOM can reduce sedimentation and total phosphorus (TP) retention in the TGR. Compared with the actual operation method (AOM), sediment outflow and sediment export ratio (Eratio) of the TGR increased about 17.13% and 1%–3% in 2015–2017, and sedimentation decreased about 3% under the TTOM. TP retention flux and retention rate (RE) decreased about 13.77% and 2%–4%. The flow velocity (V) and sediment carrying capacity (S*) increased about 40% in the local reach. Larger daily water level fluctuation at dam site is more conducive to reducing sedimentation and TP retention in the TGR. Sediment inputs from the Yangtze River, Jialing River, Wu River and other tributaries account for 59.27%, 11.21%, 3.81% and 25.70% of the total sediment inflow during 2015–2017, and TP inputs were 65.96%, 10.01%, 17.40% and 6.63%. In the paper, an innovative method is proposed to reduce sedimentation and P retention in the TGR under the given hydrodynamic conditions and related quantitative contribution driven by the proposed method is analyzed. The work is favorable for expanding the understanding of the hydrodynamic and nutrition flux changes in the TGR, and provides a new perspective for water environment protection and reasonable operation of large reservoirs.

Lamb dilatation and its hydrodynamic viscous flux in near-wall incompressible flows

In this paper, we introduce a new physical concept referred to as the wall-normal Lamb dilatation flux (WNLDF), which is defined as the wall-normal derivative of the Lamb dilatation (namely, the divergence of the Lamb vector) multiplied by the dynamic viscosity for incompressible viscous flows. It is proved that the boundary Lamb dilatation flux (BLDF, namely the wall-normal Lamb dilatation flux at the wall) is determined by the boundary enstrophy flux (BEF) and the surface curvature-induced contribution. As the first step to explore this new concept, the present study only considers flow past a stationary no-slip flat wall without curvature-fluid dynamic coupling effects. It is found that the temporal–spatial evolution rate of the WNLDF is contributed by four source terms, which can be explicitly expressed by using the fundamental surface quantities including skin friction (or surface vorticity) and surface pressure. Therefore, the instantaneous near-wall flow structures are directly related to the WNLDF in both laminar and turbulent flows. As an example, for the turbulent channel flow at Reτ=180, the intensification of the temporal–spatial evolution rate of the WNLDF is caused by the strong wall-normal velocity event (SWNVE) associated with quasi-streamwise vortices and high intermittency of the viscous sublayer. In addition, near the SWNVE, this evolution rate is mainly contributed by the dot product of skin friction and surface enstrophy gradient, as well as the coupling between the skin friction divergence and the boundary enstrophy. The exact results presented in this paper could provide new physical insights into complex near-wall flows.

Energy Flux Through the Magnetopause During Flux Transfer Events in Hybrid-Vlasov 2D Simulations.

Solar wind-magnetosphere coupling drives magnetospheric dynamic phenomena by enabling energy exchange between magnetospheric and solar wind plasmas. In this study, we examine two-dimensional noon-midnight meridional plane simulation runs of the global hybrid-Vlasov code Vlasiator with southward interplanetary magnetic field driving. We compute the energy flux, which consists of the Poynting flux and hydrodynamic energy flux components, through the Earth's magnetopause during flux transfer events (FTEs). The results demonstrate the spatiotemporal variations of the energy flux along the magnetopause during an FTE, associating the FTE leading (trailing) edge with an energy injection into (escape from) the magnetosphere on the dayside. Furthermore, FTEs traveling along the magnetopause transport energy to the nightside magnetosphere. We identify the tail lobes as a primary entry region for solar wind energy into the magnetosphere, consistent with results from global magnetohydrodynamic simulations and observations.

Spin-piston problem for a ferromagnetic thin film: Shock waves and solitons

The unsteady, nonlinear magnetization dynamics induced by spin injection in an easy-plane ferromagnetic channel subject to an external magnetic field are studied analytically. Leveraging a dispersive hydrodynamic description, the Landau-Lifshitz equation is recast in terms of hydrodynamic-like variables for the magnetization's perpendicular component (spin density) and azimuthal phase gradient (fluid velocity). Spin injection acts as a moving piston that generates nonlinear, dynamical spin textures in the ferromagnetic channel with downstream quiescent spin density set by the external field. In contrast to the classical problem of a piston accelerating a compressible gas, here, variable spin injection and field lead to a rich variety of nonlinear wave phenomena from oscillatory spin shocks to solitons and rarefaction waves. A full classification of solutions is provided using nonlinear wave modulation theory by identifying two key aspects of the fluid-like dynamics: subsonic/supersonic conditions and convex/nonconvex hydrodynamic flux. Familiar waveforms from the classical piston problem such as rarefaction (expansion) waves and shocks manifest in their spin-based counterparts as smooth and highly oscillatory transitions, respectively. The spin shock is an example of a dispersive shock wave, which arises in many physical systems. New features without a gas dynamics counterpart include composite wave complexes with "contact" spin shocks and rarefactions. Magnetic supersonic conditions lead to two pronounced piston edge behaviors including a stationary soliton and an oscillatory wavetrain. These coherent wave structures have physical implications for the generation of high frequency spin waves from pulsed injection and persistent, stable stationary and/or propagating solitons in the presence of magnetic damping. The analytical results are favorably compared with numerical simulations.

Fabrication of Polyamide-6 Membranes-The Effect of Gelation Time towards Their Morphological, Physical and Transport Properties.

Porous polyamide-6 membranes were fabricated via a non-solvent induced phase inversion method, and the influence of gelation time on the properties of the membranes was investigated. Membrane samples with various gelation times were prepared. The evaluation of the membranes’ properties was carried out by various analyses and tests, such as scanning electron microscopy, atomic force microscopy, contact angle, wet and dry thickness, mean pore size measurements, porosity, water uptake, mechanical resistance, hydrodynamic water fluxes, membrane hydrodynamic permeability, and retention testing. The scanning electron microscopy images (both surface and cross-section) demonstrated that the increase in gelation time from 0 (M0) to 10 (M10) min led to the morphological change of membranes from isotropic (M0) to anisotropic (M10). The wet and dry thickness of the membranes showed a downward tendency with increasing gelation time. The M0 membrane exhibited the lowest bubble contact angle of 60 ± 4° and the lowest average surface roughness of 124 ± 22 nm. The highest values of mean pore size and porosity were observed for the M0 sample (0.710 ± 0.06 µm and 72 ± 2%, respectively), whereas the M10 membrane demonstrated the highest tensile strength of 4.1 MPa. The membrane water uptake was diminished from 62 to 39% by increasing the gelation time from 0 to 10 min. The M0 membrane also showed the highest hydrodynamic water flux among the prepared membranes, equal to 28.6 L m−2 h−1 (at Δp = 2 bar).

The response of the suspended sediment load of the headwaters of the Brahmaputra River to climate change: Quantitative attribution to the effects of hydrological, cryospheric and vegetation controls

The response of suspended sediment load to climate change has important implications for downstream ecosystems and biogeochemical cycles. Glaciers, permafrost, vegetation and hydrological processes are rapidly changing in high-altitude regions. However, there is currently no consensus about how and to what extent these factors will affect interannual changes of fluvial sediment export. Here, we investigate the increasing trend of suspended sediment load in the headwaters of the Brahmaputra River from 1972 to 2008. The key influencing factors, including the hydrodynamic flux and erosivity of glacier runoff, precipitation, vegetation cover, and soil erodibility were quantified, and their changing trends were analyzed. Two methods were used to attribute the variation of sediment load; Partial Least Squares Path Modeling (PLS-PM) based on statistical regression and Sediment Identity (SI) based on algebraic equations. The results obtained by these two relatively independent methods were broadly consistent, and the error of the SI model was less than 8%. The sediment load showed a significant increasing trend at an average changing rate of 18.9%/10a, without any abrupt fluctuations. The increase in fluvial sediment load was primarily due to increases in both the amount (43%) and erosivity (55%) of precipitation, combined with minor influences from increased sediment supply due to earlier thawing (16%) and intensified glacier melting (14%). The increases in fluvial sediment load were partially buffered by a decrease in water yield capacity due to denser vegetation cover (−29%). This study provides valuable insights into the natural response of fluvial sediment loads to climatic and environmental change, and the findings are critical for soil and water conservation planning in cold and high-altitude regions.

Interfacial thermal resistance in phonon hydrodynamic heat conduction

Two-dimensional materials are usually predicted to have ultrahigh thermal conductivity because of the numerous phonon normal scatterings, which might cause hydrodynamic heat conduction. In addition, boundary and interface are significant in the polycrystalline structure and material contacts. Therefore, this article investigates the thermal behaviors at the boundary and interface in phonon hydrodynamics. Monte Carlo simulation is adopted to study the heat conduction phenomena in Poiseuille hydrodynamics and Ziman hydrodynamics. The concept of a boundary temperature step is defined to depict the temperature decline behaviors at the boundary in steady hydrodynamic heat conduction. Interfacial thermal behaviors can be treated as a combination of the boundary effects and phonon transmission effects, where the interface properties can be described by the interface transmissivity and the specular reflectivity. Moreover, the inverse temperature difference at the interface is observed, which means that the heat is transported from low temperature to high temperature, implying that the definition of temperature in phonon hydrodynamic heat conduction ought to be further investigated. Then, two theoretical models are proposed to describe these phenomena, namely, the particle propagation model and the dual boundary flux model. The particle propagation model tries to trace the propagation and evolution of phonons with simpler rules, and it finds that the heat flux reduction originates from the backward phonons that are scattered by the normal scattering process. The dual boundary flux model divides the whole boundary heat flux into the hydrodynamic heat flux and the diffusive heat flux, and the boundary temperature step appears in the transition between these two fluxes. These two models are compared with the results obtained by Monte Carlo simulations.

Acoustic and hydrodynamic power of wave scattering by an infinite cascade of plates in mean flow

We study the balance of outgoing and incoming power for acoustic wave scattering by a cascade of flat blades in uniform subsonic mean flow. In nonzero mean flow the Kutta condition at the trailing edge of the cascade plates leads to the production of vorticity in the form of unsteady vortex sheets attached at the trailing edges, which results in a hydrodynamic power flux and the exchange of energy between the unsteady vorticity and the acoustic field. Although the scattering problem itself has been subject to much previous research, a comprehensive study of the composition of the outgoing energy flux for cascade scattering has received little previous attention. As an application of our detailed study we show how the time-averaged energy balance can be used to explain certain symmetries of the field in the angle of incidence with respect to the cascade face when there is no mean flow and the effect of zero acoustic reflection at certain angles of incidence with mean flow. We also show that, depending on the angle of incidence of the acoustic wave and the velocity of the mean flow, the vortex sheets can either absorb or generate acoustic energy. This can, for certain parameter values, result in a net increase in outgoing sound power over the incident sound power, as we demonstrate on a number of numerical examples that are computed using the semi-analytical solution described in recent work by Maierhofer and Peake (J. Sound Vib., vol. 481 (2020), 115418).

Stormtime Energetics: Energy Transport Across the Magnetopause in a Global MHD Simulation

Coupling between the solar wind and magnetosphere can be expressed in terms of energy transfer through the separating boundary known as the magnetopause. Geospace simulation is performed using the Space Weather Modeling Framework (SWMF) of a multi-ICME impact event on February 18–20, 2014 in order to study the energy transfer through the magnetopause during storm conditions. The magnetopause boundary is identified using a modified plasma β and fully closed field line criteria to a downstream distance of −20Re. Observations from Geotail, Themis, and Cluster are used as well as the Shue 1998 model to verify the simulation field data results and magnetopause boundary location. Once the boundary is identified, energy transfer is calculated in terms of total energy flux K, Poynting flux S, and hydrodynamic flux H. Surface motion effects are considered and the regional distribution of energy transfer on the magnetopause surface is explored in terms of dayside X>0, flank X<0, and tail cross section X=Xmin regions. It is found that total integrated energy flux over the boundary is nearly balanced between injection and escape, and flank contributions dominate the Poynting flux injection. Poynting flux dominates net energy input, while hydrodynamic flux dominates energy output. Surface fluctuations contribute significantly to net energy transfer and comparison with the Shue model reveals varying levels of cylindrical asymmetry in the magnetopause flank throughout the event. Finally existing energy coupling proxies such as the Akasofu ϵ parameter and Newell coupling function are compared with the energy transfer results.

Flow-boiling canopy wick capillary-viscous limit

It was predicted that the perforated, leveed flow-boiling canopy wick (FBCW) can significantly increase the hydrodynamic instability limit of the flow boiling [1,2]. The perforations represent and modulate the vapor columns in the boiling crises hydrodynamic stability theory. The lower capillary-viscous limit, governed by the evaporator wick maximum capillary pressure and the viscous drag within the three-dimensional composite wick, may control the actual dryout performance. Saturated-water flow-boiling (one atm) experiment results are presented employing an FBCW, created from sintered-copper particles, with and without the hydrodynamic-stabilizing levees. The bimodal bilayer evaporator wick allows for sufficiently large maximum capillary pressure and permeability, compared to the monolayer wick. The experimental results indicate that the levees can increase the critical heat flux (CHF) over that without levees, whereas the existing maximum capillary pressure controls disallow achieving this larger hydrodynamic heat flux limit. This capillary-viscous dryout limit is predicted by employing the minimum surface energy principles, network models, and computational fluid dynamics (CFD), in good agreement with experiments. By employing both CFD and flow visualizations, good agreement is also established between the predicted and observed two-phase hydrodynamics above the canopy wick. They indicate that although the local dryout commences downstream; however, dryout is experimentally observed when a critical fraction of the liquid supply tracks dry downstream. The measured large thermal conductance (heat transfer coefficient) is beneficial for the predictions, verifying the vapor-occupied space between the canopy and evaporator wicks.

Dynamic soliton–mean flow interaction with non-convex flux

The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to non-convex flux is studied in the framework of the modified Korteweg–de Vries (mKdV) equation, a canonical model for internal gravity wave propagation and potential vorticity fronts in stratified fluids. The effect of large amplitude, dynamically evolving mean flows on the propagation of localised waves – essentially ‘soliton steering’ by the mean flow – is considered. A recent theoretical and experimental study of this new type of dynamic soliton–mean flow interaction for convex flux has revealed two scenarios where the soliton either transmits through the varying mean flow or remains trapped inside it. In this paper, it is demonstrated that the presence of a non-convex cubic hydrodynamic flux introduces significant modifications to the scenarios for transmission and trapping. A reduced set of Whitham modulation equations is used to formulate a general mathematical framework for soliton–mean flow interaction with non-convex flux. Solitary wave trapping is stated in terms of crossing modulation characteristics. Non-convexity and positive dispersion – common for stratified fluids – imply the existence of localised, sharp transition fronts (kinks). Kinks play dual roles as a mean flow and a wave, imparting polarity reversal to solitons and dispersive mean flows, respectively. Numerical simulations of the mKdV equation agree with modulation theory predictions. The mathematical framework developed is general, not restricted to completely integrable equations like mKdV, enabling application beyond the mKdV setting to other fluid dynamic contexts subject to non-convex flux such as strongly nonlinear internal wave propagation that is prevalent in the ocean.

Multidimensional low-Mach number time-implicit hydrodynamic simulations of convective helium shell burning in a massive star

Context. A realistic parametrization of convection and convective boundary mixing in conventional stellar evolution codes is still the subject of ongoing research. To improve the current situation, multidimensional hydrodynamic simulations are used to study convection in stellar interiors. Such simulations are numerically challenging, especially for flows at low Mach numbers which are typical for convection during early evolutionary stages. Aims. We explore the benefits of using a low-Mach hydrodynamic flux solver and demonstrate its usability for simulations in the astrophysical context. Simulations of convection for a realistic stellar profile are analyzed regarding the properties of convective boundary mixing. Methods. The time-implicit Seven-League Hydro (SLH) code was used to perform multidimensional simulations of convective helium shell burning based on a 25 M⊙ star model. The results obtained with the low-Mach AUSM+-up solver were compared to results when using its non low-Mach variant AUSMB+-up. We applied well-balancing of the gravitational source term to maintain the initial hydrostatic background stratification. The computational grids have resolutions ranging from 180 × 902 to 810 × 5402 cells and the nuclear energy release was boosted by factors of 3 × 103, 1 × 104, and 3 × 104 to study the dependence of the results on these parameters. Results. The boosted energy input results in convection at Mach numbers in the range of 10−3–10−2. Standard mixing-length theory predicts convective velocities of about 1.6 × 10−4 if no boosting is applied. The simulations with AUSM+-up show a Kolmogorov-like inertial range in the kinetic energy spectrum that extends further toward smaller scales compared with its non low-Mach variant. The kinetic energy dissipation of the AUSM+-up solver already converges at a lower resolution compared to AUSMB+-up. The extracted entrainment rates at the boundaries of the convection zone are well represented by the bulk Richardson entrainment law and the corresponding fitting parameters are in agreement with published results for carbon shell burning. However, our study needs to be validated by simulations at higher resolution. Further, we find that a general increase in the entropy in the convection zone may significantly contribute to the measured entrainment of the top boundary. Conclusion. This study demonstrates the successful application of the AUSM+-up solver to a realistic astrophysical setup. Compressible simulations of convection in early phases at nominal stellar luminosity will benefit from its low-Mach capabilities. Similar to other studies, our extrapolated entrainment rate for the helium-burning shell would lead to an unrealistic growth of the convection zone if it is applied over the lifetime of the zone. Studies at nominal stellar luminosities and different phases of the same convection zone are needed to detect a possible evolution of the entrainment rate and the impact of radiation on convective boundary mixing.

A new process combination with high water flux and superior treatment performance for stevia sugar liquor

Stevia sugar is widely used in food owing to its advantages of low calorie content and high sweetness. In the production of stevia sugar, stevia leaves are immersed in water to obtain a crude liquor, which contains many impurities, such as proteins, polysaccharides, organic acids, etc. These impurities are removed typically by coagulation and filtration. At present, suppliers of stevia mainly use a “lime + ferrous sulfate (FeSO4)” process as the pretreatment, but residual SO42− can cause hydrogen sulfide corrosion and Ca2+ can affect the safety of subsequent boilers in the process. In this study, polyaluminium ferric chloride (PAFC) was evaluated as a superior coagulant to FeSO4, together with ultrafiltration (UF) and nanofiltration to remove flocs and other organic contaminants. It was found that coagulation by PAFC out-performed FeSO4 and was unaffected by pH, while the performance of FeSO4 was sensitive to pH. At pH 10, FeSO4 had an enhanced coagulation efficiency, but formed soluble Fe2+ complexes with organic substances in the original solution, which adversely affected the quality of the filtrate. Compared with a UF membrane, the NF membrane was more effective in removing contaminants, and more tolerant to membrane fouling. In addition to hydrodynamic flux, measurement of trans-membrane electrical current were used to reflect the severity of membrane fouling. The combination of PAFC as coagulant without pH adjustment, and ultrafiltration, was found to be a simpler and superior treatment for stevia liquor with a high degree of contaminant separation, and avoidance of problems caused by SO42− and Ca2+.

Hydrodynamics across a fluctuating interface.

Understanding what happens inside the rippling and dancing surface of a liquid remains one of the great challenges of fluid dynamics. Using molecular dynamics, we can pick apart the interface structure and understand surface tension. In this work, we derive an exact mechanical formulation of hydrodynamics for a liquid-vapor interface using a control volume, which moves with the surface. This mathematical framework provides the local definition of hydrodynamic fluxes at any point on the surface. These are represented not only by the flux of molecules and intermolecular interactions acting across the surface but also as a result of the instantaneous local curvature and movement of the surface itself. By explicitly including the surface dynamics in the equations of motion, we demonstrate an exact balance between kinetic and configurational pressure normal to the surface. The hydrodynamic analysis makes no assumptions regarding the probability distribution function, so it is valid for any system arbitrarily far from thermodynamic equilibrium. The presented equations provide a theoretical basis for the study of time-evolving interface phenomena, such as bubble nucleation, droplet dynamics, and liquid-vapor instabilities.

Event-Driven Hyporheic Exchange during Single and Seasonal Rainfall in a Gaining Stream

This study combined a reach-scale field survey and numerical modelling analysis to reveal the pattern of transient hyporheic exchange driven by rainfall events in the Zhongtian River, Southeast China. Field observations revealed hydrodynamic properties and flux variations in surface water (SW)/ groundwater (GW), suggesting that the regional groundwater recharged the study reach. A two-step numerical modelling procedure, including a hydraulic surface flow model and a groundwater flow model, was then used to interpret the transient hyporheic flow system. The hyporheic exchange exhibited strong temporal evolution in the study reach, as indicated by the rainfall event-driven hyporheic exchange. The reversal of the hydraulic gradient and transient hyporheic exchange were simulated using numerical simulation. Anisotropic hydraulic conductivity is the key to generating transient hyporheic exchange. A revised conceptual model was used to interpret the observed temporal patterns in hyporheic exchange. The seasonal rainfall events generate transient hyporheic exchange, and the pattern of transient hyporheic exchange indicates that transient hyporheic exchange appears only after an increased phase of the river stage but does not last long. The temporal pattern of hyporheic exchange can significantly affect the hydrodynamic exchange and the evolution of hydrology in the hyporheic zone for a gaining stream, and these results have important guiding significance for the comprehensive management of surface water and groundwater quantity and quality.

Clustering of Janus particles in an optical potential driven by hydrodynamic fluxes.

Self-organisation is driven by the interactions between the individual components of a system mediated by the environment, and is one of the most important strategies used by many biological systems to develop complex and functional structures. Furthermore, biologically-inspired self-organisation offers opportunities to develop the next generation of materials and devices for electronics, photonics and nanotechnology. In this work, we demonstrate experimentally that a system of Janus particles (silica microspheres half-coated with gold) aggregates into clusters in the presence of a Gaussian optical potential and disaggregates when the optical potential is switched off. We show that the underlying mechanism is the existence of a hydrodynamic flow induced by a temperature gradient generated by the light absorption at the metallic patches on the Janus particles. We also perform simulations, which agree well with the experiments and whose results permit us to clarify the underlying mechanism. The possibility of hydrodynamic-flux-induced reversible clustering may have applications in the fields of drug delivery, cargo transport, bioremediation and biopatterning.

River Flow Dynamics with Two-dimensional Shallow-water Equations

This study proposes a numerical model to simulate unsteady flows in rivers and associated riverine processes. The mathematical model is based on Saint-Venant equations. The governing equations are descritized by Godunov-type finite volume method. Use of total variation diminishing (TVD) scheme and MUSCL-Hancock technique satisfies second-order temporal and spatial accuracy requirements, respectively. Hydrodynamic flux components associated are computed by Harten-Lax-van Leer (HLL) method. A modified method is introduced for approximating flow features along the shear wave. The proposed scheme is used to model an idealized dam-break case, and a case of dam-break in a channel with a 90° bend which results in bore formation and complex two-dimensional flow; the predictions show excellent agreement with available experimental results and analytical solutions.

Steel Flow into a Mold from a Submerged Nozzle with Eccentric Outputs

The flow of steel melt into a mold has not been adequately studied. In general, analysis of the melt flow is a complex mathematical problem, and accordingly numerical modeling is employed. The present work employs Odinokov’s numerical method, which is based on a finite-difference form of the initial system of equations. This method has been successfully employed in continuum mechanics; in casting to determine the stress–strain state of shell-type molds; and in solving other technological problems. That suggests its universality. In the present work, it is applied to the hydrodynamic fluxes of liquid metal when steel is cast in a mold of rectangular cross section. The use of a submerged nozzle with eccentric holes for steel supply requires a three-dimensional mathematical model describing the metal fluxes into the mold. Odyssey software is used to simulate the processes in the mold. The calculation is based on the fundamental hydrodynamic equations and the proposed numerical model. The solution is obtained numerically and takes the form of a system of differential equations. The region of interest is divided into finite elements, and the system of equations is written in difference form for each element. The result obtained is the field of metal flow velocities into the mold. A numerical approach and a corresponding algorithm are developed for solution of the system of algebraic equations obtained and are incorporated in a computation program written in Fortran-4. By means of the mathematical model, the geometric dimensions of the mold and the cross section of the exit holes in the submerged nozzle may be varied. The model clarifies the pattern of metal flows, which affects the heat transfer by the mold walls, and permits determination of the optimal parameters of metal exit from the submerged nozzle in different casting conditions. As an example, the model is applied to steel casting in a mold of rectangular cross section (height 100 cm, horizontal dimensions 2000 × 40 cm). Steel flow from the submerged nozzle is eccentric in two directions within the horizontal plane. The results of solution are presented in graphic form. The pattern of metal fluxes into the mold is shown, and the magnitude and intensity of the fluxes is determined.

Preparation and characterization of water‐soluble polymers and their utilization in chromium sorption

ABSTRACTIn this article, we study the sorption of chromium from aqueous solutions using water‐soluble polymers (WSPs): poly[2‐(acryloyloxy) ethyl] trimethylammonium chloride, P(ClAETA); poly[2‐(methacryloyloxy) ethyl] trimethylammonium methyl sulfate, P(SAETA); and poly(sodium 4‐styrenesulfonate), P(NaSS). These WSPs were obtained by radical polymerization and purified by fractionation through ultrafiltration membranes with different molar mass cut‐offs (30 and 100 kDa). The characterization was carried out by thermogravimetric analysis (TGA), FTIR, and 1H‐NMR spectroscopies and scanning electron microscopy/energy dispersive X‐ray spectroscopy. The chromium retention properties of the polymers were determined in terms of pH, optimal polymer concentration, and the effect of interfering ions. The results show yields above 80% for all of the synthesized WSPs. Characterization by spectroscopy confirmed the chemical structure of the polymers. TGA shows thermal decomposition temperatures of 264 and 324 °C for P(ClAETA) and P(SAETA), respectively. In the case of P(NaSS), the first thermal decomposition begins at approximately 450 °C. Maximum retention of Cr(VI) (100%) by the polymers P(ClAETA) and P(SAETA) was achieved at pH 9, and the maximum retention of Cr(III) (100%) was achieved by P(NaSS) at pH 3. The optimal polymer:Cr molar ratio obtained was 20:1. The retention of chromium(VI) was decreased by the presence of interfering ions, and the hydrodynamic flux was almost constant during ultrafiltration. © 2017 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2017, 134, 45355.