Flux Equations Research Articles

Multiple-beam output, high-peak power passively Q-switched Nd:YAG/Cr4+:YAG laser for ignition application - Modeling of laser pulse characteristics and thermal behavior

In this work we report on the characteristics of a passively Q-switched Nd:YAG/Cr4+:YAG laser with four beams, developed for studies on the laser ignition of different combustible mixtures. The compact laser contains a Nd:YAG/Cr4+:YAG ceramic composite medium, with a monolithic optical resonator. A configuration in which each beam contains laser pulses with an energy of 3.1 mJ and a duration of 0.9 ns (i.e. with a peak power of 3.4 MW) is discussed; operation can be performed in pulse-burst mode, with up to five pulses per pumping pulse. The characteristics of the laser pulses are described by a model based on the rate equations for the photon flux and population inversions in the Nd:YAG gain medium and in the Cr4+:YAG medium with saturable absorption. In addition, the thermal effects are simulated for the operation up to 100 Hz repetition rate, considering the Nd:YAG/Cr4+:YAG laser medium with two geometries, the first of the rod type and the second of the annular type (i.e. cylinder with a central hole) for a good management of thermal effects. This work is an important advancement towards developing a compact laser device with multiple beams, featuring laser pulses and structural properties suitable for igniting combustible mixtures in real engines.

Fluxes of carbon dioxide gases (CO2) in the mangrove soil of Passo Village, Ambon City

Mangrove ecosystems play a significant role in carbon absorption. However, the accumulation of organic matter in mangrove sediments undergoes decomposition, which triggers the release of CO2 gas flux. This study aimed to analyze the CO2 gas flux in mangrove sediments of Negeri Passo, Ambon City. Gas data collection was performed using cylinder canopies at the three observation stations. Gas was passed through the syringe five times with an interval of 30s. Gas concentration analysis was carried out using the gas chromatography method, while CO2 flux was analyzed with flux equations referring to slope regression, volume and area of the scope, temperature, molecular weight of the gas, ideal gas settings, and time constants based on gas intake intervals. The results showed that the average CO2 concentration in St. 1 was 465.14 ± 96.52 ppm, and was the lowest compared to St. 2 and St. 3 with values of 638.60 ± 90.05 ppm and 630.98 ± 54.09 ppm, respectively. Meanwhile, the average CO2 flux was 50.44 mg/m2/hour. The largest CO2 gas flux was observed at St. 2 at 103.69 mg/m2/hour. Meanwhile, the lowest flux was found at St. 3, which was 16.24 mg/m2/hour. Based on this, it can be concluded that] the mangrove ecosystem in Negeri Passo has a higher concentration of CO2 gas than the average concentration of climate change stabilization scenarios. However, the CO2 flux was lower than that at other locations in the Ambon Dalam Bay area. In addition, the potential for significant carbon sequestration based on the Tier 1 model approach indicated that mangrove ecosystems in this location play an important role in climate change mitigation.

The Effect of Grain Boundary Misorientation on Hydrogen Flux Using a Phase‐Field‐Based Diffusion and Trapping Model

Understanding hydrogen–grain boundary (GB) interactions is critical to the analysis of hydrogen embrittlement in metals. This work presents a mesoscale fully kinetic model to investigate the effect of GB misorientation on hydrogen diffusion and trapping using phase‐field‐based representative volume elements (RVEs). The flux equation consists of three terms: a diffusive term and two terms for high and low angle grain boundary (H/LAGB) trapping. Uptake simulations show that decreasing the grain size results in higher hydrogen content due to increasing the GB density. Permeation simulations show that GBs are high‐flux paths due to their higher enrichment with hydrogen. Since HAGBs have higher enrichment than LAGBs, due to their higher trap‐binding energy, they generally have the highest hydrogen flux. Nevertheless, the flux shows a convoluted behavior as it depends on the local concentration, alignment of GB with external concentration gradient as well as the GB network connectivity. Finally, decreasing the grain size resulted in a larger break‐through time and a larger steady‐state exit flux.

Validation of a Central-Difference Finite Volume Method for Analysing Wall Turbulent Heat Transfer Using Transport Equations for Turbulent Heat Fluxes

Abstract The aim of this study is to validate the finite volume method approach using the transport equations for turbulent heat fluxes in turbulent heat transfer at the wall surface. The finite volume method technique is a numerical analysis based on conservation laws of physical quantities in the flow field, with particular emphasis on the conservation of velocity fluctuation intensity in the flow field. The behaviour of turbulent heat flows is analysed in detail using this method and its validity is tested. In the study, heat transfer in channel turbulence is analysed under a constant heat flux heating condition. The individual terms of the temperature variance equation are analysed and the results are shown to be in agreement with a previous study. The residual is small, confirming the high accuracy of the results. The present study analyses the transport equation for the dissipation of the temperature variance. The transport equations for turbulent heat fluxes in the flow direction and in the vertical wall direction are analysed and show good agreement with those of the previous study. The results show that in the vertical wall direction, non-zero terms such as the pressure diffusion term appear to contribute and reduce the accuracy of the transport equations.

Extracting Thermodynamic, Kinetic, and Transport Properties from Batteries Using a Simple Analytical Pulsing Protocol

Gaining insights into the fundamental properties of lithium-ion batteries through scalable and non-destructive methods is challenging for commercial cell formats. In this work, a simple analytical pulsing protocol (APP) is performed on a commercial cell to understand its thermodynamic, kinetic, and mass transport properties. While testing procedures that rely on electrochemical pulses are well documented, the APP is novel in the level of fundamental insight that can be gained. For thermodynamics, a static-differential capacity analysis can be performed that removes the effects of kinetic and transport overpotentials and allows for the calculation of Gibbs free energy. For kinetics, the exchange current density of the cell can be calculated according to the Butler-Volmer model. For transport, a whole-cell lithium-ion diffusion coefficient can be calculated from a derivation of Fick’s second law and the generalized flux equation. Comparing the results from these properties gives an unparalleled level of mechanistic insight into battery performance from a single non-destructive technique. This APP requires no additional equipment and provides properties that can be easily correlated to materials or processing parameters. Therefore, the APP is valuable for research and development, manufacturing, quality assurance, and second-life applications, among others.

Study of salt free-diffusion by 1D transport numerical simulations and shadowgraph experiments

Applying the Nernst-Planck formalism of 1D diffusive flux equations for ionic species of salts dissolved in water, Pitzer formalism for activity coefficients and PHREEQC software for species diffusion coefficients, with ionic strength-dependence coefficients optimized, we obtained expressions for apparent diffusion coefficients of sodium chloride, calcium chloride, and sodium sulphate, prevalent in deep aquifers. PHREEQC 1D transport simulations of free-diffusion experiments yielded temporal evolution of the concentration and gradient vertical profiles. The dynamics of concentrations non-equilibrium fluctuations during the free-diffusion experiments have been recorded by shadowgraphy. Thanks to the calculated gradient profiles the thickness of the sample has been integrated in the analysis equations of the structure functions. From the earliest stages of the diffusion process we measured the diffusion coefficients for the three salts, in very good agreement with reference measurements, validating the technique and method of analysis for studying free-diffusion in reservoirs targeted for CO2 and energy vectors storage.

1,3-Dioxolane production via reactive distillation combined with vapor permeation: Experiments and modelling

To overcome the problem of difficult separation of the 1,3-dioxane (DOL)–H2O azeotrope in the production of DOL, a new process via reactive distillation (RD) combined with vapor permeation (VP) was established. The reliability of the RD model has been reported in the literature. The VP membrane used an Aspen Custom Modeler (ACM) self-built model, with the permeation parameters of H2O, DOL, and formaldehyde (FA) in the membrane flux equation fitted through the H2O-DOL-FA dehydration experiments by NaA zeolite membrane. The results showed that the relative error between the mass of H2O in a series of permeates in experiments and simulated by the VP model was mainly within 20%, and R2 is 0.94. For DOL and FA, which have very small permeation, the value was within 50%, implying the reliability of the VP model. On this basis, a two-dimensional VP model was established. Finally, a new RD-VP-D process for producing DOL was designed. The VP was used to overcome the azeotropic point of DOL and H2O by dehydration. Then 99.9 wt% DOL can be obtained at the bottom of the subsequent DOL purification column, with DOL- H2O azeotrope at the top recycled back to the VP membrane. The effects of the degree of VP dehydration and column pressure on membrane area, reboiler duty, and TAC were investigated, meanwhile introducing a new concept: logarithmic mean partial-pressure difference (LMPD). Under the optimal parameter, TAC was reduced by 20.0% compared with the process reported in the literature, and CO2 emissions were reduced by 35.8%.

Diffusion in multicomponent mixed solvent electrolyte systems

Thermodynamic ideality is often assumed in diffusion calculation, implying concentrations are assumed equal to the activities with activity coefficients of one. A general set of diffusion flux and molar flux equations are developed, incorporating equilibrium thermodynamic models for non-ideal conditions in well-known non-equilibrium diffusion theory. Derivations are based on the Maxwell-Stefan friction theory with the chemical potential as the driving force. Isothermal and isobaric conditions are assumed, and the theory does not include the Soret coefficient or the Onsager formalism.In this work, Fick's law is derived from the general diffusion theory, also as a function of activity gradients. The results illustrate that concentration cannot just be substituted by activity. Using a similar approach demonstrates that the Nernst-Planck (NP) equation may be derived from the general theory incorporating terms for migration and convection. An extended NP equation is derived including the thermodynamic correction factors (TCF) to account for the non-ideality of the liquid. The derived equation shows the improvement of the self-diffusion terms and friction coefficients in the NP equation and that the TCF enhances the observed effective diffusivities.Examples of TCF are presented using the extended UNIQUAC thermodynamic model. This is done for the CO2-NaOH-monoethyleneglycol-water system, which is typically found in wet natural gas pipeline CO2 corrosion systems. pH-stabilization is classically used in the prevention of corrosion, but the results show that glycol has a noticeable effect of the diffusion on key components in the corrosion mechanism. A TCF of 0.4 to 1.3 may be expected. This indicates that the flux calculation deviates up to 60 % from the real-life scenario if ideality is assumed. The thermodynamic model may therefore improve the diffusion calculations considerably.

Estimation of mixed-gas selectivity of membranes using the linear friction-based model

The development of analytical models for predicting the transport properties of membranes in the separation of multicomponent mixtures is highly relevant because (1) measurements of permeability and sorption of gas mixtures in membrane materials are laborious and time-consuming, and (2) estimation of transport properties based on pure gas measurements is desirable, however, rather unreliable and can lead to serious errors. In this paper, within the framework of the well-known frictional-coefficient formalism, we present nonlinear equations for the fluxes of gas mixture components through the membrane, which take into account the diffusional coupling of the fluxes compared to independent gas transport. As a result, an analytical equation for the mixed-gas selectivity was obtained. This equation has an implicit form, which is not quite convenient for practical application, since it has to be solved numerically. A simple explicit expression for mixed gas selectivity was derived using the so-called linear transport model, which is a linearized version of the original nonlinear flux equations. It is important to emphasize that a comparison of the linear model with the formally more rigorous nonlinear model showed only minor quantitative differences. The input parameters for the model are the single gas transport parameters. Testing the linear model using experimental data for the separation of hydrocarbons (n-butane/methane, n-butane/i-butane, propylene/propane) revealed an acceptable fit. A comparison of the model with recent experimental data on the separation of CO2-containing mixtures using microporous polymers is also presented. It is found that the calculated CO2/N2 mixed gas selectivity is in good agreement with the experimental data, while only a semi-quantitative agreement is observed for the CO2/CH4 separation. Besides, it is shown that the model developed for binary gas separation can be extended to ternary gas mixtures.

Technical Feasibility of Extraction of Freshwater from Produced Water with Combined Forward Osmosis and Nanofiltration.

This study assesses the technical feasibility of a forward-osmosis-based system for concentrating produced water and extracting freshwater. Forward osmosis was combined with nanofiltration, the latter system used to restore the initial osmotic pressure of the diluted draw solutions while concurrently obtaining the final freshwater product. Three draw solutions, namely, MgCl2, NaCl, and C3H5NaO2, were initially tested against a synthetic water mimicking a pretreated produced water effluent having an osmotic pressure equal to 16.3 bar. MgCl2 was thus selected for high-recovery experiments. Different combinations of draw solution osmotic pressure (30, 40, 60, 80, and 120) and draw-to-feed initial volume ratios (1, 1.6, and 2.2) were tested at the laboratory scale, achieving recovery rates between roughly 35% and 70% and water fluxes between 4 and 8 L m-2h-1. One-dimensional, system-wide simulations deploying the analytical FO water flux equation were utilized to validate the experiments, investigate co-current and counter-current configurations, and understand the system potential. The diluted draw solutions were then transferred to nanofiltration to regenerate their original osmotic pressure. There, the highest observed rejection was 96.6% with an average flux of 21 L m-2h-1, when running the system to achieve 100% relative recovery.

The influence of thermal diffusion on water migration through a porous insulation material

Excess water on pipes and equipment under porous insulation materials can lead to undesired corrosion. This work aims to clarify to what extent thermal diffusion affects the migration of water inside insulation materials subject to large temperature gradients. Since no experimental data is available on the thermal diffusion coefficients of humid air, revised Enskog theory for Mie fluids is used to estimate transport properties. Comparison to experimental data from literature shows that the theory reproduces the diffusion coefficient, viscosity and thermal conductivity of humid air within 8.7%, 5.0% and 3.5% respectively. The small discrepancies suggest that the theory can also provide reliable estimates of the thermal diffusion coefficients. In the investigated composition and temperature range, the theory predicts the Soret coefficient of water to be approximately ▪, while the Soret coefficient of oxygen varies from ▪ to +▪. A case study with heating of glass wool insulation containing humid air, encapsulating a cylindrical pipe is investigated. Non-equilibrium thermodynamics is used to consistently incorporate the Soret coefficients into the flux equations in a dynamic, non-isothermal model that includes diffusion, convection, thermal conduction and water sorption in the porous medium. With 50 K temperature difference across 5 cm of insulation, we find that at steady-state, thermal diffusion leads to a mole fraction of water in the gas phase that is about 1.5% higher at the hot location than if thermal diffusion is neglected.

Fourth-order compact block-centered splitting domain decomposition method for parabolic equations

It is of importance to develop efficient high-order solution-flux domain decomposition methods for solving the large scale time-dependent partial differential equations. In this paper, a fourth-order compact block-centered splitting domain decomposition method is developed for solving parabolic partial differential equations in large domains. Combining the time second-order splitting technique with non-overlapping domain decompositions, we propose the compact block-centered splitting domain decomposition algorithm to the solution-flux structured system of parabolic partial differential equations. The domain is divided into non-overlapping multi-block sub-domains of block-centered meshes. On the interfaces of sub-domains, the interface fluxes are predicted by the semi-implicit interface flux scheme while the solution and flux in the interiors of sub-domains are computed by the compact block-centered implicit solution-flux scheme. The feature of the method is that over block-centered grids, constructing high-order compact difference scheme to the structured system of solution and flux equations leads to efficient schemes for solving the problems over domain decompositions. Numerical experiments show that the method is of fourth-order convergence in space and of second order convergence in time and it is much efficient in parallel computing.

Accelerated particle beams in a 3D simulation of the quiet Sun

Context. Charged particles are constantly accelerated to non-thermal energies by the reconnecting magnetic field in the solar atmosphere. Our understanding of the interactions between the accelerated particles and their environment can benefit considerably from three-dimensional atmospheric simulations that account for non-thermal particle beam generation and propagation. In a previous publication, we presented the first results from such a simulation, which considers quiet Sun conditions. However, the original treatment of beam propagation ignores potentially important phenomena such as the magnetic gradient forces associated with a converging or diverging magnetic field. Aims. Here we present a more general beam propagation model incorporating magnetic gradient forces, the return current, acceleration by the ambient electric field, corrected collision rates due to the ambient temperature, and collisions with heavier elements than hydrogen and the free electrons they contribute. Neglecting collisional velocity randomisation makes the model sufficiently lightweight to simulate millions of beams. We investigate how each new physical effect in the model changes the non-thermal energy transport in a realistic three-dimensional atmosphere. Methods. We applied the method of characteristics to the steady-state continuity equation for electron flux to derive ordinary differential equations for the mean evolution of energy, pitch angle, and flux with distance. For each beam, we solved these numerically for a range of initial energies to obtain the evolving flux spectrum, from which we computed the energy deposited into the ambient plasma. Results. Magnetic gradient forces significantly influence the spatial distribution of deposited beam energy. The magnetic field converges strongly with depth in the corona above loop footpoints. This convergence leads to a small coronal peak in deposited energy followed by a heavy dip caused by the onset of magnetic mirroring. Magnetically reflected electrons carry away 5 to 10% of the injected beam energy on average. The remaining electrons are relatively energetic and produce a peak in deposited energy below the transition region a few hundred kilometres deeper than they would in a uniform magnetic field. A diverging magnetic field at the beginning of the trajectory, which is common in the simulation, enhances the subsequent impact of magnetic mirroring. The other new physical effects do not qualitatively alter the picture of non-thermal energy transport for the atmospheric conditions under consideration.

Identifying Challenges to 3D Hydrodynamic Modeling for a Small, Stratified Tropical Lake in the Philippines

Three-dimensional hydrodynamic modeling for small, stratified tropical lakes in the Philippines and in Southeast Asia in general is not deeply explored. This study pioneers investigating the hydrodynamics of a small crater lake in the Philippines with a focus on temperature simulation using a Fantom Refined 3D model that has been tested mostly for temperate and sub-tropical lakes. The lake’s monthly temperature during the dry season served as a reference for the model’s initial condition and validation. For the simulation to proceed, input data such as weather, inflow, and bathymetry were prepared. In the absence of hourly meteorological data from local weather stations, this paper adopted the satellite weather data from Solcast. Simple correlation analysis of daily weather data between local stations and Solcast showed valid and acceptable results. Inflow values were estimated using the rational method while the stream temperature was estimated from a regression equation using air temperatures as input. The validated satellite-derived data and runoff model can therefore be employed for 3D modeling. The simulations resulted in extremely higher temperatures compared with those observed when using previous default model settings. Direct modifications were then applied to weather parameters, compromising their integrity but resulting in reasonable profiles. By adding scaling factors to heat flux equations and multiplying their components by 0.75 (shortwave), 1.35 (longwave), 0.935 (air temperature), and 0.80 (wind), better results were achieved. This study identifies several challenges in performing 3D hydrodynamic modeling, such as paucity in input hydro-meteorologic and limnologic data and the need for heat flux model improvement. Overall, this study was successful in employing 3D hydrodynamic modeling in a tropical lake, which can pave directions and serve as an excellent reference for future modeling in the same region.

Mancha3D Code: Multipurpose Advanced Nonideal MHD Code for High-Resolution Simulations in Astrophysics

The Mancha3D code is a versatile tool for numerical simulations of magnetohydrodynamic (MHD) processes in solar/stellar atmospheres. The code includes nonideal physics derived from plasma partial ionization, a realistic equation of state and radiative transfer, which allows performing high-quality realistic simulations of magnetoconvection, as well as idealized simulations of particular processes, such as wave propagation, instabilities or energetic events. The paper summarizes the equations and methods used in the Mancha3D (Multifluid (-purpose -physics -dimensional) Advanced Non-ideal MHD Code for High resolution simulations in Astrophysics 3D) code. It also describes its numerical stability and parallel performance and efficiency. The code is based on a finite difference discretization and a memory-saving Runge–Kutta (RK) scheme. It handles nonideal effects through super-time-stepping and Hall diffusion schemes, and takes into account thermal conduction by solving an additional hyperbolic equation for the heat flux. The code is easily configurable to perform different kinds of simulations. Several examples of the code usage are given. It is demonstrated that splitting variables into equilibrium and perturbation parts is essential for simulations of wave propagation in a static background. A perfectly matched layer (PML) boundary condition built into the code greatly facilitates a nonreflective open boundary implementation. Spatial filtering is an important numerical remedy to eliminate grid-size perturbations enhancing the code stability. Parallel performance analysis reveals that the code is strongly memory bound, which is a natural consequence of the numerical techniques used, such as split variables and PML boundary conditions. Both strong and weak scalings show adequate performance up to several thousands of processors (CPUs).

Estimate of groundwater flow and salinity contribution to the Great Salt Lake using groundwater levels and spatial analysis

Groundwater discharge to Great Salt Lake (GSL) is difficult to quantify but represents a potentially significant source of water and salinity to the lake’s overall water budget and chemistry, respectively. Understanding groundwater and its role in the overall health of GSL is critical due to the current and historically low lake levels. We compiled existing groundwater level data in basin-fill wells around GSL and used spatial analysis methods to 1) create potentiometric-surface maps in the areas adjoining GSL, 2) calculate groundwater contributions to GSL, and 3) estimate salinity inputs from groundwater to GSL. We observed groundwater-level declines in most of the basin-fill wells from the 1980s to 2010s. These declines are consistent with historical groundwater-level trends in the Salt Lake, Tooele, Curlew, and Weber Valleys and are a consequence of aquifer overdraft associated with less than average precipitation in the basin and increased groundwater withdrawals in the GSL watershed. Using the Darcy flux equation, we calculated a groundwater flux to GSL of 313,500 acre-feet per year, substantially greater than previous estimates derived from water balance studies but consistent with estimates derived from geochemical modeling of GSL water chemistry. We calculated a salt contribution from groundwater to GSL of 1.18 million metric tons per year, which represents about 10% of the solutes derived from surface flows to GSL in 2013.

Efficient modeling and simulation of gas separations applying Maxwell-Stefan approach and Ideal Adsorbed Solution Theory

We present an efficient way to incorporate the Maxwell-Stefan approach (Krishna, 1990) for modeling gas separation processes, i.e., adsorbent particles and membranes. Ideal Adsorbed Solution Theory (Myers and Prausnitz, 1965) is applied for the description of competitive equilibria. The proposed calculation method introduces analytical expressions for the partial derivatives of the loadings w.r.t. fluid-phase concentrations, developed by Rubiera Landa et al. (2013), in the calculation of the thermodynamic correction factors, as required by Maxwell-Stefan flux equations. These expressions are solved simultaneously with transient mass balances that describe the gas separation processes in a differential-algebraic equation formulation. Robustness and advantages of the proposed solution approach are demonstrated with several examples adapted from the literature.

Thermal convection with a Cattaneo heat flux model

The problem of thermal convection in a layer of viscous incompressible fluid is analysed. The heat flux law is taken to be one of Cattaneo type. The time derivative of the heat flux is allowed to be a material derivative, or a general objective derivative. It is shown that only one objective derivative leads to results consistent with what one expects in real life. This objective derivative leads to a Cattaneo–Christov theory, and the results for linear instability theory are in agreement with those for a material derivative. It is further shown that none of the theories allow a standard nonlinear, energy stability analysis. A further heat flux due to P.M. Mariano is added and then an analysis is performed for stationary convection, oscillatory convection, and fully nonlinear theory. For the material derivative case, the analysis proceeds and global nonlinear stability is achieved. For Cattaneo–Christov theory, it appears necessary to add a regularization term in the equation for the heat flux, and even then the analysis only works in two space dimensions, and is conditional upon the size of the initial data. For the three-dimensional situation, it is shown how a nonlinear stability analysis may be achieved with a Navier–Stokes–Voigt fluid rather than a Navier–Stokes one.

Symmetry Properties of Models for Reversible and Irreversible Thermodynamic Processes

The problem of formulating variational models for irreversible processes of media deformation is considered in this paper. For reversible processes, the introduction of variational models actually comes down to defining functionals with a given list of arguments of various tensor dimensions. For irreversible processes, an algorithm based on the principle of stationarity of the functional is incorrect. In this paper, to formulate a variational model of irreversible deformation processes with an expanded range of coupled effects, an approach is developed based on the idea of the introduction of the non-integrable variational forms that clearly separate dissipative processes from reversible deformation processes. The fundamental nature of the properties of symmetry and anti-symmetry of tensors of physical properties in relation to multi-indices characterizing independent arguments of bilinear forms in the variational formulation of models of thermomechanical processes has been established. For reversible processes, physical property tensors must necessarily be symmetric with respect to multi-indices. On the contrary, for irreversible thermomechanical processes, the tensors of physical properties that determine non-integrable variational forms must be antisymmetric with respect to the permutation of multi-indices. As a result, an algorithm for obtaining variational models of dissipative irreversible processes is proposed. This algorithm is based on determining the required number of dissipative channels and adding them to the known model of a reversible process. Dissipation channels are introduced as non-integrable variational forms that are linear in the variations of the arguments. The hydrodynamic models of Darcy, Navier–Stokes, and Brinkman are considered, each of which is determined by a different set of dissipation channels. As another example, a variational model of heat transfer processes is presented. The equations of heat conduction laws are obtained as compatibility equations by excluding the introduced thermal potential from the constitutive equations for temperature and heat flux. The Fourier and Maxwell–Cattaneo equations and the generalized heat conduction laws of Gaer–Krumhansl and Jeffrey are formulated.

Coupled continuum modelling of size segregation driven by shear-strain-rate gradients and flow in dense, bidisperse granular media

Dense granular systems that consist of particles of disparate sizes segregate based on size during flow, resulting in complex, coupled segregation and flow patterns. The ability to predict how granular mixtures segregate is important in the design of industrial processes and the understanding of geophysical phenomena. The two primary drivers of size segregation are pressure gradients and shear-strain-rate gradients. In this work, we isolate size segregation driven by shear-strain-rate gradients by studying two dense granular flow geometries with constant pressure fields: gravity-driven flow down a long vertical chute with rough parallel walls and annular shear flow with rough inner and outer walls. We perform discrete element method (DEM) simulations of dense flow of bidisperse granular systems in both flow geometries, while varying system parameters, such as the flow rate, flow configuration size, fraction of large/small grains and grain-size ratio, and use DEM data to inform continuum constitutive equations for the relative flux of large and small particles. When the resulting continuum model for the dynamics of size segregation is coupled with the non-local granular fluidity model – a non-local continuum model for dense granular flow rheology – we show that both flow fields and segregation dynamics may be simultaneously captured using this coupled, continuum system of equations.