Averaged Transport Equations Research Articles

A theoretical Fourier-transformation model for the formation of X-ray time lags from black hole accretion discs

ABSTRACT X-ray emission from active galactic nuclei (AGNs) often displays complex and rapid variability, which may provide a glimpse into the detailed thermal and dynamical structure of the accreting gas near the event horizon of the central black hole. The observed variability can be analysed using Fourier transforms of the light curves in multiple energy channels, which can be used to generate Fourier phase lags, corresponding to lags in the time domain. The X-ray time lags may be either soft lags or hard lags, depending on whether the variability in the hard energy channel precedes that in the soft channel or vice versa. The physical explanation for the observed X-ray time lags from AGNs has been puzzling, and several scenarios have been proposed. In this paper, we explore the hypothesis that the X-ray time lags are produced as a result of the reprocessing of iron L-line and K-line seed photons generated via fluorescence, which is driven by a variable incident radiation field. The seed photons are reprocessed by a combination of thermal and bulk Comptonization and spatial reverberation. We assume that the inner region of the accretion flow can be approximated as a hot, geometrically thick ADAF disc. The outer radius of the ADAF region is equal to the shock formation radius, which is located just outside the centrifugal barrier. The time-dependent radiative transfer in the disc is analysed using a Fourier-transformed, vertically averaged transport equation in cylindrical coordinates. We demonstrate that the new model can successfully reproduce the complex X-ray variability data for the Seyfert 1 galaxies 1H 0707–495 and Ark 564.

A modified Darcy’s law for viscoelastic flows of highly dilute polymer solutions through porous media

Viscoelastic flows of polymer solutions in complex geometries can generate a strong localization of stress within small regions of the fluid and the formation of birefringent strands. In porous media , these localized structures of stress drive preferential flow paths and increase global dissipation. Modeling the impact of such effects at Darcy or larger scales is a daunting task—one of the reasons being the lack of approaches using homogenization theories to help figure out both the correct form of the averaged transport equations and the relevant set of effective parameters. Here we homogenize the incompressible Oldroyd-B equations at zero Reynolds number to obtain a Darcy scale model that captures the effect of localized polymeric stress. This model consists of an advection—reaction transport equation for the average conformation tensor along with a form of Darcy’s law that contains an additional forcing term associated with structures of localized stress. The derivation is based upon a limit of high dilution, a regime where the Oldroyd-B model can be transformed into a sequence of linear problems using asymptotic developments. We validate our approach in test cases corresponding to flows in a channel and through arrays of circles. Besides providing a new model for viscoelastic flows in porous media , our work also shows that modeling viscoelastic flows through porous media is not simply a matter of determining an apparent permeability tensor—the homogenized model cannot be easily reduced to a simple form of Darcy’s law—but rather requires the development of specific homogenized models that capture the coupling between the transport of the polymeric stress and momentum. • The Oldroyd-B equations are homogenized using asymptotics and volume averaging. • We derive a Darcy-scale model for viscoelastic flow in porous media. • Our model couples a form of Darcy’s law with transport of the conformation tensor. • Our model captures the effect of stress localization at pore-scale on the average flow. • The notion of effective permeability alone seems to be insufficient for viscoelastic flows.

Paul Andrews Libby Contribution to Zone-Conditioned Average Transport Equations in Intermittent Turbulent Flows

ABSTRACT Professor Paul A. Libby has made lasting contributions to the theory and modeling of turbulent premixed flames. A less known subject of his research relates to the conditioning of the local and instantaneous Navier-Stokes equations for either the turbulent or the irrotational flow regions separated by a thin interface. He postulated an evolution equation for the intermittency function, a generalized function introduced by experimentalists in the decades of 1960 and 1970, which is unity in the vorticity-containing turbulent flow and zero in the surrounding irrotational flow. This intermittency function equation allowed the derivation of transport equations for average variables conditioned either to the turbulent or the vorticity-free regions. Paul A. Libby formalism is explained in this paper. Further research triggered by Libby’s seminal work is then reported. An exact expression is obtained for his “intermittency creation” term, which allows a simple physical interpretation of the entrainment and zone interactions. This methodology is applied to an intermittent scalar in a turbulent flow of a variable density fluid. Various types of intermittent flows with zonal discrimination, based on vorticity, scalar and scalar-gradient criteria, are presented. The analogy of turbulent/nonturbulent interfaces with propagating turbulent premixed flames is remarked.

Charge transport equation for bidisperse collisional granular flows with non-equipartitioned fluctuating kinetic energy

Starting from the Boltzmann–Enskog kinetic equations, the charge transport equation for bidisperse granular flows with contact electrification is derived with separate mean velocities, total kinetic energies, charges and charge variances for each solid phase. To close locally averaged transport equations, a Maxwellian distribution is presumed for both particle velocity and charge. The hydrodynamic equations for bidisperse solid mixtures are first revisited and the resulting model consisting of the transport equations of mass, momentum, total kinetic energy, which is the sum of the granular temperature and the trace of fluctuating kinetic tensor, and charge is then presented. The charge transfer between phases and the charge build-up within a phase are modelled with local charge and effective work function differences between phases and the local electric field. The revisited hydrodynamic equations and the derived charge transport equation with constitutive relations are assessed through hard-sphere simulations of three-dimensional spatially homogeneous, quasi-one-dimensional spatially inhomogeneous bidisperse granular gases and a three-dimensional segregating bidisperse granular flow with conducting walls.

DREAM: A fluid-kinetic framework for tokamak disruption runaway electron simulations

Avoidance of the harmful effects of runaway electrons (REs) in plasma-terminating disruptions is pivotal in the design of safety systems for magnetic fusion devices. Here, we describe a computationally efficient numerical tool, that allows for self-consistent simulations of plasma cooling and associated RE dynamics during disruptions. It solves flux-surface averaged transport equations for the plasma density, temperature and poloidal flux, using a bounce-averaged kinetic equation to self-consistently provide the electron current, heat, density and RE evolution, as well as the electron distribution function. As an example, we consider disruption scenarios with material injection and compare the electron dynamics resolved with different levels of complexity, from fully kinetic to fluid modes. Program summaryProgram Title:DreamCPC Library link to program files:https://doi.org/10.17632/vs3yvnrzg6.1Developer's repository link:https://github.com/chalmersplasmatheory/DREAMLicensing provisions: MITProgramming language: C++, PythonNature of problem: Self-consistently simulates the plasma evolution in a tokamak disruption, with specific emphasis on runaway electron dynamics. The runaway electrons can be simulated either as a fluid, fully kinetically, or as a mix of the two. Plasma temperature, current density, electric field, ion density and charge states are all evolved self-consistently, where kinetic non-thermal contributions are captured using an orbit-averaged relativistic electron Fokker-Planck equation, which couples to the plasma evolution. In the typical use case, the electrons are represented by two distinct populations: a cold fluid population and a kinetic superthermal population.Solution method: The system of equations is solved using a standard multidimensional Newton's method. Partial differential equations—most prominently the bounce-averaged Fokker–Planck and current diffusion equations—are discretized using a high-resolution finite volume scheme that preserves density and positivity.

Multiscale model for flow and transport in CO2-enhanced coalbed methane recovery incorporating gas mixture adsorption effects

In this work we develop a multiscale model for flow and transport problem in CO2-enhanced coalbed methane recovery. The coalbed methane reservoir is characterized by two levels of porosity associated with nanopores in the matrix and cleat network. Mass conservation equations for fluid mixture (CH4 and CO2) in the matrix at the microscale are rigorously derived by using the formal homogenization technique taking into account the gas mixture adsorption in the nanopores. The Density Functional Theory (DFT) is used to compute the gas adsorption isotherms and the solvation force acting on the nanopore wall, showing a much more pronounced adsorption capacity of CO2 compared to CH4. The average transport equations in the matrix together with the multiphase flow problem in fracture network (gas mixture and water) are homogenized giving rise to a macroscopic model ruled by the effective conductivities, partition and transfer coefficients. The cleat permeability evolution due to deformation is taken into account through a three-scale poromechanical model reported in a previous work. Computational simulations illustrate the macroscopic laws underlying the gas pressure distributions, cleat closure phenomena and CH4 production curve enhanced by CO2 injection.

Global model for pulsed inductively coupled plasma sources: Effect of edge-to-center density ratio and electron heating

The volume-averaged global plasma model has been widely used to analyze the characteristics of plasma, although the spatial variation of plasma parameters cannot be obtained from it. It has also been used to obtain temporal plasma parameters for pulsed plasma sources. In this work, we analyzed the effect of an edge-to-center density ratio (h factor) and an electron heating model on the plasma parameters in pulsed plasma simulations using the global model for Ar discharges. In most previous pulse simulations using the global model, the h factor has been applied to pulse conditions in the same way it has been applied to radio frequency, and the power absorbed by the electrons was assumed to be equal to the applied power. However, in this work, we considered a time-varying h factor and determined the absorbed power using an analytical electron heating model, solving the spatially averaged transport equations in a self-consistent manner. We found that a decreased h factor increases the plasma density and consequently changes the time dependence of the electron temperature. In addition, the overshoot of the electron temperature is limited at the beginning of the pulse power-on in the self-consistent electron heating model. Our results are of great relevance with respect to the analysis of plasma parameters.

Simulation of ECN diesel spray A using conditional source-term estimation

A novel combustion modelling approach is proposed here to study the transient effects of diesel spray. Conditional Source-term Estimation (CSE) is a combustion model which invokes the Conditional Moment Closure (CMC) hypothesis to provide an approximation of the mean chemical source term in an averaged transport equation. Unlike CMC, where transport equations are solved for conditional moments, CSE recovers these conditional moments through the solution of an inverse problem. Integral equations are inverted for the conditional moments, by assuming spatial homogeneity in the conditional averages where Tikhonov regularisation is applied. Previous CSE studies have shown that the model is able to predict the flame characteristics successfully for both premixed and non-premixed combustion modes. However, most of these investigations were based on methane flames. This study will be the first successful application of CSE to a complex hydrocarbon fuel, n-dodecane, under the Engine Combustion Network's (ECN) ‘Spray A’ conditions. Detailed chemistry is included in tabulated form using the Flamelet Generated Manifold (FGM) methodology. The predictions of this study include both the Favre averaged conditional mass fraction of reactive species and temperature. The results are compared with available experimental data and previous numerical results. Both RANS and LES simulations are performed under the same condition. The objectives of the paper are (i) assessment of the application of CSE on igniting diesel spray (ii) comparison of the CSE numerical results with available experimental results and previous numerical simulations. Overall, the combination of a chemical mechanism that has been tuned to predict ‘Spray A’ conditions with the CSE-FGM model is able to successfully predict autoignition delay time and lift-off length of n-dodecane spray within the scatter of the experimental data. CSE-FGM offers a feasible tool for detailed combustion analysis of diesel spray flames. Both RANS and LES can give reasonably good global predictions of the flame.

Non-local entropy evolution in heat exchangers with elliptical and circular tube geometries

Numerical simulations of the entropy transportation process inside a plate and tube heat exchanger are presented. Entropy levels around tubes with circular and elliptical cross sections are analyzed and compared. The purpose of the comparison is to determine how the entropy flux contributes to increase, or decrease, the value of the entropy in certain regions of the flow field. This is accomplished by numerically integrating the Reynolds averaged transport equations for the mass, momentum, energy and entropy, in a three dimensional domain. A k-ε-2L model is used to account for the fluctuations appearing in the laminar-to-turbulent transitional regime of the flow. The importance of entropy convection, relative to the generation of entropy, is underlined by means of a direct comparison through a series of visualization maps. This approach allows some useful insights about the evolution of the entropy field in complex geometries. For example, in this case certain troublesome regions where the entropy field reaches high values can be identified and, therefore, design recommendations can be stated in order to increase the thermal efficiencies of the devices under consideration.

One- and Two-Equation Models to Simulate Ion Transport in Charged Porous Electrodes

Energy storage in porous capacitor materials, capacitive deionization (CDI) for water desalination, capacitive energy generation, geophysical applications, and removal of heavy ions from wastewater streams are some examples of processes where understanding of ionic transport processes in charged porous media is very important. In this work, one- and two-equation models are derived to simulate ionic transport processes in heterogeneous porous media comprising two different pore sizes. It is based on a theory for capacitive charging by ideally polarizable porous electrodes without Faradaic reactions or specific adsorption of ions. A two-step volume averaging technique is used to derive the averaged transport equations for multi-ionic systems without any further assumptions, such as thin electrical double layers or Donnan equilibrium. A comparison between both models is presented. The effective transport parameters for isotropic porous media are calculated by solving the corresponding closure problems. An approximate analytical procedure is proposed to solve the closure problems. Numerical and theoretical calculations show that the approximate analytical procedure yields adequate solutions. A theoretical analysis shows that the value of interphase pseudo-transport coefficients determines which model to use.

Population balance modeling of polydispersed bubbly flow in continuous casting using average bubble number density approach

A new methodology of average bubble number density (ABND) model for handling the evolution of polydispersed bubbly flow in the slab continuous casting mold is presented and evaluated. The average bubble number density transport equation coupled with the Eulerian-Eulerian two-fluid model is employed to describe size distribution of bubbles. Various interfacial forces including drag force, lift force, virtual mass force, and turbulent dispersion force are incorporated in this model. SST turbulence model is used with extra source terms introduced to account for the interaction between the bubbles and the liquid. The coalescence of bubbles is formulated according to the random collision driven by turbulence and wake entrainment, and the breakage of bubbles is formulated through considering the impact of turbulent eddies. The intermediate peak and core peak behaviors of void fraction inside the submersed entry nozzle (SEN) are captured very well. Comparisons of gas void fraction, liquid flow pattern, and local bubble size distribution profiles with experimental measurements are provided, showing the applicability and accuracy of the ABND approach in modeling the polydispersed bubbly flow in the slab continuous casting mold.

Surface transport processes in charged porous media

Surface transport processes are very important in chemistry, colloidal sciences, engineering, biology, and geophysics. Natural or externally produced charges on surfaces create electrical double layers (EDLs) at the solid-liquid interface. The existence of the EDLs produces several complex processes including bulk and surface transport of ions. In this work, a model is presented to simulate bulk and transport processes in homogeneous porous media comprising big pores. It is based on a theory for capacitive charging by ideally polarizable porous electrodes without Faradaic reactions or specific adsorption of ions. A volume averaging technique is used to derive the averaged transport equations in the limit of thin electrical double layers. Description of the EDL between the electrolyte solution and the charged wall is accomplished using the Gouy-Chapman-Stern (GCS) model. The surface transport terms enter into the average equations due to the use of boundary conditions for diffuse interfaces. Two extra surface transports terms appear in the closed average equations. One is a surface diffusion term equivalent to the transport process in non-charged porous media. The second surface transport term is a migration term unique to charged porous media. The effective bulk and transport parameters for isotropic porous media are calculated solving the corresponding closure problems.

Modeling the Capacitive Deionization Process in Dual-Porosity Electrodes

In many areas of the world, there is a need to increase water availability. Capacitive deionization (CDI) is an electrochemical water treatment process that can be a viable alternative for treating water and for saving energy. A model is presented to simulate the CDI process in heterogeneous porous media comprising two different pore sizes. It is based on a theory for capacitive charging by ideally polarizable porous electrodes without Faradaic reactions or specific adsorption of ions. A two steps volume averaging technique is used to derive the averaged transport equations in the limit of thin electrical double layers. A one-equation model based on the principle of local equilibrium is derived. The constraints determining the range of application of the one-equation model are presented. The effective transport parameters for isotropic porous media are calculated solving the corresponding closure problems. The source terms that appear in the average equations are calculated using theoretical derivations. The global diffusivity is calculated by solving the closure problem.

Numerical study of turbulent porous media combustion coupled with thermoelectric generation in a recuperative reactor

This paper presents two-dimensional simulations of an adiabatic recuperative porous media burner in turbulent regime coupled with thermoelectric elements. Time and volume averaged transport equations and second order turbulence model were employed. FLUENT was used to simulate the reactor through its UDF (User Defined Functions) and UDS (User Defined Scalars) interfaces for extra terms involving turbulence and thermoelectric quantities. The study includes the production of thermal NOx modeled by the extended Zeldovich mechanism with postprocessing computation. Flow and electric field computations were obtained considering as variables gas inlet velocity and composition, porous media thermal conductivity and inner wall insulation material. Flammability limits for stable combustion were found alongside with the electric potential generated within the system. Higher values of electrical potential, thermoelectric efficiency and flame temperatures were reached when the gas mixture energy content and heat recovery capacity of the system were increased.

Volume Averaging Study of the Capacitive Deionization Process in Homogeneous Porous Media

Ion storage in porous electrodes is important in applications such as energy storage by supercapacitors, water purification by capacitive deionization, extraction of energy from a salinity difference and heavy ion purification. A model is presented to simulate the charge process in homogeneous porous media comprising big pores. It is based on a theory for capacitive charging by ideally polarizable porous electrodes without faradaic reactions or specific adsorption of ions. A volume averaging technique is used to derive the averaged transport equations in the limit of thin electrical double layers. Transport between the electrolyte solution and the charged wall is described using the Gouy–Chapman–Stern model. The effective transport parameters for isotropic porous media are calculated solving the corresponding closure problems. The source terms that appear in the average equations are calculated using numerical computations. An alternative way to deal with the source terms is proposed.

A model study of the impact of the transport of inoculant particles on microstructure formation during solidification

We investigate the impact of the convective transport of inoculant particles on the distribution of the final microstructure (grain size) in grain-refined aluminum-alloy castings. We carry out numerical simulations of a casting experiment, considering the solidification of an Al–22 wt.%Cu alloy inoculated with Al–Ti–B in a side-cooled 76 × 76 × 254 mm sand mold. We use a fully coupled multiscale volume-averaged two-phase model. At the macroscopic scale the transport of mass, heat and solute, as well as the transport of globular equiaxed grains and of inoculant particles are described by volume averaged transport equations. At the microscopic scale, nucleation and grain growth are accounted for. Nucleation is considered to be heterogeneous and athermal. Grains nucleate on polydisperse inoculant particles at an undercooling inversely proportional to the particle size. The growth of the globular solid grains is controlled by solute diffusion. We analyze the individual roles of the phenomena of transport of inoculant particles and of equiaxed grains for a range of process parameters (initial superheat, cooling rate, growth-restriction parameter). We show that the consideration of the transport of polydisperse inoculant particles has a strong impact on the prediction of the heterogeneity of the final microstructure across the casting. The transport of the inoculant particles considerably increases the heterogeneities of the microstructure and reduces the average grain size, and the grain motion reduces the microstructure heterogeneities.

Fractional Ensemble Average Governing Equations of Transport by Time-Space Nonstationary Stochastic Fractional Advective Velocity and Fractional Dispersion. II: Numerical Investigation

AbstractIn this paper, the second in a series of two, the theory developed in the companion paper is applied to transport by stationary and nonstationary stochastic advective flow fields. A numerical solution method is presented for the resulting fractional ensemble average transport equation (fEATE), which describes the evolution of the ensemble average contaminant concentration (EACC). The derived fEATE is evaluated for three different forms: (1) purely advective form of fEATE, (2) moment form of the fractional ensemble average advection-dispersion equation (fEAADE) form of fEATE, and (3) cumulant form of the fractional ensemble average advection-dispersion equation (fEAADE) form of fEATE. The Monte Carlo analysis of the fractional governing equation is then performed in a stochastic flow field, generated by a fractional Brownian motion for the stationary and nonstationary stochastic advection, in order to provide a benchmark for the results obtained from the fEATEs. When compared to the Monte Carlo sim...

Fractional Ensemble Average Governing Equations of Transport by Time-Space Nonstationary Stochastic Fractional Advective Velocity and Fractional Dispersion. I: Theory

AbstractIn this study, starting from a time-space nonstationary general random walk formulation, the pure advection and advection-dispersion forms of the fractional ensemble average governing equations of solute transport by time-space nonstationary stochastic flow fields were developed. In the case of the purely advective fractional ensemble average equation of transport, the advection coefficient is a fractional ensemble average advective flow velocity in fractional time and space that is dependent on both space and time. As such, in this case, the time-space nonstationarity of the stochastic advective flow velocity is directly reflected in terms of its mean behavior in the fractional ensemble average transport equation. In fact, the derived purely advective form represents the Lagrangian derivation of the ensemble average mass conservation equation for solute transport in fractional time-space. In the case of the fractional ensemble average advection-dispersion transport equation, the moment and cumula...

On the Budget Terms of the Double Averaged Turbulent Stress Transport Equations in Porous Media

To provide information for modeling turbulent flows in porous media, LES (large eddy simulation) of fully developed flows in porous media is performed. The numerical scheme used for the LES is the D3Q27 (three dimensional 27 discrete velocity) MRT (multiple relaxation time) LBM (lattice Boltzmann method) coupled with the WALE (wall-adapting local eddy-viscosity) sub- grid-scale turbulence model. The considered porous medium configurations are square rod arrays, cube arrays and BCC (body centered cube) foams. Using the statistical data by the LES, the behaviors of the budget terms of the double averaged turbulent stress transport equations are investigated carefully for modeling those transport equations.

Estimation of effective diffusion coefficient and its effect on effectiveness factor for HDS catalytic process: A multi-scale approach

Effectiveness factors have great relevance in multiphase reactors modeling since they are the conventional way of incorporating the effects of intra-particle resistance reaction rate. This work determines the description level effect of catalytic pellet microstructure on mass and energy effective transport coefficients prediction, isothermal and no isothermal. For such a purpose some results about on evaluation of the effective diffusivity and conductivity with the methodology of volume averaging were applied. The obtained results along with a Langmuir–Hinshelwood/Hougen–Watson kinetic expression were applied to establish the concentration and temperature fields in a catalytic particle. The evaluation of concentration field and effectiveness factors were developed using two different models: pseudo- homogeneous mass and energy transport model for a catalytic particle with reaction in all domain, and heterogeneous mass and energy transport model with fluid-catalytic surface interphase reaction for a realistic porous structure model. The results show the differences in concentration and temperature profiles between both models and consequently in effectiveness factors. This could be ascribed to the form of evaluation of effective transport coefficients used in the pseudo-homogeneous model, and presumably to the simple shape of the unit cells used for the solution of the closure problem for the average transport equations with homogeneous reaction.