Linear Transport Model Research Articles

Convergence to Steady-state and Boundary Layer Profiles in a Linear Chromatography System

We study the hyperbolic system of equations of the so-called linear transport model in a true moving bed chromatography device with four ports. By using methods based on a suitable energy functional we show that all solutions approach exponentially a unique steady-state solution. Then, with the use of asymptotic analysis techniques we calculate the limit profiles of these steady-state solutions when the mass transfer coefficient between the liquid and solid phases tends to infinity. Along this singular limit sharp boundary layers appear near some ports. We are able to obtain explicit and simple formulas for these limit profiles.

A Generalized Linear Transport Model for Spatially Correlated Stochastic Media

We formulate a new model for transport in stochastic media with long-range spatial correlations where exponential attenuation (controlling the propagation part of the transport) becomes power law. Direct transmission over optical distance τ(s), for fixed physical distance s, thus becomes (1 + τ(s)/a)− a, with standard exponential decay recovered when a → ∞. Atmospheric turbulence phenomenology for fluctuating optical properties rationalizes this switch. Foundational equations for this generalized transport model are stated in integral form for d = 1, 2, 3 spatial dimensions. A deterministic numerical solution is developed in d = 1 using Markov Chain formalism, verified with Monte Carlo, and used to investigate internal radiation fields. Standard two-stream theory, where diffusion is exact, is recovered when a = ∞. Differential diffusion equations are not presently known when a < ∞, nor is the integro-differential form of the generalized transport equation. Monte Carlo simulations are performed in d = 2, as a model for transport on random surfaces, to explore scaling behavior of transmittance T when transport optical thickness τt ≫ 1. Random walk theory correctly predicts T∝τ− min {1, a/2}t in the absence of absorption. Finally, single scattering theory in d = 3 highlights the model’s violation of angular reciprocity when a < ∞, a desirable property at least in atmospheric applications. This violation is traced back to a key trait of generalized transport theory, namely, that we must distinguish more carefully between two kinds of propagation: one that ends in a virtual or actual detection and the other in a transition from one position to another in the medium.

Membrane-integrated physico-chemical treatment of coke-oven wastewater: transport modelling and economic evaluation.

A modelling and simulation study with economic evaluation was carried out for an advanced membrane-integrated hybrid treatment process that ensures reuse of water with recovery of ammoniacal nitrogen as struvite from coke-oven wastewater. Linearized transport model was developed based on extended Nernst-Plank and concentration polarization modulus equation. Effects of pH, transmembrane pressure and cross-flow rate of interest on membrane charge density, solute rejection and solvent flux were investigated. The membrane module was successful in yielding a pure water flux as high as 120 L m(-2) h(-1) removing more than 95 and 96% of the cyanide and phenol, respectively, while permeating more than 90% NH4 (+)-N at a transmembrane pressure of only 15 × 10(2) KPa and at a pH of 10 for a volumetric cross-flow rate of 800 L h(-1). The Fenton's reagents were used to degrade more than 99% of pollutants present in the concentrated stream. The developed model could successfully predict the plant performance as reflected in the very low relative error (0.01-0.12) and overall high correlation coefficient (R(2) > 0.96). Economic analysis indicated that such a membrane-integrated hybrid system could be quite promising in coke wastewater treatment at low cost i.e. $0.934/m(2) of wastewater.

William W. Cooper: Innovator, Fighter, and Scholar

W illiam Wager Cooper was born on July 23, 1914 in Birmingham, Alabama, where his father was a bookkeeper and later a distributor for Anheuser-Busch. Bill’s family moved to Chicago when he was three years old. His father owned a chain of gasoline stations, but lost them during the Great Depression. With his father in ill health and no family revenue, Bill left high school after his sophomore year to work at whatever jobs he could find. These included spotting pins in bowling alleys, caddying at golf courses, and even professional boxing, where his record was 58 wins, 3 losses, and 2 draws. While hitchhiking to the golf course one day, Bill met Eric L. Kohler, a principal with Arthur Andersen & Co., who became his life-long mentor. Kohler loaned him money to enroll as a non-degree student at The University of Chicago. Bill liked the academic atmosphere and took the college entrance examinations, intending to become a physical chemist. During his studies, Kohler asked him to review the mathematics used in a patent infringement suit. After Cooper found errors in the mathematics used by the plaintiff’s engineers, Andersen hired him part time during the school year and full time in the summer. This awakened Cooper’s interest in accounting, and he switched his major to economics. While at Chicago, Bill became good friends with a political science student, Herbert A. Simon. Cooper graduated Phi Beta Kappa with his A.B. in 1938.

Regolith production and transport in the Susquehanna Shale Hills Critical Zone Observatory, Part 1: Insights from U‐series isotopes

AbstractTo investigate the timescales of regolith formation on hillslopes with contrasting topographic aspect, we measured U‐series isotopes in regolith profiles from two hillslopes (north facing versus south facing) within the east‐west trending Shale Hills catchment in Pennsylvania. This catchment is developed entirely on the Fe‐rich, Silurian Rose Hill gray shale. Hillslopes exhibit a topographic asymmetry: The north‐facing hillslope has an average slope gradient of ~20°, slightly steeper than the south‐facing hillslope (~15°). The regolith samples display significant U‐series disequilibrium resulting from shale weathering. Based on the U‐series data, the rates of regolith production on the two ridgetops are indistinguishable (40 ± 22 versus 45 ± 12 m/Ma). However, when downslope positions are compared, the regolith profiles on the south‐facing hillslope are characterized by faster regolith production rates (50 ± 15 to 52 ± 15 m/Ma) and shorter durations of chemical weathering (12 ± 3 to 16 ± 5 ka) than those on the north‐facing hillslope (17 ± 14 to 18 ± 13 m/Ma and 39 ± 20 to 43 ± 20 ka). The south‐facing hillslope is also characterized by faster chemical weathering rates inferred from major element chemistry, despite lower extents of chemical depletion. These results are consistent with the influence of aspect on regolith formation at Shale Hills; we hypothesize that aspect affects such variables as temperature, moisture content, and evapotranspiration in the regolith zone, causing faster chemical weathering and regolith formation rates on the south‐facing side of the catchment. The difference in microclimate between these two hillslopes is inferred to have been especially significant during the periglacial period that occurred at Shale Hills at least ~15 ka before present. At that time, the erosion rates may also have been different from those observed today, perhaps denuding the south‐facing hillslope of regolith but not quite stripping the north‐facing hillslope. An analysis of hillslope evolution and response timescales with a linear mass transport model shows that the current landscape at Shale Hills is not in geomorphologic steady state (i.e., so‐called dynamic equilibrium) but rather is likely still responding to the climate shift from the Holocene periglacial to the modern, temperate conditions.

The logistics of supplying single vs. multi-crop cellulosic feedstocks to a biorefinery in southeast North Dakota

This study investigates the economics of supplying cellulosic crop residue feedstocks within 100mile (or 161km) radius of a potential new biorefinery in southeast North Dakota. We utilize linear programming transportation models and GIS (Geographic Information System) maps to identify optimal cellulosic feedstock supply sites for the biorefinery. Our evidence indicates that future crop residue removal policy will be important in assessing economic viability of harvesting crop residues for biofuel production. An increase in residue removal rate would narrow the size of supply areas required to provide total feedstock need for the biorefinery and consequently result in reducing transportation costs. There is an economic tradeoff between residue collection density and distance from the biorefinery. When multiple residue feedstocks are considered, this analysis is complicated further because greatest densities for each crop are not in similar regions. Overall, our findings show that higher costs savings can be achieved if multiple feedstocks are employed to produce cellulosic biofuels. In addition, feedstock supply risks can be lower with the use of multiple feedstocks.

Control of modeling errors in the coupling of linear transport and diffusion models

We consider in this paper the initial-boundary value problem for the 1D neutron transport equation with isotropic scattering, set in some bounded interval with inflow boundary conditions. The usual parabolic scaling yields the diffusive limit. A surrogate model, coupling transport and diffusion equations, is then introduced in order to accurately assess the value of specific quantities of interest. The control of the quality of the computation (with respect to such a quantity of interest) is performed by means of an approach for modeling error estimation combined with an adaptive strategy to control the surrogate model, if necessary.

Interband photorefractive effect in semiconductors with hot-electron transport at arbitrary modulation depth

Hot-electron transport in photorefractive semiconductors strongly affects the recording of photorefractive grating. However, the theoretical description of this behavior has been always made under small signal approximation. In this paper rigorous numerical investigation of stationary and transient behavior of photorefractive grating induced in the semiconductor quantum wells structure is presented. The calculations were performed in the framework of the band transport model for the case of high modulation depth. The comparison between numerical and approximated analytical solutions is given as well as between linear and nonlinear transport model. Obtained results allow to verify some statement formulated in few earlier works concerning beneficial effects of electron transport nonlinearity.

Logistics Cost Optimization for Straw Biomass Supply and its Application in Heilongjiang Province, China

The operation of many biomass-based power plants in China have demonstrated that biomass supply logistics system is a key part to achieve economic, stable, and safe operation for such plants. In this paper, the straw biomass supply system of a biomass-based power plant located in Wangkui County, Heilongjiang Province was analyzed, and a mathematical model was established to optimize the fuel supply system. In consideration of the current straw biomass fuel supply system in China, that is "fuel origin – purchasing station - power plant", appropriate hypothesis and simplification were made on the linear transportation model. The logistics parameters for the biopower plant were obtained through the model analysis. Sensitivity analyses were conducted to analyze the optimal delivered cost of biomass under different operation scenarios, which enhanced the reliability of the final results. The optimal model proposed in this paper can also be used as a reference for the other biopower plants in the other regions.

Analysis of electron heat transport with off-axis modulated ECRH in Tore Supra

Experiments to study inward heat transport phenomena have been performed in the Tore Supra tokamak with off-axis electron cyclotron resonance heating (ECRH). Both power balance and perturbation transport analysis have been done for low-frequency (1 Hz) ECRH modulation experiments. Heat diffusivity and heat pinch have been separately determined by fitting the experimental data of the amplitude and phase of the Fourier transform of the modulated temperature with a linear transport model including convection term. Comparison with the critical gradient model has shown that the heat pinch previously obtained could include a pseudo-pinch due to the non-linearity of the diffusivity and an additional non-diffusive heat pinch. The pinch effect is reduced for higher densities.

Towards a unified linear kinetic transport model with the trace ion module for EIRENE

Linear kinetic Monte Carlo particle transport models are frequently employed in fusion plasma simulations to quantify atomic and surface effects on the main plasma flow dynamics. Separate codes are used for transport of neutral particles (incl. radiation) and charged particles (trace impurity ions). Integration of both modules into main plasma fluid solvers provides then self-consistent solutions, in principle. The required interfaces are far from trivial, because rapid atomic processes in particular in the edge region of fusion plasmas require either smoothing and resampling, or frequent transfer of particles from one into the other Monte Carlo code. We propose a different scheme here, in which despite the inherently different mathematical form of kinetic equations for ions and neutrals (e.g. Fokker–Planck vs. Boltzmann collision integrals) both types of particle orbits can be integrated into one single code. We show that the approximations and shortcomings of this “single sourcing” concept (e.g., restriction to explicit ion drift orbit integration) can be fully tolerable in a wide range of typical fusion edge plasma conditions, and be overcompensated by the code-system simplicity, as well as by inherently ensured consistency in geometry (one single numerical grid only) and (the common) atomic and surface process modules.

Low-Rank Kalman Filtering for Efficient State Estimation of Subsurface Advective Contaminant Transport Models

AbstractAccurate knowledge of the movement of contaminants in porous media is essential to track their trajectory and later extract them from the aquifer. A two-dimensional flow model is implemented and then applied on a linear contaminant transport model in the same porous medium. Because of different sources of uncertainties, this coupled model might not be able to accurately track the contaminant state. Incorporating observations through the process of data assimilation can guide the model toward the true trajectory of the system. The Kalman filter (KF), or its nonlinear invariants, can be used to tackle this problem. To overcome the prohibitive computational cost of the KF, the singular evolutive Kalman filter (SEKF) and the singular fixed Kalman filter (SFKF) are used, which are variants of the KF operating with low-rank covariance matrices. Experimental results suggest that under perfect and imperfect model setups, the low-rank filters can provide estimates as accurate as the full KF but at much low...

Giant spin-dependent photo-conductivity in GaAsN dilute nitride semiconductor

A theoretical and experimental study of the spin-dependent photoconductivity in dilute Nitride GaAsN is presented. The non linear transport model we develop here is based on the rate equations for electrons, holes, deep paramagnetic and non paramagnetic centers both under CW and pulsed optical excitation. Emphasis is given to the effect of the competition between paramagnetic centers and non paramagnetic centers which allows us to reproduce the measured characteristics of the spin-dependent recombination power dependence. Particular attention is paid to the role of an external magnetic field in Voigt geometry. The photoconductivity exhibits a Hanle-type curve whereas the spin polarization of electrons shows two superimposed Lorentzian curves with different widths, respectively related to the recombination of free and trapped electrons. The model is capable of reproducing qualitatively and quantitatively the most important features of photoluminescence and photocurrent experiments and is helpful in providing insight on the various mechanisms involved in the electron spin polarization and filtering in GaAsN semiconductors.

Sorption and transport of trichloroethylene in caliche soil

Sorption of TCE to the caliche soil exhibited linear isotherm at the high TCE concentrations (Co = 122–1300 mg L −1) but Freundlich isotherm at the low concentration range (1–122 mg L −1). Sorption strength of the carbonate fraction of the soil was about 100-fold lower than the sorption strength of soil organic matter (SOM) in the caliche soil, indicating weak affinity of TCE for the carbonate fraction of the soil. Desorption of TCE from the caliche soil was initially rapid (7.6 × 10 −4 s −1), then continued at a 100-fold slower rate (7.7 × 10 −6 s −1). Predominant calcium carbonate fraction of the soil (96%) was responsible for the fast desorption of TCE while the SOM fraction (0.97%) controlled the rate-limited desorption of TCE. Transport of TCE in the caliche soil was moderately retarded with respect to the water ( R = 1.75–2.95). Flow interruption tests in the column experiments indicated that the rate-limited desorption of TCE controlled the non-ideal transport of TCE in the soil. Modeling studies showed that both linear and non-linear nonequilibrium transport models provided reasonably good match to the TCE breakthrough curves ( r 2 = 0.95–0.98). Non-linear sorption had a negligible impact on both the breakthrough curve shape and the values of sorption kinetics parameters at the high TCE concentration (Co = 1300 mg L −1). However, rate-limited sorption/desorption processes dominated at this concentration. For the low TCE concentration case (110 mg L −1), in addition to the rate-limited sorption/desorption, contribution of the non-linear sorption to the values of sorption kinetics became fairly noticeable.

Hydrodynamic model for the signal propagation along carbon nanotubes

We formulated a hydrodynamic model in order to describe the dynamical behavior of the π- electrons in single carbon nanotube shells of arbitrary chirality, either metallic or semiconducting, below terahertz frequencies, as long as only intraband transitions of the π- electrons are allowed. The hydrodynamic equations were derived in a self-consistent way from the semiclassical Boltzmann equation. The electron fluid was taken to comprise many electron species, each characterized by a different effective mass, which takes into account the interaction with the nanotube ion lattice. A linear transport model for the π- electrons was derived from the hydrodynamic equations. A transmission line model was eventually formulated to describe the propagation of an electric signal along a single-wall carbon nanotube of arbitrary chirality. The transport model formulated can be also used for analyzing electromagnetic propagation in complex structures composed of single carbon-nanotube shells with different chirality, such as bundles of single wall carbon nanotubes and multi-wall carbon nanotubes, provided that the tunneling between adjacent shells may be disregarded.

Linear transport theory for particles moving on a spherical surface

The paper deals with a linear transport model for particles moving on a spherical surface in the presence of absorption and scattering events. Starting from the analytical expression for the particle density in the case of a pure absorbing sphere, an integral equation is derived for situations in which isotropic scattering is present. The equation has been solved numerically in order to obtain the Green function of the problem. Analytical and numerical results are compared with those obtained resorting to the Monte Carlo method.

Nanofiltration of bivalent nickel cations — model parameter determination and process simulation

Nanofiltration is a pressure-driven membrane separation method that is also used for selective removal of salts from liquid streams. The separation by nanofiltration is based on both sieving and charge effects and operated typically at relatively low pressure. Although several mathematical models for nanofiltration are already known, the modeling of mass transfer in complex aqueous solutions remains still under discussion. In this work, the filtration of different single model solutions containing mono- and bivalent cations like Na + and Ni 2+ is studied with several membranes. First, the effect of solution pH on the rejection of the cations and on the permeate flux is investigated. Furthermore, the influence of pH on membrane characteristics due to change of the membrane charge is shown. The rejection of cations increases at pH values below the isoelectric point, in consequence of repelling interactions between cations and membrane which is positively charged at low pH. According to this finding, the optimal separation condition in terms of pH value for the purpose of recovery of Ni 2+ is ascertained for two membranes. From these experiments as well as from those with glycerol solution, three model parameters (pore radius, membrane charge and pore dielectric constant) are determined. These parameters are sufficient for simulating the process on the basis of a linearized transport model suggested by Bowen. The validity of this simplified approach is examined. The finally received model is applied for simulating a case study of a process integrated nickel recovery.

Comparative analysis of linear and nonlinear models for ion transport problem in chronoamperometry

Ion transport problem related to controlled potential experiments in electrochemistry is studied. The problem is assumed to be superposition of diffusion and migration under the influence of an electric field. The comparative analysis are presented for three well-known models—pure diffusive (Cottrell’s), linear diffusion-migration, and nonlinear diffusion-migration (Cohn’s) models. The nonlinear model is derived by the identification problem for a nonlinear parabolic equation with nonlocal additional condition. This problem reduced to an initial-boundary value problem for nonlinear parabolic equation. The nonlinear finite difference approximation of this problem, with an appropriate iteration algorithm is derived. The comparative numerical analysis for all three models shows an influence of the nonlinear migration term, the valences of oxidized and reduced oxidized species, also diffusivity to the value of the total charge. The obtained results permits one to estimate bounds of linear and nonlinear ion transport models.

Transport modelling

Linear and quasi-linear transport models for core transport of a tokamak plasma are reviewed. The dependence of the modelled transport on the main plasma parameters is discussed. The comparison of the models with the experiment is reviewed both directly in terms of the predicted stored energy as well as indirectly discussing key physics ingredients of the models. To cite this article: A.G. Peeters et al., C. R. Physique 7 (2006).

DIFFUSION APPROXIMATION FOR TRANSPORT PROCESSES WITH GENERAL REFLECTION BOUNDARY CONDITIONS

Diffusion approximations are obtained for space inhomogeneous linear transport models with reflection boundary conditions. The collision kernel is not required to satisfy any balance condition and the scattering kernel on the boundary is general enough to include all examples of boundary conditions known to the authors (with conservation of the number of particles) and, in addition, to model the Debye sheath. The mathematical approach does not rely on Hilbert expansions, but rather on martingale and stochastic averaging techniques.