Effective Transport Equations Research Articles

Interfacial mass transfer and axial liquid dispersion in aerated column bioreactors

The upscaling of gas–liquid mass transfer and bioreaction in aerated cultures was performed within the volume averaging theory. Effective transport equations, including their associated closure problems, were derived for 1) the main substrates and biomass (non-transferable components), and 2) the gaseous substrate (transferable component). For the transferable component, two approaches were applied: 2a) the local mass equilibrium (one equation), and 2b) the two-fluid model (two equations). The nature of the dispersion tensors in the three averaged models is discussed. Numerical computations of the axial dispersion coefficient and the volumetric mass transfer coefficient were performed by solving the associated closure problems in unit cells (0.1 ≤ Re ≤ 300; 1 ≤ Sc ≤ 550; 0.1 ≤ Pe ≤ 300; Mo = 2.5 × 1011; 0.001 ≤ εG ≤ 0.05). The results of both parameters agree with trends reported in literature; some differences in orders of magnitude for DLzz are attributable to the mismatch between the experimental conditions and the occurrence of mesoscale interactions undetected in the computational cells.

Water Mixing Approach (WMA) for reactive transport modeling

We describe a formulation to solve reactive transport (RT) problems. The basic idea is to represent transport as mixing water instead of individual solute concentrations, hence the Water Mixing Approach (WMA) name. This conceptual representation simplifies RT calculations as it decouples transport from chemistry. Transport is first solved in terms of water mixing ratios (λ), which we show is feasible for any transport formulation. Chemical calculations can then be written as a set of reactive mixing calculations, which are local, so that they do not need to iterate with transport. We have implemented the WMA for a mixed Eulerian-Lagrangian transport solver with streamline-oriented grid and constant travel time between sequential cells (isochronal grid), which is free of numerical dispersion. We test the WMA on three RT cases. First, a binary system case is used to test the accuracy of the method. Second, a calcite dissolution case is used to compare the WMA to the Direct Substitution Approach to test both accuracy and computational cost (CPU). Third, the accuracy and simplicity of the proposed approach is tested on a reactive pulse injection case, solved in 2D on a spreadsheet. Results confirm the accuracy, simplicity and efficiency (low CPU cost) that result from decoupling transport and chemical steps. Transport through highly heterogeneous media remains a challenge, but the definite separation and simplification of mixing processes in WMA opens a new path for reactive transport modeling in general, and for the search of an effective transport equation.

Taylor dispersion of elongated rods

Particles transported in fluid flows, such as cells, polymers, or nanorods, are rarely spherical. In this study, we numerically and theoretically investigate the dispersion of an initially localized patch of passive elongated Brownian particles constrained to one degree of rotational freedom in a two-dimensional Poiseuille flow, demonstrating that elongated particles exhibit an enhanced longitudinal dispersion. In a shear flow, the rods translate due to advection and diffusion and rotate due to rotational diffusion and their classical Jeffery's orbit. The magnitude of the enhanced dispersion depends on the particle's aspect ratio and the relative importance of its shear-induced rotational advection and rotational diffusivity. When rotational diffusion dominates, we recover the classical Taylor dispersion result for the longitudinal spreading rate using an orientationally averaged translational diffusivity for the rods. However, in the high-shear limit, the rods tend to align with the flow and ultimately disperse more due to their anisotropic diffusivities. Results from our Monte Carlo simulations of the particle dispersion are captured remarkably well by a simple theory inspired by Taylor's original work. For long times and large Peclet numbers, an effective one-dimensional transport equation is derived with integral expressions for the particles' longitudinal transport speed and dispersion coefficient. The enhanced dispersion coefficient can be collapsed along a single curve for particles of high aspect ratio, representing a simple correction factor that extends Taylor's original prediction to elongated particles.

Impact of network heterogeneity on electrokinetic transport in porous media

We present a numerical study of electrokinetic transport in porous media, focusing on the role of heterogeneity in a porous microstructure on ion concentration polarization and over-limiting current. For simplicity, the porous medium is modeled as a network of long, thin charged cylindrical pores, each governed by one-dimensional effective transport equations. For weak surface conduction, when sufficiently large potential is applied, we demonstrate that electrokinetic transport in a porous network can be dominated by electro-convection via internally induced flow loops, which is not properly captured by existing homogenized models. We systematically vary the topology and “accessivity” of the pore network and compare with simulations of traditional homogenized parallel-pore (capillary-bundle) models, in order to reveal the effects of regular and hierarchical connectivity. Our computational framework sheds light on the complex physics of electrokinetic phenomena in microstructures and may be used to design porous media for applications, such as water desalination and purification by shock electrodialysis.

Upscaling chemical reactions in multicontinuum systems: When might time fractional equations work?

It may often be tempting to take a fractional diffusion equation and simply create a fractional reaction-diffusion equation by including an additional reaction term of the same form as one would include for a standard Fickian system. However, in real systems, fractional diffusion equations typically arise due to complexity and heterogeneity at some unresolved scales. Thus, while transport of a conserved scalar may upscale naturally to a fractional diffusion equation, there is no guarantee that the upscaling procedure in the presence of reactions will also result in a fractional diffusion equation with such a naive reaction term added. Here we consider a multicontinuum mobile-immobile system and demonstrate that an effective transport equation for a conserved scalar can be written that is similar to a diffusion equation but with an additional term that convolves a memory function and the time derivative term. When this memory function is a power law, this equation is a time fractional dispersion equation. Including a first-order reaction in the same system, we demonstrate that the effective equation in the presence of reaction is no longer of the same time fractional form. The presence of reaction modifies the nature of the memory function, tempering it at a rate associated with the reaction. Additionally, to arrive at a consistent effective equation the memory function must also act on the reaction term in the upscaled equation. For the case of a bimolecular mixing driven reaction, the process is more complicated and the resulting memory function is no longer stationary in time or homogenous in space. This reflects the fact that memory does not just act on the evolution of the reactant, but also depends strongly on the spatio-temporal evolution and history of the other reactant. The state of mixing in all locations and all times is needed to accurately represent the evolution of reactants. Due to this it seems impossible to write a single effective transport equation in terms of a single effective concentration without having to invoke some approximation, analogous to a closure problem. For both reactive systems simply including a naive reactive term in a fractional diffusion equation results in predictions of concentrations that can be orders of magnitude different from what they should be. We demonstrate and verify this through numerical simulations. Thus, we highlight caution is needed in proposing and developing fractional reaction-diffusion equations that are consistent with the system of interest.

Effective transport equations in quasi 1D systems

The mapping methods reducing 2D or 3D transport equations in quasi 1D structures onto the longitudinal coordinate x are revisited. The general formalism based on homogenization is explained on the simplest case, diffusion in a 2D channel of varying width A(x). Then its modifications to diffusion in an external field (Smoluchowski equation), and nonzero mass of the particles (Klein-Kramers equation) are demonstrated. A special attention is payed to the role of the “natural” curvilinear coordinates, connected with the stationary flow, in the mapping and derivation of the effective equations.

Anomalous dispersion in chemically heterogeneous media induced by long-range disorder correlation

AbstractWe study transport in an idealized porous medium characterized by a spatially varying retardation factor, which models linear instantaneous chemical adsorption of a solute. Using a stochastic modelling approach, we study the impact of disorder correlation on the large-scale dispersion behaviour. We consider short, long-range and intermediate-range disorder correlations, and demonstrate that (truncated) power-law correlation causes anomalous dispersion, even in the presence of weak heterogeneity. We identify different preasymptotic and asymptotic regimes of anomalous dispersion that shed new light on the disorder and local-scale transport mechanisms leading to non-Fickian behaviour. The analytical results are complemented by numerical random walk particle tracking simulations, which are found to be in good agreement with the derived dispersion behaviour. We conclude the paper by deriving an effective transport equation for this system, which can be shown to be tied to the family of continuous-time random walk models.

Effective non-local reaction kinetics for transport in physically and chemically heterogeneous media

The correct characterization of the effective reactive transport dynamics is an important issue for modeling reactive transport on the Darcy scale, specifically in situations in which reactions are localized, that is when different reactions occur in different portions of the porous medium. Under such conditions the conventional approach of homogenizing only the porous medium chemistry is not appropriate. We consider here reactive transport in a porous medium that is characterized by mass transfer between a mobile and a distribution of immobile regions. Chemical and physical heterogeneities are reflected by distributions of kinetic reaction rate constants and residence times in the immobile zones. We derive an effective reactive transport equation for the mobile solute that is characterized by non-local physical mass transfer and reaction terms. Specifically, chemical heterogeneity is upscaled in terms of a reactive memory function that integrates both chemical and physical heterogeneity. Mass transfer limitations due to physical heterogeneity yield effective kinetic rate coefficients that can be much smaller than the volumetric average of the local scale coefficients. These results help to explain and quantify the often reported discrepancy between observed field reaction rate constants and the ones obtained under well mixed laboratory conditions. Furthermore, these results indicate that transport under physical and chemical heterogeneity cannot be upscaled separately.

Effective transport dynamics in porous media with heterogeneous retardation properties

We study a transport model that is characterized by a spatially distributed retardation coefficient. The latter accounts for linear instantaneous mass transfer between a mobile and immobile phases as well as instantaneous equilibrium adsorption. Using a Lagrangian framework we upscale the local scale transport problem to an observation scale that is much larger than the variation scale of spatial heterogeneity. The derived effective Lagrangian transport equation describes the effective particle motion as a random walk in space‐time, or in other words, a continuous time random walk, which is characterized by a joint transition length and time distribution. The transition time distribution is obtained by an exact map from the spatial distribution of retardation coefficients. The effective model compares well with numerical simulations of the small scale transport problem. For broad disorder distributions transport is highly non‐Fickian and an (constant) effective retardation coefficient may not exist at practically relevant time and length scales.

Transport of reactive solutes in unsteady annular flow subject to wall reactions

The present paper concerns with the dispersion process in steady and oscillatory flows through an annular pipe in presence of reversible and irreversible reactions at the wall. Method of homogenization, a multiple-scale method of averaging, is adopted for deriving the effective transport equations. The main objective is to look into the effect of aspect ratio of the annular pipe on the dispersion coefficient due to the combined effect of axial convection and radial diffusion in steady and oscillatory flows along the annulus, subject to the kinetic reversible phase exchange and irreversible absorption at the outer wall. Results demonstrate that upto a certain critical value of aspect ratio, dispersion coefficient increases with increase of aspect ratio when the wall is retentive, though the wall inertness may lead to decrease of dispersion coefficient with increase of aspect ratio. The results would be useful to the medical practitioners working in the domain of catheterized artery.

Effective Transport Equations and Enhanced Backscattering in Random Waveguides

In this paper we derive a general system of transport equations for the moments of reflected and transmitted mode amplitudes in a randomly perturbed waveguide, in a regime where backscattering is significant. The derivation is based on a limit theorem for the system of coupled differential equations for the mode amplitudes, in the limit where the amplitude of the random fluctuations of the medium is small, the correlation lengths in the transverse and longitudinal directions are of the same order of the wavelength, and the waveguide is long. Using this system we derive several results in specific regimes, including the enhanced backscattering phenomenon for the reflected wave: when an incoming monochromatic wave with a specific incidence angle is present, the mean reflected power has a local maximum in the backward direction twice as large as the mean reflected power in the other directions.

Solute Transport in Porous Media. Media with Capillaries as Voids

We study solute transport in porous media with periodic microstructures consisting of interconnected thin channels. We discuss a local physical mechanism that occurs at the intersections of channels and promotes mixing of the solute with the solvent (i.e., the host liquid). We identify the parameter regime, where this mechanism is the dominant cause of dispersion, and obtain the effective or macroscopic transport equation that the concentration of solute satisfies when the medium is subjected to a time periodic applied pressure gradient. We conclude with illustrative examples.

Morphogen transport in epithelia

We present a general theoretical framework to discuss mechanisms of morphogen transport and gradient formation in a cell layer. Trafficking events on the cellular scale lead to transport on larger scales. We discuss in particular the case of transcytosis where morphogens undergo repeated rounds of internalization into cells and recycling. Based on a description on the cellular scale, we derive effective nonlinear transport equations in one and two dimensions which are valid on larger scales. We derive analytic expressions for the concentration dependence of the effective diffusion coefficient and the effective degradation rate. We discuss the effects of a directional bias on morphogen transport and those of the coupling of the morphogen and receptor kinetics. Furthermore, we discuss general properties of cellular transport processes such as the robustness of gradients and relate our results to recent experiments on the morphogen Decapentaplegic (Dpp) that acts in the wing disk of the fruit fly Drosophila.

Dispersion in open-channel flow subject to the processes of sorptive exchange on the bottom and air-water exchange on the free surface

An analytical study is presented of dispersion in steady and oscillatory open-channel flow under the combined effects due to reversible sorptive phase exchange with the bed and irreversible air-Cwater exchange on the free surface. The free-surface reaction is assumed to be so weak that the asymptotic mean concentration can be developed well before undue substantial loss to the air occurs. The effective advection-Cdispersion transport equation, which is derived using a multiple-scale homogenization technique, contains formal expressions for the various components of the dispersion coefficient arising from the two chemical effects. Numerical calculations demonstrate that, as long as the bed is sorbing, the kinetics of the sorptive exchange plays a key role in controlling the overall dispersion mechanism. The otherwise weak effects of the free-surface loss can be significantly enhanced by the presence of kinetic phase exchange on the bottom.

Exact effective transport dynamics in a one-dimensional random environment

We study effective transport under linear equilibrium adsorption characterized by a spatially random retardation factor. In a stochastic framework, we present a methodology to quantify explicitly the impact of spatial disorder on effective transport dynamics. We derive an exact effective transport equation, which is equivalent to transport under linear kinetic adsorption characterized by a spectrum of adsorption times. The distribution of adsorption times is given explicitly in terms of the spatial disorder distribution. Furthermore, we find that effective transport is formally equivalent to a decoupled continuous time random walk. This observation and the explicit nature of the presented results allow for a mapping of the static disorder distribution on the transition time distribution.

Dispersion in oscillatory Couette flow with sorptive boundaries

A multiple-scale method of averaging is applied to the study of dispersion in oscillatory Couette flow where the solute may undergo reversible sorptive phase exchange with the boundary walls. On assuming that the oscillation period and the sorption reaction time are comparable with the transverse diffusion time, which is much shorter than the axial transport time scales, an effective advection-dispersion transport equation in terms of the mean concentration is deduced at the second order. Analytical expressions are obtained for the two dispersion coefficients due to the steady and oscillating components of the Couette flow, incorporating the coupling effects between the flow oscillation, sorption kinetics, and the retardation due to phase partitioning.

Robust Formation of Morphogen Gradients

We discuss the formation of graded morphogen profiles in a cell layer by nonlinear transport phenomena, important for patterning developing organisms. We focus on a process termed transcytosis, where morphogen transport results from the binding of ligands to receptors on the cell surface, incorporation into the cell, and subsequent externalization. Starting from a microscopic model, we derive effective transport equations. We show that, in contrast to morphogen transport by extracellular diffusion, transcytosis leads to robust ligand profiles which are insensitive to the rate of ligand production.

A time-varying diffusivity model for shear dispersion in oscillatory channel flow

An analysis is presented of the shear dispersion in turbulent channel flow forced by a purely oscillatory pressure gradient. The flow is described by an algebraic eddy viscosity model in which the instantaneous flow and viscosity are inter-dependent. Both eddy viscosity and eddy diffusivity contain a second-harmonic time-varying component. The effective transport equation is derived using a two-time-scale perturbation analysis, which also yields a closed-form expression for the dispersion coefficient. Numerical results demonstrate that, in this problem, the time-varying component of the eddy diffusivity can have a finite effect on the dispersion coefficient, which can be as much as a 30% difference from that obtained from a time-invariant eddy viscosity model.

Dual-Species Transport Subject to Sorptive Exchange in Pipe Flow

The transport in pipe flow of a chemical species can be materially affected by the presence of solid suspension if the species is capable of partitioning into a solute phase and a solid phase sorbed onto the suspended particles. An asymptotic analysis is used in this work to deduce the effective transport equations for the two phases, with kinetic sorptive exchange taken into account. The effects of sorption on the advection and dispersion of a sorbing chemical are discussed and illustrated with a numerical example.

Effective Equations for Two-Phase Flow with Trapping on the Micro Scale

In this paper we consider water-drive for recovering oil from a strongly heterogeneous porous column. The two-phase model uses Corey relative permeabilities and Brooks--Corey capillary pressure. The heterogeneities are perpendicular to the flow and have a periodic structure. This results in one-dimensional flow and a space periodic absolute permeability, reflecting alternating coarse and fine layers. Assuming many---or thin---layers, we use homogenization techniques to derive the effective transport equations. The form of these equations depends critically on the capillary number. The analysis is confirmed by numerical experiments.