Effective Transport Equations Research Articles

Morphological stability analysis of directional solidification into an oscillatory fluid layer

We study the stability of a planar solid-melt boundary during directional solidification of a binary alloy when the solid is being periodically vibrated in the direction parallel to the boundary (or equivalently, under a far field uniform and oscillatory flow parallel to the planar boundary). The analysis is motivated by directional solidification experiments under the low level residual acceleration field characteristic of a microgravity environment, and possible effects on crystal growth in space. It is known that periodic modulation of the solid-melt interface under the conditions stated induces second order stationary streaming flows within a boundary layer adjacent to the interface, the thickness of which is the same as the wavelength of the modulation. We derive an effective solute transport equation by averaging over the fast time scale of the oscillatory flow, and obtain the resulting dispersion relation for a small disturbance of a planar interface. We find both regions of stationary and oscillatory instability. For small ratios of the viscous to solutal layer thicknesses, s, the flow generally destabilizes the planar interface. For s≃1, the flow stabilizes the stationary branch, but it can also excite an oscillatory instability. For large s, the effect of the flow is small.

Effects of kinetic sorptive exchange on solute transport in open-channel flow

A theory is presented for the transport in open-channel flow of a chemical species under the influence of kinetic sorptive exchange between phases that are dissolved in water and sorbed onto suspended sediments. The asymptotic method of homogenization is followed to deduce effective transport equations for both phases. The transport coefficients for the solute are shown to be functions of the local sediment concentration and therefore vary with space and time. The three important controlling parameters are the suspension number, the bulk solid–water distribution ratio and the sorption kinetics parameter. It is illustrated with a numerical example that when values of these parameters are sufficiently high, the advection and dispersion of the solute cloud can be dominated by the sorption effects. The concentration distribution can exhibit an appreciable deviation from Gaussianity soon after discharge, which develops into a long tailing as the solute cloud gradually moves ahead of the sediment cloud.

Mass transport in a two-layer wave boundary layer

The transport of a chemical species under the pure action of surface progressive waves in the benthic boundary layer which is loaded with dense suspended sediments is studied theoretically. The flow structure of the boundary layer is approximated by that of a two-layer Stokes boundary layer with a sharp interface between clear water and a heavy fluid. The simplest model of constant eddy diffusivities is adopted and the exchange of matter with the bed is ignored. For a thin layer of heavy fluid, whose thickness is comparable to the surface wave amplitude and the Stokes boundary layer thickness, effective transport equations are deduced using an averaging technique based on the method of homogenization. The effective advection velocity is found to be equal to the depth-averaged mass transport velocity, while the dispersion coefficient can be shown to be positive definite. Explicit expressions for the transport coefficients are obtained as functions of fluid properties and flow kinematics. Physical discussions on their relations are also presented.

Macrodispersion in a radially diverging flow field with finite Peclet Numbers: 2. Homogenization theory approach

We study the transport behavior of a tracer in a radially diverging heterogeneous velocity field. Making use of homogenization theory, we derive effective transport equations. These effective transport equations are very similar to those defined on the local scale. However, the local transport parameters such as local dispersion coefficients are replaced by effective dispersion coefficients. For smoothly varying heterogeneous media, explicit results for effective radial dispersion coefficients are derived. Starting with the purely advective transport behavior (infinite Peclet numbers), we extend our calculations to transport with finite Peclet numbers. We find that the impact of molecular diffusion on the effective dispersivity differs from the impact of local dispersion: Including local dispersion leads to effective dispersivities which are constant and equivalent to the effective dispersivities found in uniform flow configurations. In contrast, effective dispersivities including diffusion are not constant but depend on the radial distance. We compare the results found by homogenization theory with those derived by Neuweiler et al. (this issue) by standard method of moments.

Chemical transport associated with discharge of contaminated fine particles to a steady open-channel flow

In this paper, an analytical study on the advective-dispersive transport of a chemical contaminant resulting from the discharge of contaminated fine solid particles into a two-dimensional, steady and uniform turbulent open-channel flow is presented. Because of sorptive exchange, the transport of the chemical cloud is affected by that of the suspended particulates. Such a relationship has so far not been explicitly established by intuitive arguments. The effective transport equations are formally derived by an extended method of homogenization. It is found that over a long time scale the fall velocity will delay the sediment advection, and the advection velocity and dispersion coefficient for the chemical transport will change with space and time according to the local sediment concentration. Numerical results confirm that the centers of mass of the sediment and dissolved phase clouds are not advancing at the same speed, and the dispersion of the chemical is enhanced by the local retardation factor.

Effective transport equations for non-Abelian plasmas

Starting from classical transport theory, we derive a set of covariant equations describing the dynamics of mean fields and their statistical fluctuations in a non-Abelian plasma in or out of equilibrium. A general procedure is detailed for integrating out the fluctuations as to obtain the effective transport equations for the mean fields. In this manner, collision integrals for Boltzmann equations are obtained as correlators of fluctuations. The formalism is applied to a hot non-Abelian plasma close to equilibrium. We integrate out explicitly the fluctuations with typical momenta of the Debye mass, and obtain the collision integral in a leading logarithmic approximation. We also identify a source for stochastic noise. The resulting dynamical equations are of the Boltzmann–Langevin type. While our approach is based on classical physics, we also give the necessary generalizations to study the quantum plasmas. Ultimately, the dynamical equations for soft and ultra-soft fields change only in the value for the Debye mass.

Fluctuation dynamics, thermodynamic analogies and ergodic behavior for nonequilibrium-independent rate processes with dynamical disorder

The stochastic properties of the sojourn times attached to a Markov process in continuous time and with a finite number of states are described by using a statistical ensemble approach. This approach is applied for investigating the large time behavior of independent rate processes with dynamical disorder. The large time behavior of the system is described in terms of an effective transport operator which can be expressed as a static average with respect to the stochastic properties of the sojourn times. The method is illustrated by the generalization of the Van den Broeck approach to the generalized Taylor diffusion. Explicit formulas for the effective transport coefficients and for the fluctuations of the concentration fields are derived. The results are used for extending the non-equilibrium generalized thermodynamic formalisms suggested by Keizer and by Ross, Hunt and Hunt to systems with dynamical disorder. It is shown that the logarithm of the probability density functional of concentration fluctuations is a Lyapunov functional of the effective transport equation. This Lyapunov functional plays the role of a generalized nonequilibrium thermodynamic potential which may serve as a basis for a thermodynamics description of the average behavior of the system. The existence and stability of a steady state can be expressed as an extremum condition for the Lyapunov functional. For Taylor diffusion in an external electric field different from zero the generalized potential has a structure similar to the Helmholtz free energy rather than to the entropy. A generalized chemical potential is derived as the functional derivative of the Lyapunov functional with respect to the concentration field; the gradient of this generalized chemical potential is the driving force which determines the structure of the effective transport equation.

Effects of a semipervious lens on soil vapour extraction

We describe a theory for the removal of volatile organic chemicals from an unsaturated soil stratum consisting of highly porous coarse sand layers sandwiching a thin and semipervious lens. Each soil layer is modelled as a periodic array of spherical aggregates formed by solid grains and immobile water trapped by surface tension. Volatile chemicals are vaporized in the mobile air in pores between aggregates, dissolved in the intra-aggregate water, and adsorbed on the surface of soil grains. Using the effective transport equations derived for the aggregated soils, we consider shallow layers with sharp contrast in physical properties. An asymptotic analysis is developed for an axisymmetric geometry, yielding quasi-one-dimensional governing equations for individual layers. At the leading order the flow and the vapour transport are horizontal in the coarse layers but vertical in the semipervious lens. Numerical results are presented for a simple example to demonstrate the significance of the lens permeability, diffusivity and retardation factor, and the aggregate diffusivity in the coarse layers, on the vapour transport during the stages of contamination and air-venting.

TAYLOR DISPERSION OF MULTICOMPONENT SOLUTES

Coupled transport of multicomponent solutes in globally continuous systems is considered in the framework of the Generalized Taylor dispersion theory. Coupling between transports of n different species at the local (or micro-) scale, is considered to result from first-order irreversible surface reactions occurring on the local space boundaries, or from the off-diagonal terms of the solute diffusivity matrices. General expressions are obtained for the global effective (long-time) solute dispersion matrix cofficients: mean global scalar reactivity, velocity vector and dispersivity dyadic. The effect of surface chemical reactions is to partition the matter between different solute constituents. This is manifested in a coupling of the global transport coefficients, which may be mathematically removed by a linear (canonic) transformation applied to the effective global transport equation. This type of coupling does not exist for inert solutes. The second type of the global coupling is represented by the off-diagonal terms of the global velocity and dispersivity matrices. It exists for both reactive and inert solutes. This coupling stems from the convective dispersion process (dependence or the global velocity vector on the local space coordinate). Is shown to be irremovable from the global transport equation by any linear transformation via the solute partition matrix. In the canonic form of the global equation the irremovable coupling is manifested by the traceless parts of the global solute velocity matrix and the global solute dispersivity. The solution scheme is illustrated by calculating the mean global diffusivity of a solute consisting of two components, transport of which is coupled at the microscale via the molecular diffusivity matrix. At the macroscale the coupling is shown to be represented by negative off-diagonal terms of the global diffusivity matrix,

Dispersion of chemical solutes in chromatographs and reactors

The dispersion of a chemically active solute in unidirectional laminar flow in a channel of constant cross-sectional area is considered. Adsorption/desorption of the solute at the wall or the presence of a bulk or surface chemical reaction introduce additional timescales, in addition to the diffusive and convective ones, such that, under certain conditions, the asymptotic evolution of the cross-sectional mean concentration cannot be described by a one-dimensional Taylor-Aris model. We use the centre and invariant manifold theories to establish the proper time and length scale separations necessary for the existence of an effective transport equation and to determine the dependence of the effective transport coefficients on the kinetics of adsorption/desorption and reaction. For the case of classical Taylor-Aris dispersion with no reaction, we derive the effective transport equation to infinite order in the parameter, p , representing the ratio of the characteristic time for radial molecular diffusion to that for axial convection. We show that the infinite series in the effective transport model is convergent provided p is smaller than some critical value, which depends on the initial concentration distribution. We also examine the spatial evolution of time dependent inlet conditions and show that the spatial and temporal evolutions differ at third and higher orders. It is shown that, except for slow reactions with a kinetic timescale of the same order as the transverse diffusion time, fast bulk reaction does not allow an asymptotic axial dispersion description. Slow bulk reactions do not affect dispersion but a correction to the apparent kinetics may arise due to nonlinear interaction among reaction, diffusion and convection. It is also shown that with a slow bulk reaction, steady-state dispersion due to a coupling of reaction and transverse velocity gradient can arise. Although this mechanism is distinct from the transient Taylor—Aris mechanism, the dispersion coefficient is identical to the classical unreactive Taylor—Aris coefficient. Surface reaction of any speed yields the proper asymptotic behaviour in time because the species still needs to diffuse slowly to the conduit wall. In the limit of fast surface reaction, the Taylor-Aris dispersion coefficient is reduced by a factor of 4.2, 7.1 and 4.0 for pipe, plane Poiseuille and Couette flows, respectively, as the slow-moving solutes near the wall are depleted. For the case of a linear surface reaction, we use the invariant manifold theory to derive the effective transport equation to infinite order. We also show that the radius of convergence of the invariant manifold expansion is approximately three times that of the no reaction case. We demonstrate that if adsorption/desorption is as slow as transverse diffusion an adsorption-induced dispersion, distinct from the Taylor-Aris shear dispersion, exists. While the total dispersion may increase because of the contribution of both, the Taylor-Aris component is reduced by a physical mechanism similar to surface reaction. The adsorption/desorption induced dispersion coefficient is shown to have a maximum when the adsorption equilibrium constant is exactly 2. Nonlinear Langmuir type adsorption at large concentration is shown to introduce a nonlinear drift term which causes non-Gaussian pulse responses with long tails These tails are detrimental to separation chromatography since they cause overlaps which increase with the length of the chromatograph

Analysis of effective axial diffusion in branching networks

AbstractDiffusional transport in a network of branching conduits is considered. The branching network is idealized as an ensemble of identical branching segments. Starting with the general species continuity equation, multiscale perturbation analysis is used to derive a one‐dimensional, effective transport equation for species concentration. The macrotransport equation contains an effective diffusion coefficient, D*, which arises naturally from the analysis. D* can be computed from a local diffusion problem posed on an individual branching segment of the ensemble network. The local relations defining D* bear clear similarity to their counterparts for transport in spatially periodic porous media. Although this study was originally directed toward describing gas diffusion in the peripheral airspaces of the lung, the results provide insight for other transport processes occurring in branching networks.

An Effective Equation Governing Convective Transport in Porous Media

The fine structure of disordered porous media (e.g., fully saturated randomly packed beds) causes microscopic velocity fluctuations. The effect of the spatial and temporal randomness of the interstitial velocity field on the convective transport of a scalar (heat or mass) is investigated analytically. For a uniform mean velocity profile, the effective heat transport equation is obtained as the equation governing the transport of the ensemble average of the scalar under conditions of steady or unsteady random fields (with given statistics). In both cases, it is shown that the effective transport coefficient is enhanced by a hydrodynamic dispersive component, which is an explicit function of the mean filtration velocity. The agreement with experiments is encouraging. The effective transport equation is then generalized to three-dimensional mean velocity fields for isotropic media.

A Classification Framework for Separation Science

Abstract A general framework for the development of separation science is proposed which links the criterion of merit via an effective transport equation to the molecular basis of separation processes.