Transverse Mixing Research Articles

Upscaling and reversibility of Taylor dispersion in heterogeneous porous media

Tracer flow in stratified porous media is dominated by the interaction between convective transport and transverse diffusive mixing. By averaging the tracer concentration in the transverse direction, a one-dimensional non-Fickian dispersion model is derived. The model accounts for the relaxation process that reduces the convective transport to dispersive mixing. This process is (short-) time correlated and partially reversible upon reversal of flow direction. For multiscale velocity fields, the relaxation is a multiscale process. To date only single scale processes have been successfully upscaled. Our procedure extends this to multiscale processes, using scale separation. The model parameters can be calculated a priori based on the velocity profile. For periodic flow reversal, the results are essentially the same. Despite the non-Fickian behavior during a cycle, the net contribution of each cycle to the spreading relaxes to a Fickian process in a similar way as for unidirectional flow. The cycle time averaged dispersion coefficient is a monotonically increasing function of the reversal time. It asymptotically converges towards the effective dispersion coefficient in the absence of flow reversal.

Longitudinal and transverse mixing in rotary kilns: A discrete element method approach

Longitudinal and transverse mixing in horizontal rotary kilns has been studied, using a three-dimensional Discrete Element Method approach. The focus is on the effect of the main operating conditions, i.e., the filling degree and the rotational speed of the rotary kiln, on quantitative measures of longitudinal and transverse mixing. Discrete Element Method simulations have been performed for various values of the main operating conditions. Flow regimes have been determined from visualisations of the simulations. It is shown that the mixing in the longitudinal direction can be described by a diffusion equation. The dependence of the diffusion coefficient on the operating conditions has been determined from the results of the simulations. It is found that the diffusion coefficient increases linearly with rotational speed, while the influence of the filling degree is relatively small. The mixing in the transverse plane is quantified by an entropy-like mixing index. The results of the simulations show that this mixing index varies exponentially with the number of revolutions of the rotary kiln. The dependence on the operating conditions of the transverse mixing speed, as determined from the exponential behaviour of the mixing index, has been determined. This mixing speed decreases with increasing rotational speed and with increasing filling degree. The transverse mixing speed that has been determined from the results of additional two-dimensional simulations is generally larger than that observed in the corresponding three-dimensional simulations.

Higher order regularity and approximation of solutions to the Monod biodegradation model

In this work we continue analyzing mathematically the Monod model for biochemically reacting contaminant transport in the subsurface with regard to higher order regularity of solutions as well as further establishing a higher order approximation scheme in terms of an a priori error analysis relying on the regularity results. Recently, the existence and uniqueness of nonnegative global strong solutions to the Monod model was proved by Merz in [Adv. Math. Sci. (2005), in press]. The approximation scheme, based on higher order finite element methods and backward differentiation formulae, was suggested and analyzed numerically by Bause in [Comput. Visual. Sci 7 (2004) 61; M. Feistauer et al. (Eds.), Numerical Mathematics and Advanced Applications, ENUMATH 2003, Springer, 2004, pp. 112–122]. It was successfully applied to test problems of the literature (cf. [Internat. J. Numer. Methods Fluids 40 (2002) 79; IMA J. Numer. Anal. 22 (2002) 253]) as well as to complex scenarios with an additional calculation of the flow field by solving the parabolic-elliptic degenerate Richards equation; cf. [Comput. Visual. Sci. 7 (2004) 61]. The higher order approach has shown to reduce significantly the amount of inherent numerical diffusion compared to lower order ones. Thereby an artificial transverse mixing of the species leading to a strong overestimation of the biodegradation process and wrong prediction is avoided. To illustrate our approach, in this work the movement and expansion of a BTEX plume is studied for a “real world” field scale site.

Manipulating Head Loss to Improve Disinfection Efficiency of UV Reactors

Manipulating Head Loss to Improve Disinfection Efficiency of UV ReactorsThe ideal hydraulic behavior for UV reactors is generally considered to be near plug-flow with effective lateral or transverse mixing. Typically, head loss increases as the transverse mixing increases and vice versa. Accordingly, head loss is a critical factor in determining a UV disinfection system's design application. Head loss is typically caused by friction and turbulence created as water...Author(s)Robert KellyBruno FerranShanshan JinSourceProceedings of the Water Environment FederationSubjectPostersDocument typeConference PaperPublisherWater Environment FederationPrint publication date Jan, 2005ISSN1938-6478SICI1938-6478(20050101)2005:1L.1010;1-DOI10.2175/193864705783978375Volume / Issue2005 / 1Content sourceDisinfection and Reuse SymposiumFirst / last page(s)1010 - 1021Copyright2005Word count318

On the Relationship of CMP Wafer Nanotopography to Groove-Scale Slurry Transport

AbstractMaterial removal in CMP occurs during intervals of pad-wafer contact separated by intervals of non-contact. One predictable sequence of non-contact intervals for a fixed point on the wafer is the traverse of the pad grooves, during which the wafer surface is renewed with fresh chemistry and heat is conveyed away. It is well understood that good uniformity requires machine kinematics that expose all points on the wafer to the same total contact time, mean slurry concentration, and temperature. Less widely known is that coherent structures tens to hundreds of nanometers high and matching the pitch of the pad groove pattern may be formed on an otherwise planar wafer despite multiple rotary motions. This unexpected phenomena is of interest not only because it manifests the impact of grooves and transport at scales not easily studied, but also because shrinking device architectures will ultimately disqualify even nanoscale departures from planarity. Wafer polish experiments are conducted alternately using circular, Cartesian grid, and spiral groove patterns using specialized pad conditioning and CMP recipes to amplify groove-induced nanotopography. Polish results illustrate sharp patterns in finished wafers (visible to the naked eye) that should not survive dual-axis tool kinematics. Computational 3-D model results are then presented for transient slurry mixing in the pad-wafer gap of a 200-mm polisher using the same groove patterns. A direct correspondence is found between observed wafer nanotopography and predicted groove-scale slurry mixing dynamics. In particular, the surface structures are underpolished areas traceable to intervals of non-contact protracted by depleted polish chemistry that prevails in groove segments when oriented relative to the local pad and wafer motion in a way that suspends transverse mixing in the groove crosssection. The study conclusively defines the features required in a groove pattern and polish recipe to form coherent structures matching the groove pitch. As validation of the theory, a groove pattern expected to form no surface topography is defined, experimentally tested, and shown to perform as predicted. Findings are discussed in the context of next-generation pad grooving and texturing as required for progressively more demanding applications of CMP.

MIXING CHARACTERISTICS IN TIDAL PORTION OF TAKASE RIVER

Takase River in the southeast of Aomori Prefecture is a tidal river with the compound channel. Filed measurement and laboratory experiments were carried out in order to understand mixing characteristics of river estuary in this paper. Salinities of Lake Ogawara were able to estimate from field data. Furthermore, experiments were performed in compound channel with the effects of narrow section. The transverse mixing coefficients were obtained from these results using diffusion equation, and compared with those in compound channel. It was showed that the mixing coefficients were related to the product density current parameter and cross-section of channel parameter. Therefore, it was found that Takase River estuary was influenced by compound channel from these results.

Mixing in microchannels based on hydrodynamic focusing and time-interleaved segmentation: modelling and experiment

This paper theoretically and experimentally investigates a micromixer based on combined hydrodynamic focusing and time-interleaved segmentation. Both hydrodynamic focusing and time-interleaved segmentation are used in the present study to reduce mixing path, to shorten mixing time, and to enhance mixing quality. While hydrodynamic focusing reduces the transversal mixing path, time-interleaved sequential segmentation shortens the axial mixing path. With the same viscosity in the different streams, the focused width can be adjusted by the flow rate ratio. The axial mixing path or the segment length can be controlled by the switching frequency and the mean velocity of the flow. Mixing ratio can be controlled by both flow rate ratio and pulse width modulation of the switching signal. This paper first presents a time-dependent two-dimensional analytical model for the mixing concept. The model considers an arbitrary mixing ratio between solute and solvent as well as the axial Taylor-Aris dispersion. A micromixer was designed and fabricated based on lamination of four polymer layers. The layers were machined using a CO2 laser. Time-interleaved segmentation was realized by two piezoelectric valves. The sheath streams for hydrodynamic focusing are introduced through the other two inlets. A special measurement set-up was designed with synchronization of the mixer's switching signal and the camera's trigger signal. The set-up allows a relatively slow and low-resolution CCD camera to freeze and to capture a large transient concentration field. The concentration profile along the mixing channel agrees qualitatively well with the analytical model. The analytical model and the device promise to be suitable tools for studying Taylor-Aris dispersion near the entrance of a flat microchannel.

Effects of Wall Roughness on Particle Velocities in a Turbulent Channel Flow

Abstract Experimental measurements using a laser Doppler anemometer (LDA) system have been performed on 150μm dense glass particles in a fully developed downward channel flow in air. Tests were conducted in smooth, rough development, and fully rough wall conditions with a channel Reynolds number of 13,800, corresponding to a centerline gas phase velocity of 10.5m∕s with a dilute loading of particles of 15% by mass fraction. Velocities were measured and statistics compared to see the nature of the effects of the wall roughness in a rebuilt channel facility originally used for important works including Kulick, Fessler, and Eaton, (1994, “Particle Response and Turbulence Modification in Fully-Developed Channel flow,” J. Fluid Mech., 277, pp. 109–134) and Paris (2001, “Turbulence Attenuation in a Particle-Laden Channel Flow,” Ph.D. thesis, Stanford University, Stanford, CA). Wall roughness has a substantial impact on gas phase mean velocities across most of the channel width, except very near the wall. The turbulence intensity of the gas phase is enhanced across the entire channel in the presence of fully rough walls. The rough walls have an even greater impact on the particle phase. Streamwise particle velocities are reduced up to 40%, and become quite uniform across the channel. Particle fluctuating velocities are nearly doubled near the channel centerplane. Profiles appear uniform, due in large part to strong transverse mixing induced by particle-wall collisions. Much of the data of Kulick and Paris is shown here to be strongly influenced by wall conditions with poorly defined roughness in the development region, followed by rapid flow recovery in a relatively smooth test section.

Buoyant mixing of miscible fluids in tilted tubes

Buoyant mixing of two fluids in tubes is studied experimentally as a function of the tilt angle θ from vertical, the density contrast and the common viscosity μ. At high contrasts and low θ, longitudinal mixing is macroscopically diffusive, with a diffusivity D increasing strongly with θ and μ. At lower contrasts and higher θ, a counterflow of the two fluids with little transverse mixing sets in. The transition occurs at an angle increasing with density contrast and decreasing with μ. These results are discussed in terms of the dependence of transerse mixing on θ and an analogy with the Boycott effect.

Numerical simulation of contaminant biodegradation by higher order methods and adaptive time stepping

In this work we present and analyze a reliable and robust approximation scheme for biochemically reacting transport in the subsurface following Monod type kinetics. Water flow is modeled by the Richards equation. The proposed scheme is based on higher order finite element methods for the spatial discretization and the two step backward differentiation formula for the temporal one. The resulting nonlinear algebraic systems of equations are solved by a damped version of Newton's method. For the linear problems of the Newton iteration Krylov space methods are used. In computational experiments conducted for realistic subsurface (groundwater) contamination scenarios we show that the higher order approximation scheme significantly reduces the amount of inherent numerical diffusion compared to lower order ones. Thereby an artificial transverse mixing of the species leading to a strong overestimation of the biodegradation process is avoided. Finally, we present a robust adaptive time stepping technique for the coupled flow and transport problem which allows efficient long-term predictions of biodegradation processes.

Sensitivity enhancement in two-dimensional solid-state NMR spectroscopy by transverse mixing.

The sensitivity of two-dimensional (2D) 13C-13C solid-state NMR spectroscopy under magic-angle spinning (MAS) is shown to be enhanced by the use of transverse polarization transfer in place of the conventional longitudinal polarization transfer. Experimental results are reported for 2D spectroscopy of a 20-residue, filament-forming peptide derived from the E. Coli RecA protein, containing five uniformly 13C-labeled residues, performed at 14.1 T with high-speed MAS and with finite-pulse radio-frequency-driven recoupling of dipolar interactions in the mixing period. Significant sensitivity enhancements observed at short mixing periods results from a more rapid build-up of cross-peaks under transverse mixing than under longitudinal mixing and from the 2 gain inherent in 2D measurements in which both orthogonal transverse polarization components in the t1 period contribute to each free-induction decay signal detected in the t2 period.

A particle-tracking model for simulating pollutant dispersion in the Strait of Gibraltar

A particle-tracking model to simulate the dispersion of contaminants in the Strait of Gibraltar has been developed. The model solves the hydrodynamic equations off-line and tidal analysis is carried out to determine tidal constants for the two main constituents. Tidal constants and residuals are stored in files that are read by the dispersion model. A lagrangian approach is used to solve dispersion; diffusion and decay are simulated by a Monte Carlo method. A method for assessing the areas of the Strait with higher probability of being affected by contamination occurring after an accident in the shipping routes is given. Generally speaking, the fate of a pollutant discharge strongly depends on wind conditions. Winds from the east tend to retain contamination into the Strait. As a consequence, transverse mixing occurs and both Spain and Morocco coasts are affected by contamination. Under calm conditions and west winds, contaminants are flushed out of the Strait faster and transverse mixing does not occur. Thus, only part of Morocco coast has a higher probability of being affected by contamination.

Pore-scale analysis of anaerobic halorespiring bacterial growth along the transverse mixing zone of an etched silicon pore network.

The anaerobic halorespiring microorganism, Sulfurospirillum multivorans, was observed in the pore structure of an etched silicon wafer to determine how flow hydrodynamics and mass transfer limitations along a transverse mixing zone affect biomass growth. Tetrachloroethene (PCE, an electron acceptor, 0.2 mM) and lactate (an electron donor, 2 mM) were introduced as two separate and parallel streams that mixed along a reaction line in the pore structure. The first visible biomass occupied a single line of pores in the direction of flow, a few pore bodies from the micromodel centerline. This growth was initially present as small aggregates; over time, these grew and fused to form finger-like structures with one end attached to downgradient ends of the silicon posts and the other end extending into pore bodies in the direction of flow. Biomass did not grow in pore throats as expected, presumably because shear forces were not favorable. Over the next few weeks, the line of growth migrated upward into the PCE zone and extended over a width of up to five pore spaces. When the PCE concentration was increased to 0.5 mM, the microbial biomass increased and growth migrated down toward the lactate side of the micromodel. A new analytical model was developed and used to demonstrate that transverse hydrodynamic dispersion likely caused the biomass to move in the direction observed when the PCE concentration was changed. The model was unable, however, to explain why growth migrated upward when the PCE concentration was initially constant. We postulate that this occurred because PCE, not lactate, sorbed to biofilm components and that biomass on the lactate side of the micromodel was limited in PCE. A fluorescent tracer experiment showed that biomass growth changed the water flow paths, creating a higher velocity zone in the PCE half of the micromodel. These results contribute to our understanding of biofilm growth and will help in the development of new models to describe this complex process.

Solute movement through an allophanic soil.

Allophanic soils are widespread around the world, but little research has been done on their transport properties. This study reveals the effect of two soil water potential heads and two water-flow regimes of continuous and intermittent flow on solute transport through undisturbed soil columns of Horotiu silt loam (Typic Hapludand), an allophanic soil. Two different methods--breakthrough curves (BTCs) and time domain reflectometry (TDR)--were employed to determine the extent of preferential solute transport in the topsoil. The TDR data were also used to look at the depth dependence of the transport properties. The convection-dispersion equation (CDE) with the appropriate boundary conditions adequately described the movement of both Br and Cl under the various flow conditions. Although no preferential flow was found under the imposed unsaturated flow conditions, the flow of water and transport of solute became more uniform with depth. The results show that both Br and Cl are retarded in this allophanic soil. Retardation values range from 1.5 to 1.9, and, as the TDR data showed, increase from the depth of 5.0 to 10.0 cm. Intermittent leaching results showed that there was no effect on solute concentrations in the leachate following no-flow periods. This suggests that water and solute transport in this soil were either relatively uniform or that transverse mixing during flow was already fast enough to eliminate concentration gradients between regions of different "mobility."

Effective dispersion in heterogeneous media under random transient flow conditions

Under steady state flow conditions, solute dispersion in heterogeneous porous media is much smaller in the transverse than in the longitudinal direction. This holds particularly for effective dispersion of a plume originating from a point‐like injection. The effective dispersion coefficient describes the actual dispersive mixing of solutes in the aquifer. The lack of dispersive transverse mixing may limit considerably natural attenuation of certain contaminants. Temporal fluctuations of the flow direction enhance horizontal transverse dispersion. This has been shown previously for uniform flow and for macrodispersion in stationary media. We present a linear stochastic theory for effective dispersion under quasi‐steady state flow conditions with random temporal fluctuations of the mean flow direction. As for macrodispersion, effective transverse dispersion proves to be dominated by transient flow effects. We compare semianalytical results derived from linear theory to those from particle‐tracking random‐walk simulations for a three‐dimensional test case. The parameters of the test case are similar to those obtained at the Borden site, where the mean transverse flow component fluctuated approximately by ±10°. Linear theory and particle simulations agree well.

사행하천에서 횡혼합에 관한 추적자 실험

사행이 있는 자연하천에서 횡혼합 특성을 분석하고 종ㆍ횡분산게수를 산정하기 위해 현장실험을 실시하였다. 이를 위해 남한강 지류인 섬강 및 청미천에서 사행구간을 선정하여 유속과 수심 측정 및 추적자실험을 수행하였다. 추적자로서는 미량으로도 관측이 용이한 방사성 동위원소를 사용하였으며 주입은 순간. 점주입 조건을 바랐다. 이를 통해 취득된 수리량 및 농도 자료를 이용하여 종ㆍ횡분산계수를 산정하였고, 이들을 기존의 추정식에 의한 결과와 비교, 분석하였다. 종분산계수는 해석해를 통해 산정하였는데 섬강은 약0.5 <TEX>$m^2$</TEX>/s, 청미천은 0.2 <TEX>$m^2$</TEX>/s 로 산정되었다. 횡분산계수는 해석해 및 모멘트방법을 이용하여 산정하였는데, 섬강은 약 0.01-0.06 <TEX>$m^2$</TEX>/s의 범위를 가지며 청미천은 약 0.0l-0.05 <TEX>$m^2$</TEX>/s의 범위를 갖는 것으로 산정되었다. Field tests were conducted to investigate characteristics of the transverse mixing and to evaluate the dispersion coefficients in the meandering natural streams. The Sum River and the Cheong-mi Creek, tributaries of Han River, were selected as the test site, and measurements of the hydraulic and dispersion data were performed. In the tracer tests, the radioisotope was used as a tracer and injected into a flow on the instantaneous point source. Using the measured data, the longitudinal and transverse dispersion coefficients were evaluated and compared with the previous studies. The longitudinal dispersion coefficients, which were evaluated by application of the analytical solution, were about 0.5 <TEX>$m^2$</TEX>/s at the Sum River and 0.2 <TEX>$m^2$</TEX>/s at the Cheong -mi Creek. The transverse dispersion coefficients, which were evaluated by the analytical solution and the moment method, were ranging from 0.01 to 0.06 <TEX>$m^2$</TEX>/s for the Sum River and from 0.01 to 0.05 <TEX>$m^2$</TEX>/s for the Cheong-mi Creek.

Transverse mixing in sinuous natural open channel flows

Accurate modelling of transport and mixing of solutes within natural watercourses is vital for environmental management. A lack of understanding of the dominant processes, particularly quantification of transverse mixing processes in natural channels, severely limits the accuracy and usefulness of such modelling. This paper presents results from tracer and hydrodynamic studies conducted on a 'natural' channel form under laboratory conditions. The measurements are analysed using the standard method of moments and the variation of the resulting mixing coefficients examined with respect to the variations in the natural channel geometry and hydrodynamic measurements. Normalisation of the resulting transverse mixing coefficients is investigated. The work conclusively shows that the transverse mixing coefficient varies in direct relation to channel curvature, the variation is cyclic with meander geometry.

Analysis and Prediction of Transverse Mixing Coefficients in Natural Channels

An experimental study has been undertaken using a naturally formed meandering channel to obtain unique tracer and hydrodynamic data. The velocity data presented are from laser Doppler anemometer measurements; tracer data were collected using an array of fluorometers in continuous flow through mode. Techniques for the prediction of the primary and secondary velocity flow fields are explored, and shown to be accurate. Analysis of the tracer data by the generalized method of moments using the cumulative transverse discharge approach is undertaken. The coefficient of transverse mixing is shown to exhibit considerable longitudinal variation over the meander cycle. A new integrated approach for predicting transverse mixing coefficients is developed and explored and has been validated against the data set. This approach requires only three parameters as input, namely, longitudinal planform curvature, cross-sectional shape, and total discharge, and has been shown capable of accurately predicting the longitudinal variation of the transverse mixing coefficient over the meander cycle.

Locating Outfalls on Meandering Channels to Optimise Transverse Mixing

AbstractMany sources of river pollution approximate steady‐state conditions and, under such conditions, the rate of transverse mixing is critical in determining the impact of pollutants. Results are presented from transverse‐mixing experiments which were carried out on a large‐scale laboratory channel with meander planform geometry of natural cross‐section, generated by flow over a mobile bed. Dye‐tracer measurements below three point sources within one cross‐section are presented and compared, together with hydrodynamic measurements. The results show the importance of locating outfalls to maximise mixing rates, hence minimising pollution impact (depending on the environmental need). A release on the outside of a bend is shown to result in a faster rate of transverse mixing than a release on the inside of a bend.

Interaction between water flow and spatial distribution of microbial growth in a two-dimensional flow field in saturated porous media

Bacterial growth and its interaction with water flow was investigated in a two-dimensional flow field in a saturated porous medium. A flow cell (56×44×1 cm) was filled with glass beads and operated under a continuous flow of a mineral medium containing nitrate as electron acceptor. A glucose solution was injected through an injection port, simulating a point source contamination. Visible light transmission was used to observe the distribution of the growing biomass and water flow during the experiment. At the end of the experiment (on day 31), porous medium samples were destructively collected and analyzed for abundance of total and active bacterial cells, bacterial cell volume and concentration of polysaccharides and proteins. Microbial growth was observed in two stripes along the length of the flow cell, starting at the glucose injection port, where highest biomass concentrations were obtained. The spatial distribution of biomass indicated that microbial activity was limited by transverse mixing between glucose and nitrate media, as only in the mixing zone between the media high biological activities were achieved. The ability of the biomass to change the flow pattern in the flow cell was observed, indicating that the biomass was locally reducing the hydraulic conductivity of the porous medium. This bioclogging effect became evident when the injection of the glucose solution was turned off and water flow still bypassed the area around the glucose injection port, preserving the flow pattern as it was during the injection of the glucose solution. As flow bypass was possible in this system, the average hydraulic properties of the flow cell were not affected by the produced biomass. Even in the vicinity of the injection port, the total volume of the bacterial cells remained below 0.01% of the pore space and was unlikely to be responsible for the bioclogging. However, the bacteria produced large amounts of extracellular polymeric substances (EPS), which likely caused the observed bioclogging effects.