Longitudinal Mixing Research Articles

Modelling counter-current chromatography: a chemical engineering perspective

In conventional chromatography, a solute is usually viewed to be longitudinally transported only in the mobile phase, remaining longitudinally motionless in the stationary phase. In counter-current chromatography, both phases undergo intense mixing in the variable force field of a coil planet centrifuge and longitudinal dispersion of matter in the stationary phase is not to be excluded. To take into account longitudinal mixing in both phases, a cell model of chromatographic process is proposed in which the number of perfectly mixed cells n is determined by the rates of mixing in stationary ( D S) and mobile ( D m) phases by the equation n= LF/(2A c D m ) (1+S f (λ−1)) with λ= K D D S/ D m ( F, L, A c and K D are the mobile phase flow-rate, column length, column cross-section and distribution ratio, respectively). This equation has been derived by comparing the discontinuous cell model with continuous diffusion assuming equilibrium conditions. Parameter determination and their relationships are discussed.

Experiments on chaotic mixing in a screw channel flow

AbstractMixing patterns of a passive dye were investigated experimentally in an unwound screw channel flow with periodic barriers that serve to induce a chaotic flow. Continuous cross‐sectional and longitudinal mixing patterns were observed as a function of the barrier fraction and were interpreted in terms of dynamical systems theory, along with 3‐D numerical simulations. Observations include: periodically invariant cross‐sectional mixing patterns due to the resonance bands; pile‐up of dye streaks caused by stretching and folding of manifolds near hyperbolic points; unmixed zones due to KAM tori; longitudinal deformation patterns with exit time distributions; and effects of the barrier fraction on the size of unmixed islands, the thickness of the band of dye streaks, and the distribution of exit times.

Choice of dispersion coefficients in reactive transport calculations on smoothed fields

Local dispersion dominates the mixing of compounds that are introduced separately into the subsurface and do not partition into any other than the aqueous phase. Thus, reactions between these compounds are controlled by dispersive mixing if they are not limited by kinetics. I quantify longitudinal dispersive mixing by the longitudinal effective dispersion coefficient of a conservative tracer introduced by a point-like injection [Water Resour. Res. 36 (12) (2000) 3591–3604]. In the upscaling of mixing-controlled reactive transport, I apply the mean velocity and the effective dispersion coefficient to the macroscopic transport calculations, whereas the reactive parameters on the macro-scale are identical to those on the local scale. The applicability of the approach is demonstrated for the transport of compounds undergoing a second-order irreversible bimolecular reaction. Ten realizations of a two-dimensional heterogeneous log-conductivity field are considered. Using the effective dispersion parameters, the overall mass balance is met in the ensemble average, whereas solute spreading is underestimated. I assess the lack of spreading by the difference between the expected macrodispersion and effective dispersion coefficients. I extend the approach to simulations on log-conductivity fields obtained by kriging of regularly spaced conductivity measurements. These fields contain the large-scale features of the true fields but do not resolve the small-scale variability. For the calculations on the kriged fields, the corresponding conditional covariance is substituted into the analytical expressions of effective dispersion, yielding a correction effective dispersion coefficient. The comparison between simulations on the fully resolved fields and on the kriged fields indicates that the approach is valid for wide plumes meeting the ergodicity condition. The high variability of mixing on small scales unresolved by kriging, however, leads to severe uncertainty when mixing-controlled reactions are predicted for narrow plumes.

Mixing of Particles in a Circulating Fluidized Bed

A phenomenological model of longitudinal mixing of particles in a circulating fluidized bed is formulated. The model allows for the main features of the process: ascending motion of particles in the core of the bed and their descending motion in the annular zone (internal circulation of the solid phase); considerable changes in the concentration of particles and in the values of the ascending and descending zones over the bed height; external circulation of the solid phase and the effect of the near‐bottom fluidized bed on the process as a whole. The validity of the initial proposition is confirmed by comparison of calculated and experimental curves of mixing.

POLY(ETHYLENE GLYCOL)-NONYLPHENYL ETHER EXTRACTION WITH SUPERCRITICAL CARBON DIOXIDE IN A PACKED COLUMN

A semibatch extraction of poly(ethylene glycol)nonylphenyl ether (PEG-NPE) with supercritical carbon dioxide was studied at 20,0 MPa, and 323 K. The process was carried out in a stainless steel column, filled with inert packing, wetted by the polymer. Three representative polymer fractions (degree of polymerisation 40, 50, and 60) and the total polymer amount at the column exit were monitored for a period of 4 h. Experimental data obtained were used for evaluation of process parameters, applying a model that considers the mass transfer on both phases and the longitudinal mixing in the solvent-rich phase, continuously flowing through the packed column. It was found that for the studied conditions the longitudinal mixing is very pronounced, but the active contact area is strongly reduced, supposedly as a result of solvent-rich phase channeling. After the transient initial period, the overall mass-transfer coefficient is controlled by the component diffusivity in the polymer-rich phase.

Predicting River Travel Time from Hydraulic Characteristics

Predicting the effect of a pollutant spill on downstream water quality is primarily dependent on the water velocity, longitudinal mixing, and chemical/physical reactions. Of these, velocity is the most important and difficult to predict. This paper provides guidance on extrapolating travel-time information from one within bank discharge to another. In many cases, a time series of discharge (such as provided by a U.S. Geological Survey stream gauge) will provide an excellent basis for this extrapolation. Otherwise, the accuracy of a travel-time extrapolation based on a resistance equation can be greatly improved by assuming the total flow area is composed of two parts, an active and an inactive area. For 60 reaches of 12 rivers with slopes greater than about 0.0002, travel times could be predicted to within about 10% by computing the active flow area using the Manning equation with n = 0.035 and assuming a constant inactive area for each reach. The predicted travel times were not very sensitive to the assu...

Distributive mixing in a single‐screw extruder—evaluation in the flow direction

AbstractThis study investigates distributive mixing in the flow direction for a single‐screw extruder. WIth a custom‐designed transparent extruder, an Image Analysis System, and a newly defined parameter, i.e., distribution index, the distribution mechanism is thoroughly examined with respect to various processing conditions or screw designs. Experimental results indicate that the longitudinal distribution can be enhanced with an increasing RPM, a longer metering section, or a decreasing diameter of the die. However, a plateau region occurs when an optimum condition exists for the RPM and the length of the metering section. In addition, an extruder modified with a barrier, pin‐elements, or high helix angle performs better in the longitudinal mixing than the conventional one. Our results further demonstrate that leakage flow significantly enhances mixing in the flow direction.

A mathematical model of the process of foamless bubble fractionation of substances using large gas bubbles

The process of bubble separation of substances using large gas bubbles totally closing the cross section of the column was investigated. The use of such bubbles decreases the longitudinal mixing of the solution and drastically increases the separation efficiency. A mathematical model was proposed for describing this process in the batch and continuous operation conditions. The model was tested experimentally.

Double polarization experiments at intermediate energy

At modern electron accelerators with highly polarized, intense, high duty factor beams double polarization coincidence experiments became feasible with good statistical accuracy. The strong potential towards the precise determination of small nucleon structure quantities is illustrated by two recent examples from MAMI. The measurement of $G_E^n$ in the quasifree reaction $D(\vec e, e'\vec n)p$ lead to a new parametrization of $G_E^n$ which is significantly above the previously preferred one from elastic $e-D$ scattering. A $p(\vec e, e'\vec p)\pi^0$ experiment at the energy of the $\Delta$ resonance yields preliminary results for the longitudinal quadrupole mixing. Both experimental errors and model uncertainties are complementary to unpolarized measurements.

Complex chemical reactions — A review

Complex responses have been observed in a wide range of chemical and engineering systems: In well-stirred and unstirred solution-phase, in gels, in heterogeneous catalysis and dissolution reactions, in gas- and solid-phase combustion, widely in biological systems, accompanying phase transitions, in atmospheric kinetics and even in interstellar dust clouds. Complex behaviour does not, however, imply a necessarily complex underlying chemical mechanism. The appropriate feedback mechanisms, built on chain-branching, autocatalysis or self-heating, arise quite commonly in Nature. In many instances, it is appropriate first to attempt to find reduced mechanisms giving a semi-quantitative fit to observed responses. An example of such an approach applied to a model for complex oscillations of species concentrations in the mesosphere is presented, and the reduced model used to investigate the effects of longitudinal mixing in this system. Once a general understanding of a particular reaction system has been obtained, more detailed questions can be addressed. This is exemplified by a study of the development of three-dimensional scroll waves in the Belousov–Zhabotinsky reaction. Such detailed information for a specific reaction also informs the general understanding of the class of `excitable media'.

Estuarine Adjustment to Changes in River Flow and Tidal Mixing

The adjustment of estuarine circulation and density to changes in river flow and tidal mixing is investigated using analytical and numerical models. Tidally averaged momentum and salinity equations in a rectangular estuary are vertically averaged over two levels, resulting in equations that are analytically tractable while retaining a broad range of time-dependent behavior. It is found that both strongly stratified and well-mixed estuaries respond rapidly to either type of forcing change, while those of intermediate stratification respond more slowly. Intermediate estuaries also have the greatest sensitivity to change. Exchange flow dominates the up-estuary salt flux in strongly stratified cases. Changing the river flow in such cases leads to an internal wave propagating the length of the estuary, which accomplishes much of the adjustment. The internal wave speed thus controls the adjustment time. Increased tidal mixing in strongly stratified cases initially decreases the exchange flow contribution to up-estuary salt flux by decreasing both the stratification and the vertical current shear. However, the decreased up-estuary salt flux leads to a loss of total salt in the estuary, and hence a greater longitudinal salinity gradient. The increasing gradient eventually restores the exchange-flow salt flux to near its original value. Well-mixed solutions have an advective–diffusive balance between river flow and longitudinal tidal mixing. In these cases the adjustment time corresponds to the time it takes the depth-averaged flow to travel the length scale of the salt intrusion, a result that applies to both types of changes considered. In all cases the adjustment depends upon the dynamical feedback between the longitudinal salt flux and the longitudinal salinity gradient, which varies as the estuary gains or loses total salt.

Relation entre les coefficients de m´lange longitudinal et transversal dans des cours d'eau naturels

This paper proposes a new equation for predicting the transversal mixing coefficient giving the mean values of the longitudinal mixing coefficient, the quantity of flow, the depth of flow and the ratio width to depth of the river. This equation is developped from the relation between the transversal and the longitudinal coefficient proposed by the authors in 1994 taking into account all the data found in litterature of all the mixing coefficients (transversal and longitudinal) measured simultanously with the same hydraulic conditions. The proposed formula expresses the transversal coefficient as a fonction of the longitudinal coefficient. This new formula is more representative of the experimental results and confirms the trend of recent years to use the product of coefficients (Kl Kt rather than Kl/Kt) to describe the interaction between the transversal and the longitudinal mixing coefficients.

MATHEMATICAL MODELLING AND APPARATUS ARRANGEMENTS OF DEEP DRYING PROCESS OF GRANULAR POLYMERS

ABSTRACT A computer - aided analysis is made of the factors influencing the kinetics of the continuous deep drying of granular polymers on the basis of a two-level mathematical model. In terms of the lower (microkinetic) level, which deals with the drying of separate granules, consideration is given to the accuracy of determining the moisture diffusion coefficients in polymers as functions of temperature and concentration D=f(U,t). The expedience of employing the solution of linear and non-linear differential diffusion equations in microkinetic calculations is also considered. In terms of the macrokinetic level, which takes account of the influence of the hydrodynamic and heat-and-mass transfer drying conditions in the dryer, a numerical study is undertaken of how the kinetics is influenced by the granules’ dimension nonuniformity and by the difference in the granules’ residence time in the dryer. The results are graphically presented in generalised variables. The travel of 12 granular polymers in the shaft (column) dryers is experimentally studied depending on definite determining parameters. Accordingly, some recommendations aimed at ensuring the uniform drying are given. Presented are also the experimental results on the longitudinal mixing for different polymers for a continuous shaft dryer with a ring material layer and with the drying agent performing the radial motion. The findings make it possible to determine the mixing rate and to outline some practical recommendations. The investigations carried out allow one to find the rational construction of the dryer for the deep drying of granular polymers and to refine the kinetic calculations.

Plankton development in a rapidly flushed lake in the River Spree system (Neuendorfer See, Northeast Germany)

The Neuendorfer See is a shallow lake through which the River Spree flows. In the lake, abundances of zooplankton (mainly rotifers) and phytoplankton increased exponentially over the flowing stretch between the inflow and the outflow of the river. Increase in space is enhanced by increased residence time in the lake. Sensitivity of plankton development on residence times of >8 days weakened at longer residence times. This pattern holds for all zooplankton species. Rates of change in chlorophyll and rotifers (after correction for temperature) are first enhanced, then are decreased at longer residence times. Concentrations of dissolved inorganic nutrients [Si, soluble reactive phosphorus (SRP) and NO 3 −] and the N/P ratio diminished with increasing residence time. The plankton dynamics in the lake were dependent on discharge and the lake behaved as a ‘plug flow tubular reactor’ with minor longitudinal mixing. This was not different from the transportation in large rivers.

Flow-rate dependence of the height equivalent to a theoretical plate in nitrogen isotope separation by displacement chromatography

The effect of flow-rate in nitrogen isotope separation was studied for a wide range of fluid velocities using porous microreticular cation-exchange resin. The displacement chromatography was conducted from a band velocity of 6.3×10 −3 to 12.5 cm/min (or Reynolds number of 15–30 000). A sharp adsorbed ammonium band was maintained even at a band velocity of 12.5 cm/min; on the other hand, tilted and/or channeled boundaries of the band are enlarged in the low-flow-rate region. The effect of fluid velocity on the height equivalent to a theoretical plate (HETP) of isotope separation was evaluated in terms of Reynolds number. In the present work, the HETP is proportional to the square-root of the band velocity in the turbulent flow regime and inversely proportional to the fluid velocity in the laminar flow regime due to the longitudinal mixing which is caused by tilting and/or channeling at the band boundary.

Simple model of a transport/diffusion system

The paper presents a simple transfer function model of a diffusive flow system represented by a narrow pipe with longitudinal mixing. This new model is composed of three coupled transfer functions describing the dynamics of flow, delay and diffusion, respectively. Unlike previous models, the contributions of delay and diffusion are kept separate and both depend on the hydraulic submodel. As a result, good delay approximation is achieved even with a very low-order model. The model development is justified by experimental evidence through a laboratory set-up, used to obtain experimental data and perform model testing. To determine the ‘best’ model order, the model structure is assessed from the parameter redundancy point of view and parameter calibration results are presented before model calibration.

Stochastic Theory for Irregular Stream Modeling. II: Solute Transport

A stochastic theory is developed for longitudinal dispersion in natural streams. Irregular variations in river width and bed elevation are conveniently represented as one-dimensional random fields. Longitudinal solute migration is described by a one-dimensional stochastic solute transport equation. When boundary variations are small and statistically homogeneous, the stochastic transport equation is solved in closed-form using a stochastic spectral technique. The results show that large scale longitudinal transport can be represented as a gradient dispersion process described by an effective longitudinal dispersion coefficient. The effective coefficient reflects longitudinal mixing due to flow variation both within the river cross section and along the flow and can be considerably greater than that of corresponding uniform channels. The discrepancy between uniform channels and natural rivers increases as the variances of river width and bed elevation increase, especially when the mean flow Froude number is high.

Effect of biosorption on the damping of influent fluctuation in activated sludge aeration tanks

To describe the organic removal performance in an activated sludge process, the effect of biological adsorption (biosorption, hereinafter) of both soluble and particulate organic substances (SOS and POS, respectively, hereinafter) on the rate of organic removal must be taken into account. This study quantitatively investigated the biosorption of organic pollutants in the domestic wastewater by the activated sludge. The capability of biosorption and the rate of bio-oxidation of organic pollutants were formulated based on the experimental results as functions of both COD concentration and temperature. The biosorption capacity of activated sludge was proportional to COD concentration in the mixed liquor originated from the influent. The kinetic expressions were combined with the equation of longitudinal liquid mixing to set up the mathematical model to predict the effluent COD, and thus to clarify the effect of biosorption on the damping of influent fluctuations of both COD concentration and the inflow rate. The model was also successfully applied to assessing the effectiveness of MLSS control strategies in equalizing the effluent water quality.

Effect of biosorption on the damping of influent fluctuation in activated sludge aeration tanks

To describe the organic removal performance in an activated sludge process, the effect of biological adsorption (biosorption, hereinafter) of both soluble and particulate organic substances (SOS and POS, respectively, hereinafter) on the rate of organic removal must be taken into account. This study quantitatively investigated the biosorption of organic pollutants in the domestic wastewater by the activated sludge. The capability of biosorption and the rate of bio-oxidation of organic pollutants were formulated based on the experimental results as functions of both COD concentration and temperature. The biosorption capacity of activated sludge was proportional to COD concentration in the mixed liquor originated from the influent. The kinetic expressions were combined with the equation of longitudinal liquid mixing to set up the mathematical model to predict the effluent COD, and thus to clarify the effect of biosorption on the damping of influent fluctuations of both COD concentration and the inflow rate. The model was also successfully applied to assessing the effectiveness of MLSS control strategies in equalizing the effluent water quality.

Directed TOCSY, a Method for Selection of Directed Correlations by Optimal Combinations of Isotropic and Longitudinal Mixing

The directed TOCSY pulse sequence element transfers coherence predominantly into “forward-directed” antiphase coherences while simultaneously suppressing in-phase and “backward-directed” antiphase coherences. This novel selection principle, based on the “direction” of the target coherences, provides a new approach for the simplification of crowded spectra. In this article, the theory of directed TOCSY is presented for linear spin systems that are frequently found in carbon-labeled biomolecules.