Mass transfer around prolate spheroidal drops in an extensional flow

Mass transfer in the continuous phase around a small eccentricity prolate spheroidal drop in an axisymmetric extensional creeping flow and at large Peclet numbers was investigated theoretically. The results show that, at very short times, the total quantity of solute transferred to or from the drop represents, at O(Ca¹), mass transfer by diffusion only around a sphere. For long times, or at steady‐state, the total quantity of solute transferred is, at O(Ca¹), slightly smaller than that of a spherical drop, and it decreases with an increase of the capillary number or the viscosity ratio. © 2012 Canadian Society for Chemical Engineering     

Potential Applications of Pulsed Flow for Minimizing Concentration Polarization in Ultrafiltration

Abstract

Various methods have been proposed to control or minimize concentration polarization and membrane fouling. A brief review of these alternatives is presented in this paper. Flow pulsation as a means of improving transmembrane flux has been studied experimentally by a few investigators. A mathematical model is developed to evaluate the performance of a tubular membrane module under oscillatory flow conditions. Besides the effect of osmotic pressure and axial pressure variation, the model considers the convective-diffusive mass transport without decoupling the momentum equation from the solute continuity equation. Model equations are solved by a finite difference method as part of an iterative solution. Model predictions of transmembrane flux with experimental data are found to be in good agreement. By flow pulsing, it is possible to improve the transmembrane flux by more than 68% at pulsing frequency of 60 cycles · min^?1. An analysis of extra power requirement for flow oscillation shows that the gain in transmembrane flux outweighs the cost of extra power, which is a minute fraction of the power required to maintain steady flow.     

Self‐sustained oscillations in opposed impinging jets in an enclosure

The flow of jets in confining enclosures has significant application in many engineering processes. In particular, the impingement of axisymmetric jets in a confined space has been examined using flow visualization, laser Doppler anemometry, and numerical simulations. Several flow regions were found; stable steady, regular oscillatory, and irregular oscillatory. Initially, a steady flow field existed for all arrangements for Re_d < ?90 (based on the nozzle diameter d, the fluid kinematic viscosity v and the volumetric flow rate Q through the nozzle (Q = πd²/4U_avg)) but subsequent increments in the fluid velocity caused a regularly oscillating flow field to emerge. The onset of the oscillations and the upper limit of finite oscillations were found to be a function of the Re_d, and the nozzle diameter to chamber dimension ratio. Steady numerical simulations predicted the steady flow field well and good agreement was obtained in unsteady simulations of the oscillating flow field. The oscillating flow field is considered to be a class of self‐sustaining oscillations where instabilities in the jet shear layer are amplified because of feed back from pressure disturbances in the impingement region.     

Liquid backmixing in oscillatory flow through a periodically constricted meso-tube

This paper deals with the measurement and modelling of axial liquid dispersion in a 4.5 mm internal diameter tube provided with smooth-periodic constrictions (meso-tube) in steady and oscillatory flow conditions. The residence time distribution (RTD) in the meso-tube was monitored for a range of fluid oscillation frequency (f) and amplitude (x₀) at laminar flow. The RTD response was modelled with three hydrodynamic models: (i) tanks-in-series, (ii) tanks-in-series with backflow and (iii) plug flow with axial dispersion. The steady flow through the meso-tube at flow rates up to 21.30 ml/min resulted in broad RTDs, mainly due to the parabolic velocity profile. The use of fluid oscillations allowed a fine-control of the axial liquid dispersion in the meso-tube due to generation of secondary flow in the regions between the constrictions. The axial dispersion coefficient D was reduced by up to 13-fold in comparison with the steady flow situation. Values of x₀ ≤ 1 mm and f = 10 Hz generally resulted in a maximum reduction in axial dispersion through, therefore maximum improvements in RTD. The tanks-in-series model was generally not capable of predicting RTDs in the meso-tube. The potential of this platform for the continuous, sustainable production of added-value products is herein demonstrated.     

An estimate of the axial dispersion during flow through a compressible wood‐chip bed

In this work, we present a methodology to characterise the axial dispersion of a solute during steady‐flow through a compressible bed of wood chips under mechanical load. We use a non‐invasive imaging technique, namely electrical resistance tomography (ERT), to visualise the uniaxial displacement of a salt solution. Here we demonstrate that under two limiting cases the porosity of the porous bed varies slowly in the flow‐direction and to the lowest order can be considered a constant. This simplified the optimisation routine we used to match the experimental data to the numerical results of the advection–diffusion equation. Using this, a methodology to estimate the axial dispersion is given by a minimisation scheme. In the experimental portion of the work we measure the axial dispersion of a bed of hemlock wood chips at three different kappa numbers κ, and compacted to three different compaction pressures p_c. We find that the dispersion D_e in the axial direction varies as a function of the porosity ε, according to , where a = 7.2 × 10^?4 m²/s and b = ?3.8 for hemlock wood chips. © 2011 Canadian Society for Chemical Engineering     

Flow characteristics of large oscillating drops in liquid-liquid systems

A study was made of the flow characteristics of large oscillating drops of pure liquid-liquid systems, using a thermostatically-controlled, rising drop column, 50 mm in diameter and 1000 mm in length. Mirrors in the jacket enabled front and side views of drops to be photographed simultaneously. Single drops in the size range 5–10 mm were investigated with both mutually-saturated phases and when the solute was being transferred from the dispersed phase. The systems studied were (1) toluene and acetone (dispersed)-water (continuous), and (2) n-heptane and acetone (dispersed)-water (continuous). Acetone concentrations were varied up to 3.75 kmol/m³. The oscillations of a travelling drop were asymmetrical; therefore, the amplitude cannot be expressed accurately in terms of only two axes. The area change of the drop compared to that of a sphere of equal volume ‘ε’, was shown to represent the amplitude accurately. The periods of droplet oscillation were uniform for the mutually saturated systems of constant physical and flow properties but changed when mass transfer was taking place. The interfacial tension exerted a marked effect on the amplitude, which also depended upon the oscillation frequency. The amplitude changed with droplet size in a similar manner to the terminal velocity, i.e. it increased with increasing size until it reached a maximum, subsequently decreasing less rapidly. The drag coefficient increased with increasing rate of mass transfer from the drop. Correlation of the results and the area eccentricity ‘ε’ by dimensional analysis embracing all possible parameters and physical properties affecting drop oscillation, resulted in the correlation ε = 0.22 Sr^0.42 We^?0.53 M^0.13 with a mean deviation of ± 14%. This will facilitate more accurate prediction of the interfacial area for mass transfer calculations, relating to equipment containing droplets in the oscillating regime.     

DISPERSION IN DISORDERED POROUS MEDIA

We report results of Monte Carlo investigations of dispersion in one- and two-phase flow through disordered porous media represented by square and simple cubic networks of pores of random radii. Dispersion results from the different flow paths and consequent different transit limes available to tracer particles crossing from one plane to another in a porous medium. Dispersion is found to be diffusive for the process simulated, i.e., a concentration front of solute particles can be described macroscopically by a convective diffusion equation. Dispersivity in the direction of mean flow, i.e., longitudinal dispersivity, is found to be an order of magnitude larger than dispersivity transverse to the direction of mean flow. In two-phase flow, longitudinal dispersivity in a given phase increases greatly as the saturation of that phase approaches its percolation threshold; transverse dispersivity also increases, but more slowly. As the percolation threshold is neared, the backbone of the sublatlice occupied by the phase becomes increasingly tortuous, with numerous subloops which provide alternate particle paths that are evidently highly effective in dispersing a concentration front of tracer particles.     

Numerical simulation of aggregate dispersion in different flow fields using discrete element method

An attempt was made to study the aggregate dispersion process in three different flow fields namely; steady shear, elongation flow, and combined shear and elongational flows using the discrete element method. The simulation was performed on two aggregate structures characterized by their fractal dimensions. The predicted results showed that the aggregate break‐up process evaluated in terms of weighted average fragment size 〈w〉 follows a power–law type relation as 〈w〉 = kt^?m in all the three flow fields. The dispersion performance of different flow fields evaluated by dispersing rate and a final steady‐state fragment size was found to be dependent upon the extent of applied stress and flow fields such that at low applied stress levels much smaller steady state values of 〈w〉 could be obtained for the elongational flow. The aggregate structure, characterized by its fractal dimension, was found to have different effects on the aggregate dispersion process depending on the flow field and applied stress level. The results predicted from this simulation could be explained in terms of ability of flow fields in rotating the aggregates and fragments in appropriate position to be broken up and the fractal dimensions of aggregates. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2010     

HYDRODYNAMIC CHROMATOGRAPHY: THREE DIMENSIONAL LAMINAR DISPERSION IN RECTANGULAR CONDUITS WITH TRANSVERSE FLOW

Experimental observations^1,9 indicate much poorer separations than are predicted by two dimensional theory. The purpose of this work is to explain these differences and suggest ways in which system performance can be improved.

The large effect of span-wise variation in axial velocity caused by side walls on hydrodynamic separations carried out in rectangular conduits with transverse flow is studied theoretically. As the aspect ratio increases, the steady stale retentivity (convection coefficient) approaches an asymptotic value obtained by neglecting side wall effects. However, the dispersion coefficient does not reduce to that for a flow with no side walls. Indeed, the asymptotic steady state dispersion coefficient is at least six times larger than that obtained by two dimensional theory which neglects side wall effects. As the transverse Peclet number increases, the effect of side walls on the dispersion coefficient becomes much larger.

The present three dimensional theoretical predictions, in contrast to two dimensional ones, are in good agreement with the experimental data of Caldwell, et al.⁹ and Kesner, et al.¹ on electrical field flow fractionation. The results indicate that side wall effects may be of major importance in hydrodynamic chromatography even when the aspect ratio is 70 or more.

The adverse effect of side walls may be avoided by having the membranes enclose thin annular regions rather than rectangular conduits. This should improve performance significantly.     

Toward Understanding of Two‐Phase Eccentric Helical Reactor Performance

Mass transfer investigations in a two‐phase gas‐liquid Couette‐Taylor flow (CTF) reactor and a numerical flow simulation are reported. The CTF reactor is characterized by high values of the mass transfer parameters. Previous mass transfer investigations have yielded high values of the volumetric mass transfer coefficients (of the order of 10^–1 s^–1) and the specific interfacial area, compared to those obtained in a stirred tank (10³ m² m^–3). In order to intensify mass transfer in the CTF reactor, an eccentric rotor (rotating inner cylinder) was used. In the eccentric annulus with rotating inner cylinder, due to frequent variation of the hydrodynamic flow field parameters, nonlinear hydrodynamic conditions occurred. These conditions can influence the rate of mass transfer. The experimental results of benzaldehyde oxidation in an eccentric CTF reactor confirmed an increase in mass transfer, as against a concentric CTF reactor. Numerical simulation of the Couette‐Taylor (helical) flow was performed in a concentric and in an eccentric annulus. Calculation of parameters such as velocity, static pressure, kinetic energy and energy dissipation rate revealed a significant effect of gap eccentricity on the flow behavior.     

Average Crossflow Velocity in Laminar Flow Systems with Periodic Finned Surfaces

Experimental and CFD studies of the enhanced average crossflow velocity in a laminar flow system were performed. The experiments were carried out using working fluid with a kinematic viscosity of 1.8·10^–6 m²/s. A steady flow Reynolds number in the laminar range of 0 < Re < 400 and oscillation Reynolds number in the range 0 < Re_osc < 1000 were studied. The range of oscillation amplitude and frequency were 0.2 mm < A < 1.0 mm and 5 Hz < f < 90 Hz respectively. Three experimental configurations were studied, i.e., oscillating finned surface in a fluid at rest, which is similar to a batch configuration, steady finned flow and oscillating finned flow configurations. The acquired images were analyzed using particle image velocimetry (PIV) software. The study is also supported by CFD simulations using the software suit CFX 11.0 from ANSYS GmbH, Germany. The results of the flow visualization and PIV analyses reveal the formation of periodic vortices and increased transverse transport. The maximum enhancement of the average crossflow velocity was obtained at κ = 3. The oscillation parameters and shape of the fins have a significant influence on the flow patterns and the crossflow effects. A triangular finned geometry gives better performance considering the enhanced average crossflow velocity. In general, efficient fluid mixing is possible due to the complex flow structures generated.     

Small scale variability in the flow of water and solutes, and implications for lysimeter studies of solute leaching

This paper discusses results from an experiment in which the fluxes of non-reactive solutes and water were monitored in an 8×8 array of adjacent collectors (each of 36 cm² area, covering a total area of 0.23 m²) located at 1m depth in a poorly-structured sandy loam soil profile. Water was applied uniformly to the soil surface at constant rates of either 4.3 or 19 mm/h, and a pulse of non-reactive solute (chloride) was added once the flows of water had become steady. Water continued to be applied at steady rate until all of the applied solute had been leached. The breakthrough curves for individual cells were analysed to determine the mean travel velocity and dispersivity.The water fluxes in individual collectors were very stable, but varied by over an order of magnitude, with collectors showing particularly rapid flow tending to be clustered. About 80% of the total flow was collected from 40% of the overall area of the array of collectors. However, there was only two-fold variation between cells in mean travel time velocity. This, coupled with a large mobile water content (equivalent to about 70% of the porosity) implies that rapid flow through a relatively small volume of macropores was not responsible for transporting a large proportion of the water and solutes, and was not a major factor in the spatially variable discharge.We conclude that only a small increase in water-filled pore space was required to conduct the extra water applied at the faster application rate. The small amount of extra water-filled porosity brought into play at the higher flow rate served to increase the flow velocities through the matrix of water pathways that were conducting water at the slower application rate, rather than acting as a bypass giving rise to very rapid flow velocities.Analysis of the breakthrough curves suggests that small scale hydrodynamic dispersion was the dominant contributor to dispersion at the lysimeter scale.The results have implications for the design of and interpretation of lysimeter experiments and the interpretation of measurements of contaminant fluxes made using drainage samplers. We conclude that in the case of structureless sandy soils, lysimeters of the order of 1m deep and 1 m diameter are sufficiently large to be considered representative of a field soil at the 1 m scale at least in situations where macropore flow is not an important mechanisms of solute transport. Comparison of these results with other lysimeter studies on the same soil concluded that the nature of the lower boundary of lysimeters has substantial influence on the flow pathways and the consequent breakthrough curves. Drainage samplers which have collection areas much less than 0.1 m² are likely to collect water and any dissolved contaminents at rates very different from the average flux densities measured at much larger scales, and so require careful interpretation.Finally, various hypotheses are considered to explain the lateral redistribution of water that occurred in the light of the experimental results.     

Effect of pore distribution and flow segregation on dispersion in porous media ag

In order to study the effect of the pore size distribution and flow segregation on dispersion in a porous media, we consider the dispersion of solute in an array of parallel pores. Equations are obtained for the dispersion coefficient in laminar and turbulent flow, as a function of the particle Peclet number. The theory fits quite well cumulative experimental data from various researchees in Peclet number range from 10^?3 to 10⁶. The model also predicts some trends, backed by experimental data, regarding the effect of particle size, particle size distribution and fluid velocity on dispersion.     

Linear viscoelasticity of discotic mesophases

The small-amplitude oscillatory capillary Poiseuille flow of uniaxial incompressible discotic nematic liquid crystals, representative of discotic mesophases, is characterized using analytical, computational, and scaling methods. Linear viscoelastic material functions are calculated and discussed in terms of fundamental anisotropic viscoelastic processes. The role of orientation to generate flow and store elastic energy is discussed. Viscoelastic behavior is found only at frequencies of similar magnitude to the single orientation relaxation time. In the terminal small-frequency zone the storage modulus scale as G^′∼ω², and the loss modulus as G^″∼ω, typical of viscous fluids. Comparisons between steady and oscillatory Poiseuille flow shows that the Cox-Merz rule is not obeyed, but that as expected the steady and complex viscosities in the terminal zone are identical. A remarkable and useful correspondence between the stored elastic energy under steady flow and the storage modulus G^′ has been discovered.     

An investigation of stress oscillation in poly(butylene succinate‐co‐cyclohexanedimethylene succinate)

Biodegradable poly(butylene succinate‐co‐cyclohexanedimethylene succinate) (PBCS) was synthesized via polycondensation. The composition of PBCS was characterized by ¹H‐nuclear magnetic resonance (¹H‐NMR). By incorporation of 20 wt% of cyclohexanedimethylene succinate, stress oscillation behavior in stress–strain curve was observed during cold drawing. The stress was no longer stable, but exhibited constantly periodic fluctuations. Morphology and crystalline structure of oscillatory portion were evaluated by scanning electron microscopy, and X‐ray diffraction (XRD), respectively. The melting behavior was studied using differential scanning calorimetry (DSC). The surface morphology of oscillatory portion presented alternating opaque and transparent bands, which were perpendicular to the cold‐drawing direction. By comprehensive analysis of stress–strain curve, surface morphology, XRD, and DSC, the formation of stress oscillation was discussed. To describe stress oscillation, a new mathematical model was proposed to simulate stress oscillation and its viscoelastic behaviors were well explained. The novelty of the mathematical model was that new oscillatory cells consisting of a dashpot with free inextensible strings were adopted to instead of fiber bundle cells. POLYM. ENG. SCI., 55:966–974, 2015. © 2014 Society of Plastics Engineers     

SINGLE FLUID FLOW IN POROUS MEDIA

A comprehensive study on single fluid flow in porous media is carried out. The volume averaging technique is applied to derive the governing flow equations. Additional terms appear in the averaged governed equations related to porosity ε, tortuosity τ, shear factor F and hydraulic dispersivity D _h. These four parameters are uniquely contained in the volume averaged Navier-Stokes equation and not all of them are independent. The tortuosity can be related to porosity through the Brudgemann equation, for example, for unconsolidated porous media.

The shear factor models are reviewed and some new results are obtained concerning high porosity cases and for turbulent flows. It is known that there are four regions of flow in porous media: pre-Darcy's flow, Darcy's flow, Forchheimer flow and turbulent flow. The transitions between these regions arc smooth. The first region, the pre-Darcy's flow region represents the surface-interactive flows and hence is strongly dependent on the porous media and the flowing fluid. The other flow regions are governed by the flow strength of inertia. For Darcy's flow, the pressure gradient is found to be proportional to the flow rate. The Forchheimer flow, however, is identified by a strong inertia! effects and the pressure gradient is a parabolic function of flow rate. Turbulent flow is unstable and unsteady flow characterized by chaotic flow patterns. The pressure drop is slightly lower than that predicted using the laminar flow equation.

The hydraulic dispersivity is a property of the porous media. It may be considered as the connectivity of the pores in a porous medium. It characterizes the dispersion of mementum, heat and mass transfer. In this paper, only the dispersion of momentum is studied.

Single fluid flow through cylindrical beds of fibrous mats and spherical particles has been used to show how to solve the single fluid flow problems in porous media utilizing the knowledge developed in this communication. Both the pressure drop and axial flow velocity profiles are computed using the developed shear factor and hydraulic dispersion models. Both the predicted velocity profile and pressure drop compare fairly well with the published experimental data.     

Axial dispersion in laminar flow in helical coils

The results of an experimental study of dispersion of solute in helical coils under laminar flow conditions are reported. The ranges of variables covered are 10< λ < 280, 10 <N_Re < 1600 and 1·5 × 10³ <N_Sc < 8·7 × 10³. Under the conditions where the dispersion model holds the data are correlated with a standard deviation of 19·6 per cent. The conditions under which the dispersion model holds for helical coils are found to be less severe than those for straight tubes. It is shown that coiling can reduce axial dispersion under laminar flow conditions by a factor of upto 500.     

A NEW CORRELATION FOR TWO-PHASE PRESSURE DROPS IN FULLY DEVELOPED DILUTE SLURRY UP-FLOWS THROUGH AN ANNULUS

The prediction of two-phase pressure drops is important in the design of fluidized beds, heat exchangers, slurry transport lines, annular risers in reactors, driers, and other applications of particulate flows. Experimental data for liquid-solid up-flows through annuli are very limited. In this study, a correlation was developed for predicting two-phase pressure drops via Euler number (Eu _tp ) for liquid-solid suspension up-flows in the fully developed flow region of an annulus in terms of mixture Reynolds number (Re _m ), solid particle Froude number (Fr _p ), ratio of particle diameter to hydraulic diameter, and ratio of area-averaged solids density in the tested cross-sectional flow area of the annulus to the average density of the feed slurry entering the annulus. The proposed correlation, having an accuracy of almost ±25%, was obtained for water-feldspar dilute suspension up-flows in an annulus with a hydraulic diameter of 79.8 mm and with feldspar particles (2400 kg/m³ in material density and 0.072–0.138 mm in average diameter) at low concentrations.     

DISPERSION OF MATTER IN FLOW OF A BINGHAM PLASTIC IN A TUBE

The paper is concerned with the dispersion of a solute in a Bingham plastic fluid flowing in a pipe or a parallel plate channel. For pipe flow, the dispersion coefficient K ₂ first increases with ξ₀ (the dimensionless radius of the plug flow region), reaches a maximum and then decreases. But in a channel flow, K ₂ decreases monotonically with increasing ξ₀. Further K ₂ for channel flow is found to be larger than that for pipe flow for all values of ξ₀ except 0.8≤ξ₀≤1.