Steady incompressible fluid flow over a bundle of cylinders at moderate reynolds numbers

The complete Navier-Stokes equations describing the steady flow of incompressible Newtonian fluids normal to an array of long cylinders have been solved numerically using the finite difference method in terms of the stream function and vorticity variables. The inter-cylinder interactions have been mimicked using the well known free surface cell model. Extensive information on the detailed structure of the flow field in terms of the surface vorticity distribution, stagnation pressure, stream line and iso-vorticity line plots, as well as on the values of the integral quantities, such as pressure, friction and total drag coefficients, have been obtained under wide ranges of conditions as follows: 0.3 ≤ ? 0.99 and 0.01 ≤ Re ≤ 100. The numerical results presented herein have been validated using the appropriate theoretical and experimental results available in the literature; the match between the present predictions and the scant experimental results is good.     

Steady Two‐Dimensional Non‐Newtonian Flow Past an Array of Long Circular Cylinders up to Reynolds number 500: A Numerical Study

The free surface and zero vorticity cell models have been combined with the equations of motion to investigate numerically the steady flow of incompressible power‐law (shear‐thinning and shear‐thickening) fluids across banks of long cylinders. The equations of motion in the stream function/vorticity formulation have been solved numerically using a second order accurate finite difference method to obtain extensive information on the behaviour of the drag coefficient, surface vorticity distribution, streamlines and iso‐vorticity patterns, for high Reynolds numbers (Re = 50 500) and using a wide range of power‐law index (0.3 ≤ n ≤ 2.0), and porosity (0.4 ≤ e ≤0.9) values. The behaviour of the aforementioned parameters at low Reynolds numbers has also been investigated and validated using theoretical and numerical work from the literature. The results reported here enable extension of the limits of creeping flow behaviour up to Re = 50 for fluids with highly shear‐thickening characteristics under low porosity conditions.     

Frictional Pressure Drop of Air Non‐Newtonian Liquid Flow through Helical Coils in Horizontal Orientation

Investigations have been carried out to evaluate the two‐phase frictional pressure drop for air non‐Newtonian liquid flow through helical coils in horizontal orientation. The experiments performed using 36 different helical coils and 4 different concentrations of sodium salt of carboxymethyl—cellulose (SCMC) as non‐Newtonian liquids. The effects of air and liquid flow rate, coil diameter, helix angle and liquid properties‐ on two‐phase frictional pressure drop have been discussed. An attempt has been made to fit the experimental two‐phase frictional pressure drop data by the Lockhart and Martinelli, Chem. Eng. Prog. 45 , 39–48 (1949) correlation and the modified Lockhart‐Martinelli correlation as presented by different authors. In another approach, friction factor method was adopted to correlate the experimental data by dimensional analysis. The correlation developed predicts the two‐phase frictional pressure drop with acceptable statistical accuracy.     

Pressure drop and mixing behaviour of non‐Newtonian fluids in a static mixing unit

Static or motionless mixers have received wide application in chemical and allied industries due to their low cost and high efficiency. The pressure drop and mixing behaviour of such mixers have been widely studied. However, the available information for non‐Newtonian fluids is scanty. The results of pressure drop and mixing studies conducted with a locally made motionless mixer (MALAVIYA mixer) and four non‐Newtonian fluids—aq. CMC, PVA, and PEG solutions are reported in this article. The new mixer causes less pressure drop compared to some of the commercial mixers. Mixing behaviour of the unit is more closer to plug flow and a two‐parameter model correlates the dispersion data.     

Volumetric Mass Transfer Coefficient in non‐Newtonian Fluids in Spout Fluid Beds

The volumetric liquid‐phase mass transfer coefficient, k_La, was determined by absorption of oxygen in air using six different carboxy‐methyl cellulose (CMC) solutions with different rheological values in three phase spout‐fluid beds operated continuously with respect to both gas and liquid. Three cylindrical columns of 7.4 cm, 11.4 cm, and 14.4 cm diameters were used. Gas velocity was varied between 0.00154–0.00563 m/s, liquid velocity between 0.0116–0.0387 m/s, surface tension between 0.00416–0.0189 N/m, static bed height between 6.0–10.8 cm, and spherical glass particles of 1.75 mm diameter were used as packing material. A single nozzle sparger of 1.0 cm diameter was used in the spouting line. The volumetric mass transfer coefficient was found to increase with gas velocity, liquid velocity, and static bed height and to decrease with the increase of the effective liquid viscosity of the CMC solution. A dimensionless correlation was developed and compared with those listed in the literature.     

Coaxial Mixer Hydrodynamics with Newtonian and non‐Newtonian Fluids

The performance of several combinations of a wall scraping impeller and dispersing impellers in a coaxial mixer operated in counter‐ and co‐rotating mode were assessed with Newtonian and non‐Newtonian fluids. Using the power consumption and the mixing time as the efficiency criteria, impellers in co‐rotating mode were found to be a better choice for Newtonian and non‐Newtonian fluids. The hybrid impeller‐anchor combination was found to be the most efficient for mixing in counter‐rotating or co‐rotating mode regardless of the fluid rheology. For both rotating modes, it was shown that the anchor speed does not have any effect on the power draw of the dispersing turbines. However, the impeller speed was shown to affect the anchor power consumption. The determination of the minimum agitation conditions to achieve the just suspended state of solid particles (N_js) was also determined. It was found that N_js had lower values with the impellers having the best axial pumping capabilities.     

Flow of power law liquids through particle assemblages at intermediate reynolds numbers

The complete equations of motion and continuity have been solved numerically using the finite element method for the flow of power law liquids through assemblages of rigid spherical particles. The inter-particle interactions have been simulated using the free surface cell model. Extensive results on drag coefficients have been obtained under a wide range of physical and operating conditions (0.9999 ≥ 0.3), 1 ≥ n ≥ 0.4 and 20 ≥ Re ≥ 1. The observed dependence of drag coefficient on voidage and non-Newtonian flow behaviour index have been explained qualitatively with the aid of order of magnitude considerations. Finally, the theoretical predictions have been validated using suitable experimental data available in the literature.     

Flow Characteristics in Slot‐Rectangular Spouted Beds with Draft Plates

An experimental investigation was conducted of slot‐rectangular spouted beds with air entry slots spanning the full thickness of the column and vertical draft plates intended to help control the solids circulation rate. With increasing superficial gas velocity, the flow between the draft plates changed from bubbling to slugging and then to spouting with dilute pneumatic between the plates and moving‐bed downward motion on both sides. However, there was difficulty maintaining stability and symmetrical flow on the two sides. Once spouting is established, pressure drops and local voidages vary with gas velocity, particle size and gas entry size in broadly similar manners as for conventional spouted beds     

Prediction of Two‐Phase Frictional Pressure Drop for the Concurrent Gas and Newtonian Liquids Upflow through Packed Beds

Correlations were developed to predict frictional pressure drop for concurrent gas‐liquid upflow through packed beds covering all the three identified flow regimes, i.e. bubble flow, pulse flow and spray flow. The observation that the gas and liquid flow rates have different influences on the two‐phase pressure drop in different flow regimes, was taken into consideration in the development of these correlations. More than 600 experimental pressure drop data from the present study and literature covering a wide range in gas‐liquid systems, flow rates and column packing were used.     

The effect of surfactants on deformation of falling non‐Newtonian drops in a Newtonian liquid

Deformation of settling non‐Newtonian ellipsoidal drops in a Newtonian liquid was experimentally observed. Corn oil was used as the Newtonian phase and solutions of polyacrylamide in aqueous glycerine as the non‐Newtonian phase. The shear‐thinning behaviour of the drops fluid was controlled by the amount of polymer dissolved, while the effect of interfacial tension was examined using different concentrations of sodium dodecyl sulphate (SDS). In the range of 1 < E < 2.9, 0.2 < Eo, < 23, and 0 < Ma < 17.2, drop eccentricity increased linearly with a modified Eötvös number taking into account the effect of surfactants. For the range of experimental conditions tested, drop deformation was mainly controlled by viscous and interfacial tension forces, while shear‐thinning and inertia effects were negligible.     

Flow of Newtonian and power law liquids in tube bundles

In this work, the annular (tangential) flow of Newtonian and non‐Newtonian fluids in tube bundles has been studied experimentally. Extensive pressure drop data has been obtained embracing wide ranges of the Reynolds number (13–6600) and for two test modules of different geometrical arrangements, but of similar overall void fraction. Preliminary experiments suggest that the pressure drop is mainly determined by the overall void fraction of the bundle and is relatively insensitive to the detailed geometrical configuration of the bundle. A simple predictive correlation has been developed which reconciles the present results for Newtonian and power law fluids with acceptable levels of reliability.     

Effect of Non‐Newtonian Characteristics on Convective Liquid‐Solid Heat Transfer in Packed and Fluidised Beds of Spherical Particles

In this work, the field (continuity, momentum and thermal energy) equations togetherwith a cell model have been solved numerically to elucidate the influence of non‐Newtonian (Power law rheology) liquid characteristics on liquid‐solid heat transfer in packed and fluidised beds of spherical particles. The results presented herein relate to wide ranges of conditions of bed voidage, power‐law index and Peclet number but are limited to low Reynolds number (≤1) flow conditions. Within the range of conditions, the effect of power‐law index is found to be small and this is also consistent with the available experimental results on liquid‐solid mass transfer in these systems.     

Pressure drop for single and two‐phase flow of non‐newtonian liquids in helical coils

New experimental results on pressure loss for the single and two‐phase gas‐liquid flow with non‐Newtonian liquids in helical coils are reported. For a constant value of the curvature ratio, the value of the helix angle of the coils is varied from 2.56° to 9.37°. For single phase flow, the effect of helix angle on pressure loss is found to be negligible in laminar flow regime but pressure loss increases with the increasing value of helix angle in turbulent flow conditions. On the other hand, for the two‐phase flow, the well‐known Lockhart‐Martinelli method correlates the present results for all values of helix angle (2.56‐9.37°) satisfactorily under turbulent/laminar and turbulent/turbulent conditions over the following ranges of variables as: 0.57 ≤ n′ ≤ 1; Re′ < 4000; Re_l < 4000; Re_g < 8000; 8 ≤ x ≤ 1000 and 0.2 ≤ De′ ≤ 1000.     

An experimental investigation of the cross‐flow of power law liquids past a bundle of cylinders and in a bed of stacked screens

The steady flow of power‐law polymer solutions normal to arrays of cylinders and in a bed of screens has been investigated experimentally. Extensive pressure drop measurements have been made for three different test cells (two tube bundles and one bed made of screens) of different voidages for a series of inelastic carboxymethyl cellulose and sodium alginate aqueous solutions. The resulting values of friction factor correlate well with a modified definition of the Reynolds number based on the capillary bundle representation of the bed. The new data extend considerably the range of voidage values from ～0.6 to ～0.87. Extensive comparisons have also been made between the present experimental data and our previous calculations based on the use of simple cell models. The close correspondence between the two demonstrates the utility of such idealized analyses. All in all, the present results embrace the following ranges of physical and kinematic conditions: Reynolds number: 0.01 to ～1200; power‐law flow behaviour index: 0.38 to 1; and three values of voidage, namely, 0.74, 0.78 and 0.87.     

Gas Holdup and Solid‐Liquid Mass Transfer in Newtonian and non‐Newtonian Fluids in Bubble Columns

Gas holdup and surface‐liquid mass transfer rate in a bubble column have been experimentally investigated. De‐mineralized water, 0.5 and 1.0% aqueous solutions of carboxy methyl cellulose (CMC), and 60% aqueous propylene glycol have been used as the test liquids. Effects of column diameter, liquid height to column diameter ratio, superficial gas velocity and liquid phase viscosity on gas holdup and mass transfer rate are studied. Generalized correlations for the average gas holdup and wall to liquid heat and mass transfer coefficients are proposed. These are valid for both Newtonian and pseudoplastic non‐Newtonian fluids.     

Hydrodynamic Model for Horizontal Two‐phase Flow Through Porous Media

A unidirectional, two‐fluid model based on the volume‐average mass and momentum balance equations was developed for the prediction of two‐phase pressure drop and external liquid hold‐up in horizontally positioned packed beds experiencing stratified, annular and dispersed bubble flow regimes. The so‐called slit model drag force closures were used for the stratified and annular flow regimes. In the case of dispersed bubble flow regime, the liquid‐solid interaction force was formulated on the basis of the Kozeny‐Carman equation by taking into account the presence of bubbles in reducing the available volume for the flowing liquid. The gas‐liquid interaction force was evaluated by using the respective solutions of drag coefficient for an isolated bubble in viscous and turbulent flows. The proposed drag force expressions for the different flow patterns occurring in the bed associated with the two‐fluid model resulted in a predictive method requiring no adjustable parameter to describe the hydrodynamics for horizontal two‐phase flow in packed beds.