Flow of non‐Newtonian polymeric solutions through fibrous media

The equations of motion (continuity and momentum) describing the steady flow of incompressible power law liquids in a model porous medium consisting of an assemblage of long cylinders have been solved numerically using the finite difference method. The field equations as well as the pertinent boundary conditions have been re‐cast in terms of the stream function and vorticity. The inter‐cylinder interactions have been simulated using a simple “concentric cylinders” cell model. Extensive information on the detailed structure of the flow field in terms of the surface vorticity distribution, streamlines, and viscosity distribution on the surface of the solid cylinder as well as on the values of the pressure and friction drag coefficients under wide ranges of physical (0.4 ≤ ϵ ≤ 0.95; 1 ≥ n ≥ 0.4) and kinematic (0.01 ≤ Re ≤ 10) conditions have been obtained. The numerical results presented herein have been validated using the experimental results for the flow of Newtonian and power law fluids available in the literature; the match between the present predictions and the experiments was found to be satisfactory. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 76: 1171–1185, 2000     

A numerical study of steady incompressible newtonian fluid flow over a disk at moderate reynolds numbers

The hydrodynamic behaviour of a single thin disk settling steadily, broad face-wise, in an incompressible Newtonian fluid has been investigated by solving the Navier-Stokes equations numerically using an iterative procedure. The finite difference method has been used to map out the complete flow domain in terms of the values of the stream function and vorticity. Finally, the drag coefficient was evaluated by considering the energy dissipation. Extensive results on drag for a disk of thickness to diameter ratio of 0.05 show excellent agreement with the experimental, as well as, previous scant numerical results in the range 1 ≤ Re ≤ 100.     

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.     

Low Reynolds number flow of power-law fluids over two square cylinders in tandem

The governing partial differential equations have been solved numerically for the 2-D and steady powerlaw fluid flow over two square cylinders in tandem arrangement. Extensive numerical results spanning wide ranges of the governing parameters as Reynolds number (0.1≤Re≤40), power-law index (0.2≤n≤1) and inter-cylinder spacing (2≤L/d≤6) are presented herein; limited results for L/d=24 are also obtained to approach the single cylinder behavior. The detailed flow visualization is done by means of the streamline and vorticity contours in the vicinity of two cylinders. The global characteristics are analyzed in terms of the surface pressure distribution and pressure drag coefficient. The drag coefficient shows the classical inverse dependence on the Reynolds number irrespective of the value of the powerlaw index; the drag on the upstream cylinder is always greater than that for the downstream cylinder.     

Theoretical and experimental study of the molten polymer flow in the calender bank

The flow of molten polymers in the calender bank has been computed using a finite-element method with stream function and vorticity. Two nonsymmetrical-recirculating regions have been obtained fully in agreement with the experimental observations on poly(vinyl chloride) melt banks. The pressure distribution along the flow axis is very close to the one obtained using the classical-lubrication approximation.     

Steady Flow of Power Law Fluids across a Circular Cylinder

The momentum equations describing the steady cross‐flow of power law fluids past an unconfined circular cylinder have been solved numerically using a semi‐implicit finite volume method. The numerical results highlighting the roles of Reynolds number and power law index on the global and detailed flow characteristics have been presented over wide ranges of conditions as 5 ≤ Re ≤ 40 and 0.6 ≤ n ≤ 2. The shear‐thinning behaviour (n < 1) of the fluid decreases the size of recirculation zone and also delays the separation; on the other hand, the shear‐thickening fluids (n > 1) show the opposite behaviour. Furthermore, while the wake size shows non‐monotonous variation with the power law index, but it does not seem to influence the values of drag coefficient. The stagnation pressure coefficient and drag coefficient also show a complex dependence on the power law index and Reynolds number. In addition, the pressure coefficient, vorticity and viscosity distributions on the surface of the cylinder have also been presented to gain further physical insights into the detailed flow kinematics.     

Pressure drop for the slow flow of dilatant fluids through a fixed bed of spherical particles

The equations of motion in conjunction with the free surface cell model have been solved numerically for the creeping flow of dilatant fluids through a fixed bed of spheres. The effect of the flow behaviour index (1 ≤ n ≤ 2) on the individual as well as the total drag coefficients is elucidated. Finally, the theoretical predictions have been validated using the scant experimental results available in the literature.     

Effects of Viscous Dissipation on Heat Transfer between an Array of Long Circular Cylinders and Power Law Fluids

The free surface model has been combined with the equations of motion and of thermal energy to investigate the role of viscous dissipation on heat transfer between banks of long cylinders and power law (shear‐thinning and shear‐thickening) fluids. The equations of motion cast 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 local and surface‐averaged Nusselt numbers over a range of Reynolds numbers 1 – 500, for a wide range of power law indices (0.4 ≤ n ≤ 2.0), Brinkman numbers (0 ≤ Br ≤ 5) and Prandtl numbers (Pr = 1, 1000) at two representative solid volume fractions corresponding to the porosities of e = 0.4 and 0.9. Two different thermal boundary conditions are considered at the cylinder surface: constant temperature (CT) and constant heat flux (CHF). The results presented herein provide a fundamental knowledge about the influence of viscous dissipation on the heat transfer characteristics. The results reported herein further show that the effect of Brinkman number on heat transfer is strongly conditioned by the thermal boundary condition, Prandtl number and the power law index.     

Numerical studies of viscous stagnation point flows

A rapid finite-difference procedure for computing stagnation point flows at a given Reynolds number in terms of stream function and vorticity distributions is described. The method is illustrated for spherical and cylindrical curvature for Reynolds number ranges of 1 ≤ Re ≤ 400 and 4 ≤ Re ≤ 100 respectively. Predicted radial and angular velocity distributions in the frontal region are compared with results from numerical solutions of the full elliptic equations of motion. The use of a modified potential flow outer boundary condition reduces the region of numerical computation.     

Simulation bidimensionnelle de phénomènes de convection naturelle et forcée en formulation primitive

The air circulation, and the temperature and pressure profiles were simulated in two dimensions using a hybrid finite difference method which combines the MAC method and the stream function/vorticity approach. Three different flow problems were solved: (i) forced convection in a closed cavity; (ii) natural convection in a closed cavity; and (iii) forced and natural convection in a cavity with localised openings and hot sources. The results for the first two problems were compared to the literature and the results for the third problem were compared to data obtained from an experimental apparatus.     

Slow non-newtonian flow through packed beds: Effect of zero shear viscosity

The slow non-Newtonian (inelastic) flow through packed beds of mono-size spherical particles has been simulated by solving the equations of motion numerically. The inter-particle interactions have been modelled by using a simple cell model. Theoretical estimates of pressure, friction and total drag coefficients as function of the pertinent physical (l≥n≥ 0.2; 0.3 ≤ e ≤ 0.5) and kinematic parameters (0.01 ≤ '≤ 100) for a fixed value of Reynolds number {Re = 0.001) have been obtained. The theoretical predictions reported herein have been validated using the suitable experimental results available in the literature, and the importance of including the zero shear viscosity in analyses for the creeping flow problems is convincingly demonstrated.     

A comparative study on optimised and non‐optimised axial flow,spherical reactors in naphtha reforming process

In this study, the operating conditions of an axial flow spherical reactor have been optimised using a reliable optimisation technique and the results are compared with the results of non‐optimised conditions. The dynamic behaviour of the reactor has been considered in the optimisation process and orthogonal collocation method has been used in order to solve the obtained equations from mathematical modelling of the process. The goal of this study is to maximise the aromatics and hydrogen production rate. Therefore, the objective function is the combination of two terms which include the production rate of the mentioned components. The catalyst distribution for each reactor, the inlet pressure of the system, Length per radius for each reactor, the naphtha feed molar flow rate and the hydrogen mole fraction in the recycle stream as well as the inlet temperature of each reactor have been optimised in this study. © 2011 Canadian Society for Chemical Engineering     

THE TRANSIENT MOTION OF A SPHERICAL FLUID DROPLET

A numerical method is developed for investigation of the unsteady motion of a spherical fluid droplet under the influence of gravity. This study extends previous work valid for creeping flow to moderate Reynolds number. The unsteady flow fields inside and outside of the fluid sphere are described by the two-dimensional, axisymmetric Navier-Stokes equations in the form of vorticity and stream function, along with the equation of motion of the droplet. The governing equations are approximated by a central difference and a second-order upwind difference, and are solved iteratively using the Gauss-Siedel and secant methods. Numerical results of the time-dependent vorticity, stream function and drop velocity are presented for a water droplet moving through air and for an air bubble rising in water. The steady state drop velocity and the drag coefficient at various Reynolds numbers are examined, and they are shown to agree very well with previous results.     

Behavior of the vorticity vector in a supersonic nonuniform swirl flow of a combustible gas behind a moving curvilinear shock or detonation wave

Expressions for components of the vorticity vector behind a curvilinear shock or detonation wave propagating in a supersonic nonuniform flow of a combustible gas are derived. Plane and axisymmetric gas flows are considered. The free stream in the general case is a vortex flow with a specified distribution of parameters. Formulas for the vorticity components in the plane of the flow for axisymmetric flows are found to have the same form as formulas for steady axisymmetric flows. As in the case of steady flows, the normal-to-wave component of vorticity is demonstrated to remain continuous across the discontinuity surface; in the case of axisymmetric flows, the ratio of the tangential component of vorticity aligned in the plane of the flow to density also remains continuous, though the quantities themselves become discontinuous. __________ Translated from Fizika Goreniya i Vzryva, Vol. 45, No. 2, pp. 68–75, March–April, 2008.     

用改进的单元胞模型数值模拟液体穿过球形颗粒群的流动

The cell model developed since 1950s is a useful tool for exploring the behavior of particle assemblages,but it demands further careful development of the outer cell boundary conditions so that interaction in a particle swarm is better repressented.In this paper,the cell model and its development were reviewed,and the modifications of outer cell boundary conditions were suggested.Athe cell outer boundary,the restriction of uniform liquid flow was removed in our simulation conducted in the reference frame fixed with the particle.Zero shear stress condition was used to evaluate the outer boundary value of the stream function.Boundary vorticity was allowed to evolve to values compatible to existing stream function at the free shear outer boundary.The fore-aft symmetry of vorticity distribution at the outer boundary is thought critical to ensure the continuity of inflow and outflow between touching neighbor cells,and is also tested in the modified cell model.Numerical simulation in terms of stream function and vorticity based on the modified cell models was carried out to shed light on the interaction between liquid and particles.Lower predicted drag coefficient by the modified cell models was interpreted with the feature of flow structure.The drag coefficient from the simulation was also compared with correlations of drag coefficient reported in literature.It is found that the modified cell model with the uniformity of external flow relaxed and the fore-aft symmetry of boundary vorticity enforced was the most satisfactory of the overall performance of prediction.     

Agitation of yield stress fluids by gas injection

This study is in line with two previous studies by the same authors on gas injection in yield stress fluids. Gas is injected toward the bottom wall of a prismatic tank containing a yield stress fluid. When rising toward the free surface, trains of bubbles generate fluid recirculation in the tank. Two experimental colorimetric methods are introduced and validated in order to quantify the recirculation liquid flow rate as well as the time evolution of the extent and shape of the mixed volume. The influences of the injection flow rate, fluid rheology, and reactor size have been quantified. Correlations based on the characteristic nondimensional numbers of the flow have been developed to predict the downward liquid flow rate as well as the mixed volume. A model for estimating the mixing time is also developed and compared to experimental results.     

Application of CFD for simulation of a baffled tubular membrane

Computational fluid dynamics (CFD) investigation of a tubular membrane channel containing a set of baffles was conducted for predicting turbulent flow. Simulation was performed using an array of baffles oriented either in the flow or in the reverse direction. A range of local parameters such as stream function, velocity, static pressure, wall shear stress, turbulent kinetic energy, and turbulent dissipation energy on the membrane surface was computed using CFD code FLUENT. The simulation results indicate that the presence of baffle can improve the local shear stress on the membrane surface and produces eddy activities which enhance the filtration performance. The observed flux enhancement can be attributed to the intense fluctuations of wall velocity and shear stress which can disrupt the growth of boundary layer on the membrane surface. The experimental evaluation was performed through cross flow microfiltration of titanium dioxide suspension which showed an acceptable agreement with the CFD predictions.     

Pressure drop in two phase cocurrent upward flow in packed beds: Air/non-newtonian liquid systems

New data on the two phase pressure drop for the concurrent upflow of air-liquid (Newtonian and non-Newtonian) mixtures through packed beds of spherical and non-spherical particles are presented. The results for single phase flows and for the air-Newtonian liquid mixtures have been used both to gauge the overall accuracy of the present experimental methods and to evaluate the validity of the predictive expressions available in the literature. The two phase pressure drop has been measured as a function of the liquid and gas flow rates, column diameter and the power law model constants. Depending upon a suitable combination of the gas and liquid fluxes and the power law index, the two phase pressure drop may be less than its value for the flow of liquid alone. A simple expression is proposed which correlates the present set of experiments (nearly 500 data points) with satisfactory levels of accuracy over the following ranges of conditions: 0.54 ≤ n ≤ 1; 0.001 ≤ Re_L^* ≤ 50; 3.7 ≤ Re_G ≤ 177 and 0.9 ≤χ (Lockhart-Martinelli parameter) ≤ 104.     

FLOW OF NON-NEWTONIAN POWER LAW LIQUIDS THROUGH PACKED AND FLUIDIZED BEDS

The creeping flow of power law liquids through assemblages of spherical particles has been studied theoretically. The inter-particle interactions are modelled via the zero vorticity cell model. The governing equations have been solved numerically to obtain the theoretical estimates of the drag force experienced by an assemblage placed in a streaming power law fluid. The results reported herein encompass wide ranges of fluid behaviour (values of power law index) and bed voidage thereby covering packed and fluidized bed conditions. Detailed comparisons with experimental data suggest that this theory can be used to predict pressure drop for power law fluid flow in packed beds as well as velocity-bed expansion characteristics for a fluidized bed. By analogy with the Newtonian case, intuitively, one would expect these results to be applicable to hindered settling in power law fluids, and indeed this is borne out by the limited amount of data covering the range 1≥n≥0.8 available in the literature.     

Steady flow of power-law fluids across an unconfined elliptical cylinder

The momentum transfer characteristics of the power-law fluid flow past an unconfined elliptic cylinder is investigated numerically by solving continuity and momentum equations using FLUENT (version 6.2) in the two-dimensional steady cross-flow regime. The influence of the power-law index (0.2?n?1.8), Reynolds number (0.01?Re?40) and the aspect ratio of the elliptic cylinder (0.2?E?5) on the local and global flow characteristics has been studied. In addition, flow patterns showing streamline and vorticity profiles, and the pressure distribution on the surface of the cylinder have also been presented to provide further physical insights into the detailed flow kinematics. For shear-thinning (n<1) behaviour and the aspect ratio E>1, flow separation is somewhat delayed and the resulting wake is also shorter; on the other hand, for shear-thickening (n>1) fluid behaviour and for E<1, the opposite behaviour is obtained. The pressure coefficient and drag coefficient show a complex dependence on the Reynolds number and power-law index. The decrease in the degree of shear-thinning behaviour increases the drag coefficient, especially at low Reynolds numbers. While the aspect ratio of the cylinder exerts significant influence on the detailed flow characteristics, the total drag coefficient is only weakly dependent on the aspect ratio in shear-thickening fluids. The effect of the flow behaviour index, however, diminishes gradually with the increasing Reynolds number. The numerical results have also been presented in terms of closure relations for easy use in a new application.