Design and analysis of a dual-cavity coat-hanger die

A method for the design and analysis of a dual-cavity coat-hanger die is presented in this paper. A macroscopic material balance and a microscopic flow analysis using the finite element method are combined to simulate polymeric fluid flow inside the die. Leonard's macroscopic procedure was adopted to include inertial, gravitational, and viscous effects, and the finite element method was then applied to estimate the contributions of inertial and viscous terms. In addition, the flow patterns in the outer cavity were computed by the finite element method so that the appearance of an undesirable vortex could be predicted. The residence time distributions for flow in the die were approximated by a simple, statistical approach. It was found through a case study that a dual-cavity coathanger die can effectively reduce the flow non-uniformities caused by fluid inertia and viscosity variations.     

THE LOCAL VOLUME-AVERAGED EQUATIONS OF MOTION FOR A SUSPENSION OF NON-NEUTRALLY BUOYANT SPHERES

The local volume averages of the equations of motion as well as the appropriate boundary conditions are developed for a flowing suspension of non-neutrally buoyant, uniform spheres in an incompressible Newtonian fluid under conditions such that inertial effects can be neglected. These equations do not represent an asymptotic theory with respect to the volume fraction of solids. Higher order terms have been retained everywhere, except where it has been necessary to estimate the velocity distribution within the immediate neighborhood of each sphere by neglecting the effects of the other spheres present. The resulting local volume-averaged equations of motion and boundary conditions involve no free or undetermined parameters.

For the special case of a very dilute suspension of neutrally buoyant spheres, the total local volume average of Cauchy's first law reduces to the form of the Navier-Stokes equation with the effective viscosity computed by Einstein (1906, 1956).

In two succeeding papers, we demonstrate for several flows [in vertical tubes, in a cone-plate viscometer, between rotating concentric cylinders (Couette flow), and between rotating parallel plates] that our general theory describes more concentrated neutrally buoyant suspensions than does its limiting case of very dilute suspensions.     

Dynamic simulation of flow-induced fiber fracture

A dynamic simulation method has been developed of the fracture process of a fiber in a flow field using the particle simulation method proposed i a previous paper. The fiber is modeled with bonded spheres as a fiber model. The flexibility of the fiber model is altered by changing three parameters of the stretching, bending, and twisting constants between adjacent spheres. The stress induced in each bond of the fiber model as a result of deformation is formulated using displacement of the bodn distanc, bond angle, and torsion angle fr each pair of spheres. After deformation, the fiber model fractures at the bond at which the stress surpasses the strength of the fiber. The motion of the fiber model in a flow field is determined by solving the translational and rotational motion equations for individual spheres under the hydrodynamic force and torque exerted on them. The correctness of the method and formulation was verified by comparing the simulated deflection curve of a cantilever beam (with a concentrated load at the end) with the theoretical curve. Good agreement was found in both the deflection and slope of the beam. The fracture process of a fiber after bending deformation in a two-dimensional siimple shear flow was simulated under assumptions of an infinitely dilute system, no hydrodynamic interaction, and a low Reynolds number of a particle. The calculated critical conditions of the flow field for fiber fracture were compared with Forgacs and Mason's theoretical ones. Simulated values of the fracture condition of the fluid shear stress related to the Young's modulus of a fiber agree with theoretical ones over an aspect ratio of 15.     

Investigation of the convective motion through a staggered herringbone micromixer at low Reynolds number flow

A computational study is presented of the complex flow through a staggered herringbone micromixer (SHM), which utilises sequences of asymmetrical herringbone grooves in cycles where a set of topologically similar grooves represent a half cycle. It was analysed using finite-element (method) based software to elucidate the fluid flow within the channel and characterise the effect of the grooves at moving fluid across the channel thus creating non-axial fluid movement. Three separate physical systems were modelled: a channel containing a single groove, a half cycle of infinite grooves and an infinite system with one groove per half cycle. A range of groove heights were investigated for the single groove for the Reynolds number range 0-15 to identify the mechanics through which fluid is transported across the channel by the grooves, the effect that inertial and viscous forces have on the process and to identify a groove height range for optimised cross channel fluid transfer. The flow field within the grooves at various heights was analysed and their relationship with non-axial flow within the bulk channel identified. The culminating effect of increasing grooves per half cycle on their ability to transport fluid across the channel is analysed by comparing the entrainment of fluid into and across the groove for both a single and infinite grooves. The maximum increase in fluid entrainment per groove for the addition of extra grooves to a cycle was found to be 14%. The helicity (or swirl) of the flow within the channel is found to be small for all three systems, while increased helicity within the flow was found to correspond to an increase in energy dissipation.     

An experimental study on fluid flow characteristics of superposed porous and fluid layers

In the present study, fluid flow characteristics of a porous layer overlaid by a fluid layer were investigated through experiments. The experimental results were analyzed in comparison with theoretical results of a porous medium bounded by impermeable walls. With spheres, the slip coefficient was found to be 0.0107 for Poiseuille flow over a porous layer. As the permeability decreased, the experimental results approached the values calculated by Darcy’s law and Forchheimer’s equation. In addition, the effects of the presence of a fluid layer over a porous medium were examined in terms of the friction factor. The present experimental data placed in the range of the Darcy to the non-Darcy region are shown to be in reasonable agreement with the proposed correlation.     

Numerical Investigations of Fluid Flow and Lateral Fluid Dispersion in Bounded Granular Beds in a Cylindrical Coordinates System

Results are presented from a numerical study examining the flow of a viscous, incompressible fluid through a random packing of non‐overlapping spheres at moderate Reynolds numbers, spanning a wide range of flow conditions for porous media. By using a laminar model including inertial terms and assuming rough walls, numerical solutions of the Navier‐Stokes equations in three‐dimensional porous packed beds resulted in dimensionless pressure drops in excellent agreement with those reported in a previous study. This observation suggests that no transition to turbulence could occur in the range of the Reynolds number studied. For flows in the Forchheimer regime, numerical results are presented of the lateral dispersivity of solute continuously injected into a three‐dimensional bounded granular bed at moderate Peclet numbers. In addition to numerical calculations, to describe the concentration profile of solute, an approximate solution for the mass transport equation in a bounded granular bed in a cylindrical coordinates system is proposed. Lateral fluid dispersion coefficients are then calculated by fitting the concentration profiles obtained from numerical and analytical methods. Comparing the present numerical results with data available in the literature, no evidence has been found to support the speculations by others for a transition from laminar to turbulent regimes in porous media at a critical Reynolds number.     

Slip in Flow through Assemblages of Spherical Particles at Low to Moderate Reynolds Numbers

Effects of slip velocity and volume fraction of slip spheres on the momentum transfer characteristics of assemblages of slip spheres are numerically investigated. The fluid slip along the surface of the sphere is considered by Navier's linear slip model. The dimensionless governing continuity and momentum equations are solved using a semi‐implicit marker and cell method implemented on a staggered grid arrangement in spherical coordinates. The convection and viscous terms of momentum equations are discretized by means of the QUICK scheme and a second‐order central differencing scheme, respectively. The present numerical solver is benchmarked via grid independence and comparisons with the existing literature values. Results were obtained over a wide range of pertinent dimensionless numbers such as the Reynolds number, volume fraction of the dispersed phase, and dimensionless slip parameter.     

A sinusoidal pressure response method for determining the properties of a porous medium and its in-situ fluid

Frequency response data from sinusoidal pressure experiments using liquids can be used to characterize the porous medium and its fluid properties. This paper examines the theory and presents correlations to help design experiments. Experiments were designed to check the theory. Measurements were made using a synthetic oil in an unconsolidated medium. Equipment was designed to generate pressure sine waves whose upper frequency limit was in excess of 1,000 rad/s. The theory indicates that there are definite frequency limits which may be used if the equations are to adequately describe the data. As the frequency and/or permeability increases, the inertial terms become important. This study shows that practical use can be made of existing theory to increase our ability to describe the flow of fluids in porous media.     

Oscillatory power number,power density model,and effect of restriction size for a moving-baffle oscillatory baffled column using CFD modelling

Although single-hole oscillatory columns have been studied since the 1990s, to this day there is an absence of appropriate dimensionless groups to express the hydrodynamic conditions and power requirement for the moving-baffle oscillatory baffled column (OBC). This paper uses computational fluid dynamic (CFD) software coupled with moving overset meshing to aid in the derivation of the first dimensionless oscillatory power number for OBCs. In terms of the moving-baffle OBC, this work marks the first time a power density equation has been derived specifically to account for this column's unique hydrodynamic profile. Equations for period-averaged Reynolds number and period-averaged Strouhal numbers were developed to better estimate the fluid intensity within these moving-baffle columns. This work serves as an example of how complex and challenging flow regimes, such as periodically oscillating flow, can be simplified and analyzed to produce appropriate design equations.     

Mathematical model for gas flow through a packed bed in the presence of sources and sinks

Flow within a packed bed is normally calculated by attempting to simultaneously satisfy the continuity and Ergun equations. However, the presence of gas sources/sinks within the bed escalates the complexity of the problem, particularly when the flow is two-dimensional and a solution to the full Ergun equation is required. In quest of an efficient and dependable algorithm for the calculation of gas flow, a critical review of existing solution methods was undertaken and a new method, ‘FLOW’, is now proposed. The technique retains the viscous and inertial pressure gradient terms of the Ergun equation, and both are treated as linear functions of the flow. Solutions are approached iteratively; using finite difference techniques, the continuity and linearized Ergun equations are solved for the pressure field; a new flow field is then calculated from which is derived an adjustment to the inertial resistance term of the Ergun equation. The sequence is repeated until satisfactory convergence is obtained. Relatively few iterations are normally required and, for the case of negligible inertial pressure drop, one calculation cycle is sufficient. A comparison of results obtained using the ‘FLOW’, modified ‘SIMPLE’ and vorticity procedures is presented. The proposed method allow flexibility in the specification of boundary conditions and can be applied to compressible or incompressible flow, as well as for the case of nonisothermal beds.     

Rheological phenomena during the polymerization of styrene in a stirred tank

An experimental study of the bulk polymerization of styrene has been carried out. The rate of polymerization, molecular weight, and flow patterns around agitators as a function of conversion have been observed. The rate data are interpreted in terms of free-radical polymerization kinetics. Duerksen and Hamielec's modified termination rate constant and initiator efficiency relations are generally consistent with our data. The flow patterns were interpreted in terms of the mechanics of viscoelastic liquids and observations for nonreacting systems. Specifically, at low conversions for spheres and similar agitators, the flow patterns are dominated by centrifugal forces with fluid drawn in at the poles and expelled at the equator for spherical agitators. At higher conversions, normal stresses interact with the centrifugal forces causing segregated secondary flow regions adjacent to the sphere. At very high concentrations, normal stresses dominate, yielding Weissenberg effects.     

Inertial Deposition of Aerosol Particles in a Periodic Row of Porous Cylinders

The aerosol flow through a periodic row of parallel porous cylinders is investigated. The air flow field outside the cylinders is described by the Navier–Stokes equations of viscous incompressible fluid. The extended Darcy–Brinkman equations are used to calculate the flow velocity inside a porous cylinder. The dependence of the efficiency of the deposition of aerosol particles by inertial impaction and interception on the Stokes number for various values of the Darcy number is studied. Comparison of the results obtained from the numerical model and an approximate analytical model is given. The combined approximate formula for the deposition efficiency of a cylindrical fiber in a parallel array proposed by Müller et al. (2014) is extended for the porous cylindrical fiber. The aerosol flow through the porous body composed by a random array of cylinders is calculated to estimate the interior deposition.

Copyright 2015 American Association for Aerosol Research     

CREEPING FLOW OF VISCOELASTIC FLUIDS THROUGH TAPERED SLIT DIES: AN ANALYTICAL SOLUTION

In the present study, it will be shown that at vanishingly small Reynolds numbers the extensional behavior of a fluid has a strong effect on its kinematics when flowing through a tapered slit die. To show this, the Cauchy equations of motion were simplified using the creeping flow approximation. Assuming the flow to be steady, laminar, two-dimensional, and incompressible, the governing PDEs were reduced to a set of coupled ODEs using a series solution in terms of the powers of 1/r. Two different constitutive equations, both incorporating an extensional parameter, were used in the analysis: (i) the Giesekus model and (ii) the Simplified Phan-Thien-Tanner (SPTT). An analytical solution was found for each fluid model clearly depicting the strong influence of the extensional parameter on the flow becoming non-radial within the channel.     

Geometry of bluff body wakes

The attached recirculating wake zone has been experimentally investigated for spheres, spherical caps and circular discs normal to the flow. Tracer photography in a water tunnel was employed for the measurements. Results are presented over the range of Reynolds number from 20 to 400. The following wake properties are presented as a function of Reynolds number: angle of separation for spheres; wake length for spheres and discs, wake volume for all three types of body. The results for separation angle are in good agreement with the data available from the literature, both experimental and theoretical. Wake length results for spheres are also in agreement with published values and with values obtained from published numerical solutions of the equations of motion. Wake volume results for spheres are similarly in agreement with values calculated from published wake dimensions obtained both experimentally and theoretically. Spherical caps are found to have the wake volume either of discs if they are thin (h/D <0.3) or of spheres if they are thick (h/D > 0.6). Caps of intermediate thickness to diameter ratio show a transition between sphere-like and disc-like behavior. This behavior is explained in terms of the separation angle on spheres.     

FLOW OF A SUSPENSION THROUGH A VERTICAL TUBE

Jiang el at. (1986) have developed the local volume averages of the equations of motion as well as the appropriate boundary conditions for a flowing suspension of non-neutrally buoyant, uniform spheres in an incompressible Newtonian fluid under conditions such that inertial effects can be neglected. Neither the equations nor the boundary conditions involve any free or undetermined parameters.

These local volume-averaged equations of motion have been solved for flow through a vertical tube and the results compared with available measurements of velocity profiles and of apparent viscosities. The predictions of measured apparent viscosities are in error by no more than 3%, when the volume fraction of solids is less than 0.20. The errors in representing the velocity profiles are comparable, but based upon fewer data.     

Laminar flow past a permeable sphere

Flow past an isolated permeable sphere has been studied. The complete Navier-Stokes equation governs the fluid motion outside the sphere, while Brinkman's extension of Darcy's Law is assumed to hold within the porous sphere. The Navier-Stokes equation is solved using a finite difference scheme. The flow within the porous sphere is solved in two different ways, each being efficient over a particular range of Reynolds number. Drag Coefficients are presented for dimensionless permeability, β, of 5, 10, 15, and 30 and for Reynolds numbers up to 50. The computed drag coefficients are within 10% of the experimental values observed by Masliyah and Polikar for 15 < β > 33, the range covered in their work. Separation was observed only for β > 10. The onset of separation is delayed considerably in porous spheres.     

The effect of surfactant on terminal velocity of and mass transfer from a fluid sphere in a non-newtonian fluid

The effect of interfacial tension gradients on creeping flow past a fluid sphere (bubble or drop) in a non-Newtonian fluid is investigated. The drag force and the Sherwood number for contaminated fluid spheres moving in a non-Newtonian fluid are obtained by an approximate solution of equations of motion in the creeping flow regime. It has been found that surface-active agents decrease both the terminal velocity and the mass transfer rate. The influence of the flow index of the power-law on both the drag coefficient and the Sherwood number is superimposed over the influence of the interfacial tension gradient. Comparisons between the present analysis and the available experimental data in the literature show reasonable agreement for the terminal velocity and the Sherwood number.     

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.     

Hydrodynamic drag forces on two porous spheres moving along their centerline

This paper numerically evaluates the hydrodynamic drag force exerted on two highly porous spheres moving steadily along their centerline (sphere #1 and sphere #2) through a quiescent Newtonian fluid over a Reynolds number ranging from 0.1 to 40. At creeping flow limit, the drag forces exerted on both spheres were identical. At higher Reynolds numbers the drag force on sphere #1 was higher than sphere #2, revealing the shading effects produced by sphere #1 on sphere #2. At dimensionless diameter (β, =d_f/2k^0.5, d_f and k are floc diameter and interior permeability, respectively) >20, the spheres can be regarded nonporous. At β<20, the drag forces dropped. At β<2, the drag forces approached “no-spheres” limit. An increased size ratio of two spheres (d_f1/d_f2) would increase the drag force on sphere #1 and reduce that on sphere #2. At increasing β for both spheres, the drag force on sphere #2 was increased because of the more difficult advective flow through its interior, and at the same time the drag was reduced owing to the stronger wake flow produced by the denser sphere #1. The competition between these two effects leads to complicated dependence of drag force on sphere #2 on β value. These effects were minimal when β became low. Two identical spheres could move steadily along their centerline. At higher Reynolds number, the two spheres would move closer because of the incorporation of inertia force. For spheres of different diameters, the sphere # 2 would move faster than sphere #1 regardless of their size ratio and β value. This occurrence yielded efficient coagulation when two porous spheres were moving in-line.     

TURBULENT RESISTANCE OF COMPLEX BED STRUCTURES

The influence of complex packing geometry on its frictional/hydrodynamic resistance was investigated both experimentally and theoretically. The complex bed structures were modelled using the cubic packing of spheres and different number of thin bands inserted between the spheres. The turbulent flow resistance of the model systems was determined experimentally on the basis of pressure drop and air flow rate measurements. In the theoretical approach these systems were regarded as compositions of two Representative Elementary Units (REU's) which contribute to the overall pressure drop within the apparatus. The values of the overall coefficient f₀ , characterising resistance of the complex geometry structures were correlated with the values of the local coefficients f_i , describing resistance of the particular REU?s. The latter ones were independently estimated using the Computational Fluid Dynamics (CFD) code.