An analytical model for steady coextrusion of viscoplastic fluids in thin slit dies with wall slip

Coextrusion is widely used to fabricate multilayered products with each layer providing a separate functionality, including barrier resistance to gases, strength, and printability. Here an analytical model of the coextrusion die flow of two incompressible, viscoplastic fluids in a slit die, subject to nonlinear wall slip and under fully developed and isothermal conditions, is developed to allow the prediction of the steady‐state velocity and shear stress distributions and the flow rate versus pressure gradient relationship. The resulting model is applied to the coextrusion of two layers of viscoplastic fluids in a thin rectangular slit die (slit gap, h ? slit width, W). The analytical solution recognizes a number of distinct flow conditions (eleven cases) that need to be treated separately. The solutions for all eleven cases are provided along with an apriori identification methodology for the determination of the applicable case, given the shear viscosity and wall slip parameters of the two viscoplastic fluids, the slit geometry and the flow conditions. Simplifications of the model would provide the solutions for the fully developed and isothermal coextrusion flows of any combination of Hershel‐Bulkley, Bingham, power‐law and Newtonian fluids with or without wall slip at one or both walls of the slit die. POLYM. ENG. SCI., 2010. © 2009 Society of Plastics Engineers     

Modelling laminar pulsed flow for the enhancement of cleaning

A Green function method is presented which enables computation of laminar flow of an incompressible Newtonian fluid in circular and annular pipes, subject to an arbitrary forcing periodic pressure gradient, in terms of Bessel functions. The response to a step change in pressure gradient in an annular pipe is presented. The method allows direct calculation of wall shear stress and flow rates generated by pulsed flows, which are of interest in fouling mitigation and cleaning-in-place systems.     

Non-newtonian flow through open channels

A general method is proposed for prediction of the flow rate and maximum velocity in the isothermal, steady, uniform, laminar flow of any incompressible, time-independent non-Newtonian fluid in straight open channels of arbitrary cross section. The method requires only a knowledge of two geometric coefficients and a function of shear stress, used to characterize the behavior of the fluid model. The slip effect at the solid boundary has been taken into consideration. Numerical values of the geometric parameters have been determined for flow through an inclined plane of infinite width, semi-circular, semi-elliptical, rectangular, and 90° and 60° symmetric triangular open channels. Applications have been made to various non-Newtonian fluid models such as the power-law, Bingham, Ellis, Meter and the Reiner-Rivlin general model. Numerical examples are presented. A generalization of the Fanning friction factor — Reynolds number is also presented. The problem of determining the point of transition from laminar to turbulent flow in the general case is examined, as is the problem of prediction of the friction factor in turbulent flow.     

Non-isothermal laminar channel flow

This article deals with the non-isothermal, laminar flow of a power-law fluid between parallel plates with a constant plate temperature. The fluids are assumed to be incompressible and to have constant physical properties except for the consistency-index, which latter quantity is assumed to vary exponentially with temperature. Calculations are performed on the “non-isoviscous” hydrodynamics and heat transfer rate. The results are presented as logarithmic mean Nusselt-numbers and dimensionless velocity distributions and pressure drops as a function of the dimensionless axial length in the channel and as a function of a quantity Q which measures the deviation of “isoviscous” fluid behaviour.The heat transfer rate appears to be a unique function of the velocity gradient at the wall, independent of the fluid rheology.All results have been summarized in a few practical approximations and are verified experimentally with pressure drop and heat transfer measurements of laminar, newtonian channel flow. It turns out that the results may be applied to flow between rectangular ducts up to aspect ratios H/B = 0·25.     

Forced convection heat transfer to a power-law fluid in arbitrary cross-section ducts

A numerical method is developed to predict the three-dimensional forced convection laminar incompressible flow of a power law fluid in arbitrary cross-section straight ducts. The continuity equation and boundary layer forms of the energy and momentum equations in rectangular coordinates are transformed into new orthogonal coordinates with boundaries coinciding with the coordinate surfaces. The resulting equations are solved using the finite difference technique. The numerical scheme is capable of handling different hydrodynamic and thermal entry boundary conditions but results are only presented for uniform inlet velocity and temperature profiles and isothermal wall. To demonstrate the wider applicability of the method local heat transfer coefficients and pressure drop in square, trapezoidal and regular pentagonal ducts are computed as functions of pertinent thermal and hydrodynamic parameters.     

Eccentric laminar couette flow of long cylindrical capsules

The equation of motion for laminar Couette flow of an incompressible Newtonian fluid, induced by the eccentric longitudinal motion of a long circular cylinder in a circular pipe, is solved analytically. Expressions are obtained and typical plots presented for local velocity distribution, local shear stress distributions on both the surface of the pipe wall and the surface of the cylindrical core, total drag force on both pipe wall and moving core, twisting moment on the core, and volumetric flow rate of the liquid. As the eccentricity approaches unity, the solution for flow rate is shown to coincide exactly with a practical formula developed by Kruyer and Ellis to represent the Couette flow induced by the motion of cylindrical capsules.     

EXTRUSION AND LUBRICATION FLOWS OF VISCOPLASTIC FLUIDS WITH WALL SLIP

The simplest model flow which approximates the extrusion (shallow screw channels) and lubrication flow is the steady, laminar flow occurring between two infinitely long parallel plates i.e., the generalized plane Couette flow. Here we develop an analytical model of the generalized plane Couette flow of viscoplastic fluids. The deformation and flow behavior of viscoplastic fluids can be realistically represented with the Herschel-Bulkley constitutive equation, which we have utilized as the basis for the development of our analytical model. Furthermore, as also demonstrated here, the deformation behavior of viscoplastic fluids is generally complicated by the presence of wall slip at solid walls, which occurs as a function of the wall shear stress. The wall slip versus the wall shear stress behavior of viscoplastic fluids can be experimentally characterized using viscomelric flows, including steady torsional and capillary flows. Thus determined Navier's wall slip coefficient can then be utilized in modeling of processing flows. In our analytical model of the generalized plane Couette flow of viscoplastic fluids the Navier's wall slip boundary condition was included. This model should be an important engineering tool, which provides design expressions for the extrusion and lubrication flows of viscoplastic fluids, with or without wall slip occurring at the walls. @KEYWORDS:Extrusion, lubrication, flow, viscoplastic, slip.     

What is the role of “pressure” in the use of capillary and slit flows to determine the shear‐rate dependent viscosity of a viscoelastic fluid?

The purpose of this review article is to clear the confusion created by some investigators, who erroneously thought that the pressure transducers mounted on the wall of a capillary or slit die measured a quantity that could meaningfully be called “pressure,” accurately stated “indeterminate isotropic contribution to the total stress,” and then reported on the effect of “pressure” on the shear‐rate dependent viscosity of a viscoelastic fluid. On the other hand, reference to such a quantity is not needed to calculate the wall shear stress and thus shear viscosity in fully developed flow of incompressible, viscoelastic polymer melts in a capillary or slit die; instead only information on the gradient of the total wall normal stress is needed. Further, it is pointed out that much of the literature discussing “pressure shift factor” to describe the effect of “pressure” on the viscosity of polymer melts in flow through a capillary or slit die is based on an erroneous belief that there exists a physically meaningful isotropic “pressure” that can be measured. POLYM. ENG. SCI., 2008. © 2008 Society of Plastics Engineers.     

Convective diffusion in a parallel plate duct with one catalytic wall—laminar flow—first order reaction: Part II: Experimental

The catalytic reaction rate of combustion of hydrogen in air on platinized alumina was measured in a flat rectangular flow reactor with one catalytic wall in laminar flow. Since the rate of reaction is very high, this measurement normally must be made in turbulent flow. However, with the aid of the measured concentration profiles and the analysis presented in Part I, it was possible to ascertain the reaction rate in the laminar flow regime. The rate constants compare well with literature values which were measured in the turbulent flow regime. Average Nusselt numbers for mass transfer calculated for the diffusionally limited cases were in good agreement with those obtained from Part I. A narrow rectangular channel (aspect ratio = 8:1) with 16-in. of one wall comprising the reactive section was constructed for this investigation. The channel has a long preheat section, an adiabatic section with multiple heaters, and thermocouples for maintaining isothermal conditions. Two micrometer mounted probes are inserted through the wall opposite the replaceable catalytic section. Concentrations are measured at the inlet, inside the reactive section, and at the outlet in a mixing chamber.     

NONSIMILAR SOLUTIONS FOR HEAT TRANSFER IN WALL JET FLOWS

An analysis is presented to investigate the flow and heat transfer characteristics of a laminar plane wall jet with non-isothermal wall as well as uniform suction and blowing at the surface and a laminar cylindrical wall jet. The approach used is local nonsimilarity method, wherein, the nonsimilanty terms appearing in the momentum and energy equations are retained and simplifications are introduced only in the auxiliary system of equations. To insure the accuracy of the results, solutions are obtained for three levels of truncation of the governing equations. For the case of plane wall jet problem, both a series solution as well as local nonsimilarity solution have been given and the agreement between the two is found to be very good. For the cylindrical wall jet problem, the results obtained by the local nonsimilarity approach in the present paper have been compared with the series solution results. Numerical results for the wall shear stress, velocity distribution, wall heat transfer rate and temperature field are presented.     

HEAT TRANSFER IN A LAMINAR CYLINDRICAL WALL JET WITH UNIFORM SURFACE HEAT FLUX

The steady state flow and heat transfer characteristics of a laminar cylindrical wall jet are obtained for uniform surface heat flux conditions. Local nonsimilarity solutions as well as series solutions are presented for the velocity and thermal fields. Numerical results are given for the wall shear stress, surface temperature variation and temperature field for a Prandtl number of 0.73.     

Polarization-optical investigation of normal and shear stresses in flow of polymers

A method has been suggested for calculating the first difference of normal stresses characterizing the flow of polymers at high shear stresses. The calculations are based on the results of rheooptical measurements in a slit of rectangular cross section. It has been found, for several samples of high molecular weight polybutadienes and polyisoprenes, that the flow behavior of the representatives of the given polymer homologous series having different molecular weights is characterized by a general relationship between the first normal stress differences and the shear stresses in those cases where the polymers are characterized by narrow molecular weight distributions. It has also been established that the first normal stress difference sharply increases in the region of shear stresses which immediately precedes the spurt—a jumpwise increase of the flow rate at a certain critical value of shear stress; while for polymers of wide molecular weight distribution the increase of the normal stress difference in the region of high values of shear stresses is retarded. Equilibrium swell of the extrudate has been measured and the first normal stress difference determined by the rheo-optical method has been found to agree satisfactorily with the values calculated from the swelling ratios according to theoretical models.     

Effect of filler content and temperature on steady‐state shear flow of wood/high density polyethylene composites

Steady state shear flow of wood/high density polyethylene composites is investigated through capillary rheometry to gain better insight into rheology, extrudate distortions, and wall slip phenomena of wood/polymer composite melts. Effects of filler content and temperature on onset and end of stick‐slip transition, in terms of shear rate and shear stress, are also studied. Results show that shear rates at stick‐slip transition decrease while corresponding shear stresses increase with the addition of filler. Furthermore, temperature raises the shear rate and the shear stress at which the transition occurs. It is observed a log‐linear relationship in the plots of wall slip versus the shear stress, in particular, increasing the filler content and decreasing the temperature, these plots are shifted to higher shear stress, as a consequence of viscosity increment. Wall slip and filler content play a fundamental role in surface morphology; specifically, extrudates become smoother with increasing filler content and shear rate, whose increment always results in a rise of the wall slip. POLYM. COMPOS., 2011. © 2011 Society of Plastics Engineers     

Flocculation in laminar tube flow

The extent of orthokinetic flocculation of suspensions as a result of laminar tube flow has been calculated using Smoluchowski theory and shown to depend primarily on the tube dimensions and not on the flow rate. When a correction factor allowing for hydrodynamic resistance between approaching particles is introduced, the calculated flocculation rate can be much lower than the Smoluchowski value and depends inversely on flow rate and particle size. A further effect which has to be included is the non-uniformity of shear rate in the tube, which leads to less flocculation than expected on the basis of a uniform shear rate throughout the tube. Coiling of the tube should given an extra degree of mixing to the flowing suspension and hence a more uniform shear rate and an enhanced flocculation.Preliminary experiments using model suspensions flowing through straight and coiled tubes have shown fair agreement with these predictions.It is suggested that laminar flow through coiled tubes could provide the basis of a practical flocculation test method.     

Shear degradation of polymer solutions

Shear degradation of polyisobutylene solutions was studied in laminar flow through capillaries. A special apparatus was designed with a view to a controlled shear history. The various effects of initial molecular weight, concentration, temperature, and shear stress at the wall on the final degradation product are combined in a single parameter representing the minimum force required for degrading a polymer sample to its final state. This work is based on the M.Sc. Thesis of A. Kadim, submitted at the Department of Chemical Engineering, Technion—Israel Institute of Technolog, Haifa, Israel.     

HOMOTOPY ANALYSIS OF FREE CONVECTION BOUNDARY LAYER FLOW WITH HEAT AND MASS TRANSFER

A steady two-dimensional laminar boundary layer flow of an incompressible viscous fluid over a semi-infinite surface is considered to investigate the accuracy of the homotopy analysis method. The governing coupled nonlinear system of differential equations is solved by means of the HAM approach. Explicit analytical series solutions are obtained and compared with numerical solutions. Good agreement is observed between the numerical results and HAM analytical solutions.     

FINITE ELEMENT ANALYSIS FOR LAMINAR FLOW AND HEAT TRANSFER IN FINITE ROD BUNDLES

This investigation deals with the application of finite element method to solve the thermohydraulic problem of laminar fully developed flow in the interior and wall sub-channels of finite fuel rod bundles. A variational principle has been used for the solution of the momentum and energy equations. Wall shear stress and temperature distributions, ?Re and Nusselt numbers are obtained for the sub-channels of different configurations. The results are compared with solutions generated by collocation and finite difference methods.     

FINITE ELEMENT ANALYSIS FOR LAMINAR FLOW AND HEAT TRANSFER IN FINITE ROD BUNDLES

This investigation deals with the application of finite element method to solve the thermohydraulic problem of laminar fully developed flow in the interior and wall sub-channels of finite fuel rod bundles. A variational principle has been used for the solution of the momentum and energy equations. Wall shear stress and temperature distributions, ƒRe and Nusselt numbers are obtained for the sub-channels of different configurations. The results are compared with solutions generated by collocation and finite difference methods.     

Liquid wall friction in two-phase turbulent gas laminar liquid stratified pipe flow

The main result of the present work is an analytic expression for the mean liquid wall shear stress in two-phase turbulent gas/laminar liquid stratified pipe flow. The Navier-Stokes equations are solved assuming a flat fluid-fluid interface subject to a constant interfacial shear — approximating the interfacial drag exerted by the gas. The effect of a pipe inclination is accounted for, thereby retaining the interesting two-phase phenomenon of backflow in upwardly inclined pipes. The corresponding expression for the wall shear stress distribution is obtained by formal differentiation. Its limiting behaviour in the triple points, where the fluid-fluid interface meets the pipe wall, is determined by residue calculus. The expression for the mean wall shear stress is given by integration. It is found to be a linear combination of two terms. The first term accounts for the free surface liquid flow in the absence of the gas. The corresponding approximate hydraulic diameter model is found to be in surprisingly good agreement with this term. The second term represents the shear flow associated with the interfacial drag exerted by the gas (not accounted for by the hydraulic diameter approximation). The shear flow increases the flow rate near the interface on behalf of the flow rate near the pipe wall, thus reducing the wall shear stress below the free surface flow value. Expedient evaluation of the expression for the mean wall shear stress, suitable for use in a 1-D multiphase pipe flow simulator, is facilitated by replacing certain one-parameter integrals with highly accurate rational approximations.     

Analysis of Shear Stress Growth Experiments for Linear Constitutive Equations

Shear stress growth curves for viscoelastic fluids at low shear rates are analyzed using two linear rheological constitutive equations, an integral constitutive equation and a mixed type constitutive equation. It is shown that some published solutions do not satisfy all of the pertinent boundary conditions. For the low shear rate region, available experimental shear stress curves show a monotonic increase with decreasing slope in the shear stress. Shear stress curves calculated using a mixed type constitutive equation are found to exhibit this type of behavior while curves calculated using an integral constitutive equation do not. For the mixed type constitutive equation, the calculated developing velocity distribution is used to examine its effect on the developing shear stress distribution. For low values of E (the elasticity number), there is a moderate effect, but, for sufficiently large values of E, the developing velocity distribution has a negligible effect. It is also shown that results consistent with experimental data obtained at low shear rates can be attained using a single relaxation time. Additionally, incompressible Newtonian fluids are considered, and it is found that there can be single maxima in some shear stress curves with no maxima occurring in the velocity curves. Multiple maxima were not obtained in the Newtonian shear stress results unlike some published results.