Unified entry length for newtonian and power–law fluids in laminar pipe flow

A numerical method based on finite differencing is used for investigating the steady–state entrance region laminar flow of incompressible Newtonian and power–law fluids in a circular pipe. The Solution method is validated by comparing the results for Newtonian fluids with those reported in the literature. For power–law fluids, the entry length results are compared with other approximate solutions in the literature. On the basis of the calculated results, a generalized entry length ξ99 = 0.056 is shown to be valid for the laminar flow at Re > 200 of both Newtonian and power–law fluids with 0.75 < n < 1.5.     

Drag on chains and agglomerates of spheres in viscous newtonian and power law fluids

New experimental data on the free settling velocity of straight chains (up to twenty spheres) and planar clusters of touching spheres in Newionian and power law media are reported. The results embrace the following ranges of conditions: 0.65 ≤ n ≤ 1; Re < - 2.5 and 1.22 < m < 48.87 Pa·sⁿ. The straight chain drag measurements are in line with theoretical predictions for Newtonian fluids. The present results in power law fluids seem to suggest that it is possible to express the drag on a straight chain of spheres in terms of that on a single sphere of equal volume. Limited results with planar clusters are satisfactorily correlated using a volume equivalent sphere diameter.     

Three‐Dimensional Simulation of Bubble Formation Through a Microchannel T‐Junction

Three‐dimensional simulations of bubble formation in Newtonian and non‐Newtonian fluids through a microchannel T‐junction are conducted by the volume‐of‐fluid method. For Newtonian fluids, the critical capillary number Ca for the transition of the bubble breakup mechanism is dependent on the velocity ratio between the two phases and the microchannel dimension. For the power law fluid, the bubble diameter decreases and the generation frequency increases with higher viscosity parameter K and power law index n. For a Bingham fluid, the viscous force plays a more important role in microbubble formation. Due to the yield stress τ_y, a high‐viscous region is developed in the central area of the channel and bubbles deform to a flat ellipsoid shape in this region. The bubble diameter and generation frequency are almost independent of K.     

Power‐Law Fluid Flow Over a Sphere: Average Shear Rate and Drag Coefficient

Based on the consideration of the rate of mechanical energy dissipation, an expression for the average shear rate for a sphere falling in a power‐law fluid in the creeping flow regime has been deduced. The average shear rate in a power‐law fluid (n<1) appears to be higher than that in an equivalent Newtonian fluid. This in turn has been combined with the numerical predictions of drag coefficient (up to Reynolds number of 100) of a sphere to develop a generalized drag correlation for power‐law liquids encompassing both n > 1 and n < 1 which appears to apply up to much higher values of the Reynolds number. The available experimental data have been used to demonstrate the reliability and accuracy of the new correlation for shearthinning liquids. Also, in the limit of n = 1, this expression reproduces the standard drag curve with a very high accuracy.     

Hindered settling in non-newtonian power law liquids

New experimental results on the hindered settling of model glass bead suspensions in non-Newtonian suspending media are reported. The data presented encompass the following ranges of variables: 7.38 × 10^?4 ≤ Re_1∞ ≤ 2; 0.0083 ≤ d/D ≤ 0.0703; 0.13 ≤ C ≤ 0.43 and 1 ≥ n ≥ 0.8. In these ranges of conditions, the dependence of the hindered settling velocity on concentration is adequately represented by the corresponding Newtonian expressions available in the literature. The influence of the power law flow behaviour index is completely embodied in the modified definition of the Reynolds number used for power law liquids.     

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.     

Flow through porous media of a shear-thinning liquid with yield stress

Darcy's law for the laminar flow of Newtonian fluids through porous media has been modified to a more general form which will describe the flow through porous media of fluids whose flow behavior can be characterized by the Herschel-Bulkley model. The model covers the flow of homogeneous fluids with a yield value and a power law flow behavior. Experiments in packed beds of sand were carried out with solutions of paraffin wax in two oils and with a crude oil from the Peace River area of Canada. The model fitted the data well. A sensitivity analysis of the fitting parameters showed that the model fit was very sensitive to errors in the flow behavior index, n , of the Herschel-Bulkley model. A comparison of the “n” values calculated from viscometer measurements and from flow measurements agreed well. A more general Reynolds number for flow through porous media, which includes a fluid yield value, was developed. The data were fitted to a Kozeny-Carman type equation using this Reynolds number. The constant in the Kozeny-Carman equation was determined for the two packed beds studied using Newtonian oils. The data could all be represented, within the experimental error, by the relationship f^* = 150/Re^*. Since the mean volume to surface diameter of the packing was determined by the measurement of its permeability to a Newtonian oil, assuming C' = 150, the new definition of the Reynolds number allows the direct use of the Kozeny-Carman equation with Herschel-Bulkley type fluids.     

Oxygen transfer into newtonian and non-newtonian fluids in mechanically agitated vessels

Oxygen transfer into Newtonian and non-Newtonian fluids was studied in stirred tank vessels of 0.0018, 0.006 and 0.036 m³ operating volumes. Emphasis was given to the rheological effects of the pseudoplastic medium on the volumetric oxygen transfer coefficient, K_La. Results indicate that K_La is a strong function of gassed power input per unit volume (P_g/V) for the Newtonian fluid, but a weak function of P_g/V for the non-Newtonian fluid, and a strong function of superficial gas velocity (V_s) for both fluids for paddle-type impellers. K_La is found to decrease rapidly with an increase in apparent viscosity for values of μ_a greater than 2.0 Pa · s. In addition to various correlations, a dimen-sionless correlation including the impeller Reynolds number, impeller Weber number, and the aeration number is presented for the prediction of K_La in a gas-liquid non-Newtonian system.     

Effect of blockage on heat transfer from a cylinder to power law liquids

The influence of planar confining walls on the steady forced convection heat transfer from a cylinder to power-law fluids has been investigated numerically by solving the field equations using FLUENT (version 6.2). Extensive results highlighting the effects of the Reynolds number (1?Re?40), power-law index (0.2?n?1.8), Prandtl number (1?Pr?100) and the blockage ratio (β=4 and 1.6) on the average Nusselt number have been presented. For a fixed value of the blockage ratio, the heat transfer is enhanced with the increasing degree of shear-thinning behaviour of the fluid, while an opposite trend was observed in shear-thickening fluids. Due to the modifications of the flow and temperature fields close to the cylinder, the closely placed walls (i.e., decreasing value of the blockage ratio) further enhance the rate of heat transfer as the fluid behaviour changes from Newtonian to shear-thickening fluids (n>1), the opposite influence is seen with the decreasing value of the flow behaviour index (n) in shear-thinning (n<1) fluids. Finally, the functional dependence of the present numerical results on the relevant dimensionless parameters has been presented in the form of closure relationships for their easy use in a new application.     

Experimental and CFD investigation of convective heat transfer in helically coiled tube heat exchanger

Experimental studies on isothermal steady state and non-isothermal unsteady state conditions were carried out in helical coils for Newtonian as well as for non-Newtonian fluids. Water and glycerol–water mixture (10 and 20% glycerol) were used as Newtonian, and 0.5–1% (w/w) dilute aqueous polymer solutions of Sodium Carboxy Methyl Cellulose (SCMC) and Sodium Alginate (SA) as non-Newtonian fluids are used in this study. These experiments were performed for coil curvature ratios as δ = 0.0757, 0.064 and 0.055 in laminar and turbulent flow regimes (total 258 tests). The CFD analyses for laminar and turbulent flow were carried out using FLUENT 12.0.16 solver of CFD package. The CFD calculation results (Nu_i, U, T₂ and Tw_o) for laminar and turbulent flow are compared with the experimental results and the work of earlier investigators which were found to be in good agreement. For the first time, an innovative approach of correlating Nusselt number to dimensionless number, ‘M’, Prandtl number and coil curvature ratio using least-squares power law fit is presented in this paper which is not available in the literature. Several other correlations for calculation of Nusselt number for Newtonian and non-Newtonian fluids, and two correlations for friction factor in non-Newtonian fluids (based on 78 tests and 138 tests) are proposed. These developed correlations were compared with the work of earlier investigators and are found to be in good agreement.     

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.     

The Effects of Power Law Fluid Rheology on Coil Heat Transfer for Agitated Vessel in the Transitional Flow Regime

A CFD model of heat transfer from power‐law fluids to helical cooling coils in the transitional flow regime of a baffled tank mixed with a pitched blade turbine was developed with Fluent^TM. The model captured local temperature and velocity gradients. Simulations were run, varying Re, Pr, K and n. The results indicate that a Sieder‐Tate type correlation, with the exponent on and the coefficient in front of the Reynolds number being a function of n, is recommended for estimating ho. Also, a new two coil bank design was found to be more efficient when 450 < Re < 650.     

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     

Natural convection in a vertical porous cavity filled with a non‐newtonian binary fluid

Numerical and analytical study of natural convection in a vertical porous cavity filled with a non‐Newtonian binary fluid is presented. The density variation is taken into account by the Boussinesq approximation. A power‐law model is used to characterize the non‐Newtonian fluid behavior. Neumann boundary conditions for temperature are applied to the vertical walls of the enclosure, while the two horizontal ones are assumed impermeable and insulated. Both double‐diffusive convection (a = 0) and Soret‐induced convection (a = 1) are considered. Scale analysis is presented for the two extreme cases of heat‐driven and solute‐driven natural convection. For convection in a thin vertical layer (A ? 1), a semianalytical solution for the stream function, temperature, and solute fields, Nusselt and Sherwood numbers are obtained using a parallel flow approximation in the core region of the cavity and an integral form of the energy and constituent equations. Numerical results of the full governing equations show the effects of the governing parameters, namely the thermal Rayleigh number, R_T, the Lewis number, Le, the buoyancy ratio, φ, the power‐law index, n, and the integer number a. A good agreement between the analytical predictions and the numerical simulations is obtained. © 2012 American Institute of Chemical Engineers AIChE J, 58: 1704–1716, 2012     

Generalized slit flow of an ellis fluid

Isothermal, steady‐state and fully developed flows of Ellis fluids in planar and annular slits are discussed. The flow equations derived for Ellis fluids describe also the flows of Newtonian and power law fluids as specific cases. The most important flows resulting from thp general theory, i.e., the pressure flows in flat and annular slits for stationary channel walls and at transverse and longitudinal movements of a wall, are analyzed in detail. Numerical verification of the results leads to the conclusion that the planar and annular flows of Ellis and power law fluids are qualitatively similar The quantitative differences resulting from the slit curvature and the type of constitutive equation are relatively large only for flows of strongly non‐New‐tonian liquids.     

The influence of fluid elasticity on wall effects for creeping sphere motion in cylindrical tubes

Experimental results are presented of wall effect for the slow motion of spheres in elastic, constant-viscosity liquids. The results are correlated in terms of diameter ratio for d/D < 0.3, and Weissenberg number We < 5. Weissenberg number is defined as We = 2θV_m/d, with θ the Maxwellian relaxation time (θ = N₁/2τγ). The wall effect is found to be adequately described by Newtonian expressions for small Weissenberg number, We < 0.01. For larger values of the Weissenberg number, We > 0.2, virtually no wall effect is discernible; the small effect observed is correlated by the wall factor expression The wall effect observed is ascribed to the influence of fluid elasticity alone, since all the fluids used were elastic to a greater or lesser extent, but showed no shear thinning.      

A numerical study of the accelerating motion of a dense rigid sphere in non-newtonian power law fluids

The equations of motion of an accelerating sphere falling through non-Newtonian fluids with power law index n in the range 0.2 ≤ n ≤ 1.8 were integrated numerically using the assumption that the drag on the sphere was a function of both power law index and terminal Reynolds number, Re_t For 10^?2 ≤ Re_t ≤ 10³ both dimensionless time and distance travelled by the sphere under transient conditions showed a much stronger dependence on the flow behaviour index, n, for shear-thinning than for shear-thickening fluids. The form of this dependence is investigated here. Furthermore, results in four typical shear-thinning fluids suggested a strong correlation between the distance and time travelled by the sphere under transient conditions and the value of the fluid consistency index. The analysis reported herein is, however, restricted to dense spheres falling in less dense fluids, when additional effects arising from the Basset forces can be neelected.     

Some qualitative and quantitative features of die swell

Upper and lower bounds for the die swell ratio are derived for all simple fluids in the presence of surface tension and under the assumptions that the flow remains viscometric up to a finite distance from the exit and that bounds on the shear stress distribution in this length exist. Both circular and plane jets are considered and the bounds so derived depend on the normal stress on the wall in the viscometric region. For Newtonian and power law fluids, this stress can be determined, a priori, from existing theories. However, for viscoelastic liquids, it is an open question at present.     

High speed wire coating by withdrawal from a bath of viscoelastic liquid

Data are presented for the thickness H_∞ of liquid coating entrained by continuous withdrawal at speed U of a wire of radius R from the free surface of a large bath. For Newtonian fluids of viscosity μ, density ρ and surface tension σ, the data are carried out to coating speeds beyond the applicability of current theories, to Capillary numbers of nearly one hundred. In the high speed range the data, which cover several orders of magnitude in viscosity, can be well represented by the equation for U_μ/σ = N_ca > 3. All data presented are at an essentially constant Goucher number of 0.08, where N_Go ≡ R(ρg/2σ)^1/2. Data for viscoelastic fluids show phenomena quite distinct, qualitatively and quantitatively, from Newtonian observations. In particular, strongly elastic fluids show a markedly reduced ability to be entrained onto the wire. Further, the coating thickness appears to become independent of Capillary number at high speed.     

Momentum transfer to Newtonian and non-Newtonian fluids flowing through packed and fluidized beds

This paper deals with the flow behaviour of Newtonian and non-Newtonian fluids flowing through packed and expanded beds. With the help of tube-bundle theory a generalized average shear-stress—shear-rate relationship is derived and found to predict the flow behaviour of power law as well as non-power law fluids. Polyvinyl alcohol solutions in water, a representative of power law fluids, and grease in kerosene, a representative of nonpower law fluids, are studied. The present investigation covers the range of Reynolds number from 10^?3 to 10³. An expression for average shear-stress at minimum fluidization velocity is derived and found to agree with that of our experiment. The generalized frictional Reynolds number is defined and a design chart is also presented for the evaluation of fluidizing velocities.