Predictions of heat transfer and pressure drop for a modified power law fluid flow in a square duct

Numerical Solutions for the Nusselt Numbers (CHF and CWT) and the Friction Factor times Reynolds Number have been obtained for fully developed laminar flow of a MPL (Modified Power Law) fluid within a square duct. The solutions are applicable to pseudoplastic fluids over a wide shear rate range from Newtonian at low shear rates through a transition region to power law behavior at higher shear rates. A shear rate parameter is identified, which allows the prediction of the shear rate range for a specified set of operating conditions. Numerical results of the Nusselt numbers (CHF and CWT) and the Friction factors times Reynolds number for the Newtonian and power law regions are compared with previous published results, showing agreement with 0.02% in Newtonian region and 4.0% in power law region.     

Power‐law polymer solution flow in a converging annular spinneret: Analytical approximation and numerical computation

A simplified analytical solution for the flow of power‐law liquids through a conical annulus was derived to estimate the flow profile, wall shear rate, and elongation rate in the spinneret during the spinning of hollow fiber membranes. The velocity profiles and shear and elongation rates of the power‐law fluid showed good agreement with those obtained from computational fluid dynamic simulations. Although the actual spinneret is characterized by an annulus with a converging cross section, most studies to date have used a geometrical concentric annulus for the sake of simplicity. The results of the current work indicate that neglecting the converging characteristics of the actual spinneret can lead to significant underestimation of the wall shear rate. Using the equations derived in our work, we were able to predict not only the velocity profile but also the wall shear rate and the elongation rate; the influence of the spinneret design on the membrane morphology and properties were also examined. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

Predictions of pressure drop for modified power law fluids in conduits of three different cross-sectional shapes

Numerical solutions are presented for fully developed laminar flow for a modified power law fluid (MPL) in conduits of arbitrary cross sections. The solutions are applicable to pseudoplastic fluids over a wide shear rate range from Newtonian behavior at low shear rates, through a transition region, to power law behavior at higher shear rates.The analysis identified a dimensionless shear rate parameter which, for a given set of operating conditions, specifies where in the shear rate range a particular system is operating, i.e. in the Newtonian, transition, or power law regions.The numerical results of the friction factor times Reynolds number for the Newtonian and power law region are compared with previously published results showing agreement within 0.05% in the Newtonian region, and 0.9% and 5.1% in the power law region.     

Rheological behavior and processability of neoprene and acrylic rubbers

Rheological behavior of neoprene and acrylic rubbers and their carbon blackfilled compounds has been studied using a Monsanto Processability Tester. These systems show pseudoplastic flow behavior. Die swell increases with increase in shear rate up to a limit beyond which it decreases. It also decreases with increase in filler loading and temperature. The maximum recoverable deformation has been calculated by assuming that the viscous response of the rubbers obeys the power law model and elastic behavior is described by Hooke's law and has been correlated with die swell. A linear generalized relationship has been obtained between the die swell and the recoverable deformation. The principal normal stress difference has been found to increase nonlinearly with shear stress. Activation energy of melt flow process increases with increase in shear rate up to a limit, after which it decreases for acrylic rubber and 10 phr filled neoprene rubber.     

A local power‐law approximation to a smooth viscosity curve with application to flow in conduits and coating dies

Coating dies distribute liquid into a uniform layer for coating onto a moving substrate. A die comprises one or two cavities spanning the coating width and adjoining narrow slots of much higher resistance to flow. In modeling coating dies, the flows in the slots and cavities are often approximated as one‐dimensional to achieve a fully geometrically parameterized model of low computational load suitable for optimizing design for multiple liquids and flow rates. The power‐law model is mathematically efficient for one‐dimensional flows of shear‐thinning liquids but does not include limiting viscosities at low and high shear rates that are frequently present. In previous work, the truncated power‐law model, which is terminated at the limiting Newtonian viscosities, was used to alleviate this shortcoming without sacrificing the mathematical advantages. In this work, the Carreau–Yasuda model replaces the truncated power‐law model as an improvement. For flows in slots and cavities, the Carreau–Yasuda model can be approximated accurately by a local power‐law model with little increase in computational load over the truncated power‐law model. In the transition regions of the Carreau–Yasuda model between Newtonian and power‐law behavior, the local power‐law model gives more accurate results than the truncated power‐law model. POLYM. ENG. SCI., 54:2301–2309, 2014. © 2013 Society of Plastics Engineers     

Melt transformation coextrusion. II: Flow analysis

Melt Transformation Coextrusion for power law fluids was analyzed with particular attention being paid to the influence of the die's geometrical shape on flow patterns, and to skin layer effect on the velocity profile and shear stress at the interface between layers in a three-layered system. Volumetric flow rate expressions were also generated for the aforementioned system, and stream and potential functions were derived for a free boundary converging film geometry. Newtonian as well as non-Newtonian fluids were considered for this analysis in which the streamline shape changes their convergence downstream into the land section. Also, a finite element solution technique was utilized to reveal shear stress profiles that arise from the flow of high density polyethylene at 200°C in a converging fixed boundary geometry.     

Characterisation of inelastic power-law fluids using falling sphere data

Two simple methods are presented for the characterization of inelastic power law fluids from falling sphere data. The methods involve the application of shear rate or shear stress correction factors which have been derived theoretically using Slattery's solution for creeping flow about spheres. Flow curves obtained using these methods are in excellent agreement with those measured on a Weissenberg rheogoniometer for 0.83 ≤ n ≤ 1.0. The experimentally determined drag coefficients are found to be in good agreement with the predictions of Slattery's creeping flow first approximation solution. The wall correction factors of Faxen and Francis appear to be valid for inelastic non-Newtonian fluids up to a diameter ratio of at least 0.08.     

Rheological behaviour of cement and silica suspensions: Particle aggregation modelling

Cement and silica suspension rheological behaviour is modelled by supposing that particle aggregation occurs. With two independent parameters, the variation of the shear viscosity as a function of the shear rate for cement or silica suspensions having different solid volume fractions can be predicted. The model is in good agreement with experimental data. Cement suspensions are initially very similar to unreactive silica suspensions. The values of adhesion forces obtained seem to show that the effective interaction areas are very small. It reflects the fact that in the dormant period the particles are contacting at points, which become progressively higher in surface as C-S-H precipitates at these contacts. The model used is consistent with the observed power law variation of static and dynamic yield stresses with solid volume fraction. The aggregates formed under shear seem to be very compact, which is presumably related to the large solid volume fractions of the studied suspensions.     

Characterisation of inelastic poweriaw fluids using falling sphere data

Two simple methods are presented for the characterization of inelastic power law fluids from falling sphere data. The methods involve the application of shear rate or shear stress correction factors which have been derived theoretically using Slattery's solution for creeping flow about spheres. Flow curves obtained using these methods are in excellent agreement with those measured on a Weissenberg rheogoniometer for 0.83 > n > 1.0. The experimentally determined drag coefficients are found to be in good agreement with the predictions of Slattery's creeping flow first approximation solution. The wall correction factors of Faxen and Francis appear to be valid for inelastic non-Newtonian fluids up to a diameter ratio of at least 0.08.     

Approximate calculation and measurement of the pressure distribution in radial flow of molten polymers between parallel discs

The known generalized Newtonian fluid “power law” solution of the radial flow between parallel discus has been used to estimate the normal stress, the magnitude of inertia, and the temperature changes due to viscous dissipation. The flow near the wall has been found to be “nearly steady shear flow;” thus the three viscometric functions can be expected to describe the stress at the wall. Further away from the walls, however, the flow is very different from “steady shear flow.” The temperature field in the radial flow section depends on the dimensionless parameters Nahme number, Graetz number, and ratio of inner to outer radius, as well as on the thermal initial and boundary conditions. Experimentally the radial pressure profiles for flow of three different polyethylenes and of one polystyrene have been studied. The measured pressure profiles are about 20 percent lower than the calculated ones from the “power law” solution. This discrepancy cannot yet be explained; the effects of normal stresses, of inertia, or of viscous heating in these experiments are too small to give a measurable effect.     

Effect of elongational viscosity on axisymmetric entrance flow of polymers

A finite element simulation of the flow in a channel with an abrupt contraction is presented. The effects of the shear and elongational viscosities of a polymer on the entrance flow are analyzed employing a truncated power‐law model. The power‐law index and the strain rate characterizing the transition from Newtonian to power‐law behavior for the elongational viscosity are treated as being independent of the values of these two parameters for the shear viscosity. The effect of flow rate on entrance flow is also analyzed. It is confirmed that the Trouton ratio is important in determining the recirculating vortex and the extra pressure loss in entrance flow. Extra pressure loss and vortex length predicted by a finite element simulation of entrance loss are compared with the corresponding predictions from Binding's approximate analysis.     

Shear and skin friction on particles in power-law fluids agitated by flat-blade and fluid foil impellers

The shear rates that exert angular deformation on spherical particles have been measured. The particles are mimiced by a spherical probe. The probe has been immersed in various impeller-agitated power law fluids. The fluids are aqueous dispersions of polymers, e.g. CMC, xanthan gum and starch. The probe has been positioned in various points of a stirred vessel and at various angles. Angle-averaged shear rate distributions were produced. The distributions obtained are characteristic for the specific impeller flow patterns. The flow patterns have been identified by computational fluid dynamics (CFD). Two types of impellers representative for the flat and the fluid-foil blade design, i.e., a Rushton flat-blade turbine (RT) and a Narcissus impeller (NS) are studied. The effects of rheological properties and blade design on the ‘shear-rate-on-particles’ distribution are examined. The local shear field non-uniformity has been uncovered and compared in terms of the CFD-generated time-averaged velocity and deformation rate profiles. The ‘shear-rate-on-particles’ distribution apart from the impeller is found to follow qualitatively the time-averaged inner flow shear rate distribution. Referring to impeller speed 5-12.5 Hz, the dimensionless wall shear rate varied between 200 and 1000. In power law fluids, the shear rate on particles decreased up to 50%. The fluid-foil NS-generated shear field was found comparable to the shear field induced by conventional flat-blade turbines and appeared in cases less sensitive to polymer presence. The shear rate produced by the fluid-foil impeller in the highly shear-thinning model solution (n∼0.4) exceeded the flat-blade RT-imposed shear rate. The analysis has been extended to skin friction drag on particles. It is shown that, while exerting an undoubtedly greater angular deformation in water-like fluids, in polymer presence the conventional flat-blade turbine introduces a flow geometry that imposes particle drag that is close or in some cases even less than the one generated by the fluid-foil impeller. The fact implies a weak shape effect of radial turbines on shear-sensitive particles or particle dispersions in power law liquids.     

A rheological study of concentrated aqueous nanotube dispersions

The rheological behaviour of aqueously dispersed oxidised nanotubes has been studied at concentrations at which the nanotubes interacted with each other. The dispersed nanotubes represented a high aspect ratio system with a ratio of 80. Dynamic and steady shear tests were applied to the dispersions using a cone and plate rheometer. The system was found to behave as a reversibly flocculated dispersion. The structure of the dispersions was highly strain-sensitive with the linear viscoelastic region (LVR) extending to strains of 1%. The moduli within the LVR were independent of frequency and scaled with concentration by a power law. Under steady shear the dispersions rapidly shear thinned up to a Peclet number of 1 to 10. At higher Peclet numbers the shear thinning behaviour followed the Ostwald–de Waele power law. The dispersions were thixotropic and recovered their structure, and hence viscosity, upon standing.     

Gas displacing liquids from non-circular tubes: high capillary number flow of a shear-thinning liquid

Transient, three-dimensional finite element analysis has been used to investigate the displacement of a shear thinning liquid from prismatic channels of square, rectangular and trapezoidal cross sections. Inertia, gravity and surface tension effects are neglected and the results therefore apply in the limits of low Reynolds and high capillary numbers. The analysis is carried out in a fixed frame of reference and gas penetration is modelled as the bubble moves down the tube, which is long relative to its transverse dimension. Results are provided for the thickness of the layers left on the channel walls under developed conditions, and the fraction of the cross section occupied by liquid, as a function of the channel cross-sectional geometry and the degree of shear thinning, modelled using the power law. Interface contours on the channel cross sections are displayed. It is found for the Newtonian liquid that the fingering instability arises in the rectangular channel when the aspect ratio reaches about five. Shear thinning delays the onset of the instability to higher aspect ratios. The results are systematized, and insights gained into the influence of channel geometry and shear thinning, by noting a qualitative, inverse relationship between the deposited layer thickness and the shear rate at the wall in the flow ahead of the bubble.     

Rheological Behavior of Wastewater Sludge Following Cationic Polyelectrolyte Flocculation

The rheological characteristics of the wastewater sludge were investigated by using a Haake Rheostress RS 75 rheometer. The shear creep compliance experiments and the dynamic viscosity measurements were conducted. The shear creep compliance experiments indicate the addition of polymer coagulants to the sludge samples will form more rigid structures. The elastic solid-like behaviors were always observed in the samples with polymers. The Voigt model was successfully employed in modeling the viscoelastic retardation behavior of sludge samples in the shear creep compliance tests. Moreover, the dynamic viscosity curves of the sludge samples with/without polymer could be described by the power law model of Ostwald and de Waele at the medium shear rates, ca. 100–300 s^?1. Consequently, addition of polymer to the sludge tends to extend the applicable ranges of the shear rates for the power law model as well as to decrease the power law index.     

Melt rheology of high impact polystyrenes at high shear stresses

A study of the theological properties of commercial polystyrenes, (PS), and high impact polystyrenes (HIPS), is made in the range of shear stresses of practical interest in industrial polymer processing. A general viscosity-shear rate-temperature relationship is defined for these products, with a power law exponent of n = 0.3 and an activation energy of 27 Kcal/mol. The fluid elasticity is studied in terms of steady state shear compliance. An expression relating shear compliance, viscosity and molecular weight distribution is obtained for HIPS. As in other two-phase systems, a decrease in elasticity with viscosity is observed.     

Rheological Behavior of Wastewater Sludge Following Cationic Polyelectrolyte Flocculation

The rheological characteristics of the wastewater sludge were investigated by using a Haake Rheostress RS 75 rheometer. The shear creep compliance experiments and the dynamic viscosity measurements were conducted. The shear creep compliance experiments indicate the addition of polymer coagulants to the sludge samples will form more rigid structures. The elastic solid-like behaviors were always observed in the samples with polymers. The Voigt model was successfully employed in modeling the viscoelastic retardation behavior of sludge samples in the shear creep compliance tests. Moreover, the dynamic viscosity curves of the sludge samples with/without polymer could be described by the power law model of Ostwald and de Waele at the medium shear rates, ca. 100-300 s^-1. Consequently, addition of polymer to the sludge tends to extend the applicable ranges of the shear rates for the power law model as well as to decrease the power law index.     

CFD analysis of several design parameters affecting the performance of the Maxblend impeller

A CFD characterization of the hydrodynamics of the Maxblend impeller with experimental validations has been carried out with viscous Newtonian and non-Newtonian inelastic fluids. The mixing cases investigated were the non-baffled configuration with Newtonian and shear-thinning fluids, and the baffled configuration with only Newtonian fluids. The study focused on the effect of the impeller bottom clearance and the Reynolds number on the power characteristics, the distribution of shear rates and the overall flow conditions in the vessel. It was found that the bottom clearance plays a significant role on the power consumption, and that the value of the Reynolds number and the power law index strongly affect the axial pumping efficiency and the shear rate profile. The best performance was obtained when the impeller Reynolds number is superior to 10.     

Investigations of heat transfer and pressure drop between parallel channels with pseudoplastic and dilatant fluids

Central to the problem of heat exchangers design is the prediction of pressure drop and heat transfer in the noncircular exchanger duct passages such as parallel channels. Numerical solutions for laminar fully developed flow are presented for the pressure drop (friction factor times Reynolds number) and heat transfer (Nusselt numbers) with thermal boundary conditions [constant heat flux (CHF) and constant wall temperature (CWT) ] for a pseudoplastic and dilatant non‐Newtonian fluid flowing between infinite parallel channels. A shear rate parameter could be used for the prediction of the shear rate range for a specified set of operating conditions that has Newtonian behavior at low shear rates, power law behavior at high shear rates, and a transition region in between. Numerical results of the Nusselt number [constant heat flux (CHF) and constant wall temperature (CWT) ] and the product of the friction factor and Reynolds number for the Newtonian region were compared with the literature values showing agreement within 0.36% in the Newtonian region. For pseudoplastic and dilatant non‐Newtonian fluids, the modified power law model is recommended to use because the fluid properties have big discrepancies between the power law model and the actual values in low and medium range of shear rates. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 89: 3601–3608, 2003     

Shear‐rate‐dependent rheology effects on mass transport and surface reactions in biomicrofluidic devices

Consideration is given to shear‐rate‐dependent rheology effects on mass transport in a heterogeneous microreactor of rectangular cross section, utilizing both numerical and analytical approaches. The carrier liquid obeys the power‐law viscosity model and is actuated primarily by an electrokinetic pumping mechanism. It is discovered that, considering the shear‐thinning biofluids to be Newtonian fluids gives rise to an overestimation of the saturation time. The degree of overestimation is higher in the presence of large Damkohler numbers and electric double layer thicknesses. It is also increased by the application of a favorable pressure gradient, whereas the opposite is true when an opposed pressure gradient is applied. In addition, a channel of square cross section corresponds to the maximum fluid rheology effects. Finally, the numerical results indicate the existence of a concentration wave when using long channels. This is confirmed by analytical solutions, providing a closed form solution for wave propagation speed. © 2015 American Institute of Chemical Engineers AIChE J, 61: 1912–1924, 2015