Increase In Power-law Index Research Articles

Influence of power-law index on an unsteady exothermic reaction

This study presents the solution of an unsteady Arrhenius exothermic reaction where we reduced the exponential term to a power-law approximation. A numerical solution of the problem is obtained using shooting technique with second order Runge-Kuta scheme. It is shown that the temperature of the reactant depends on the power-law index. The temperature of the reactant increases as the power-law index increases. JONAMP Vol. 11 2007: pp. 545-548

Mass transfer from a single fluid sphere to power-law liquids at moderate Reynolds numbers

The steady-state convective inter-phase mass transfer from a single Newtonian fluid sphere (free from surfactants) to a continuous phase with power-law viscosity has been studied at moderate Reynolds and Schmidt numbers under the conditions when the resistance to mass transfer in the dispersed phase is negligible. The species continuity equation, segregated from the momentum equations of both phases, has been numerically solved using a finite difference method. The effects of the Reynolds number ( Re o ) , power-law index ( n o ) , internal to external fluid characteristic viscosity ratio ( k ) and Schmidt number ( Sc) on the local and average Sherwood number ( Sh) have been analysed over the following ranges of conditions: 5 ⩽ Re o ⩽ 200 , 0.6 ⩽ n o ⩽ 1.6 , 0.1 ⩽ k ⩽ 50 and 1 ⩽ Sc ⩽ 1000 . It has been observed that irrespective of the values of the Reynolds number and of the power-law index, as the value of k increases the average Sherwood number decreases for intermediate to large values of the Peclet number. As the value of the power-law index increases, the rate of mass transfer decreases for all values of the Reynolds number and the characteristic viscosity ratio thereby suggesting that shear-thinning behaviour facilitates mass transfer, whereas shear-thickening behaviour impedes it. Based on the present numerical results, a simple predictive correlation is proposed which can be used to estimate the rate of inter-phase mass transfer of a fluid sphere sedimenting in power-law liquids.

Numerical study of slip flow heat transfer of non-Newtonian fluids in circular microchannels

In the present investigation, numerical simulations are performed to study flow and thermal fields of non-Newtonian fluids in circular microchannels. The flow is considered to be slip, axisymmetric, steady, incompressible, laminar, and power law model is used to characterize the behavior of the non-Newtonian fluid. The constant wall heat flux and constant wall temperature are employed as thermal boundary conditions. The set of governing dimensionless differential equations with appropriate boundary conditions are solved together using control volume finite difference method. Results indicate the increasing slip coefficient decreases the product of friction factor and Reynolds number and centerline velocity while results in increasing of local Nusselt numbers. Slip effect enhances with increasing power law index. Also slip effect in constant wall heat flux is more than that of constant wall temperature boundary conditions.

Drag of a Spherical Bubble Rising in Power Law Fluids at Intermediate Reynolds Numbers

The steady rise of a spherical bubble through an incompressible quiescent power law fluid has been studied numerically in the 2-D axisymmetric flow regime using the finite volume method. Based on the present numerical results, a predictive correlation in terms of Reynolds number (Re) and power law index (n) is proposed which enables the prediction of the total drag coefficient for the ranges of conditions as 5 ≤ Re ≤ 500 and 0.5 ≤ n ≤ 2 (hence covering both shear-thinning and shear-thickening type of fluid behavior). For n < 1, the drag is reduced below the corresponding Newtonian value, whereas it increases above its Newtonian value in shear-thickening fluids (n > 1). Thus, the drag coefficient increases with the increasing power law index for all values of the Reynolds number. The contribution of the pressure drag also increases with the increasing Reynolds number, though the shear-thickening behavior (n > 1) seems to suppress this tendency. In addition to the drag behavior, streamline and constant vorticity plots are presented to show the detailed nature of the flow and the effect of Reynolds number and power law index on the flow characteristics.

Rheological behavior of ethylene–octene copolymer vulcanizates: Effect of blowing agent and precipitated silica filler

AbstractAn experimental study of the rheological behavior of ethylene–octene copolymer vulcanizates in extrusion containing blowing agent has been carried out. The cell morphology development has been studied through a scanning electron microscope. Rheological properties of unfilled and precipitated silica‐filled systems with variations of blowing agent, extrusion temperature, and shear rate have been studied by using a Monsanto processibility tester (MPT). The total extrusion pressure (PT), apparent shear stress (τwa), apparent viscosity (ηa), and die swell (%) of the unfilled and silica‐filled compounds have been determined by using MPT. The effect of blowing agent (ADC) on the rheological properties of the vulcanizates has also been investigated. There is a reduction of stress and viscosity with blowing agent loading. It was observed that the incorporation of a blowing agent led to decreased shear thinning behavior resulting in an increase in power law index. The viscosity reduction factor (VRF) of unfilled vulcanizates is found to be dependent on the concentration of the blowing agent, shear rate, and temperature, whereas VRF of silica‐filled vulcanizates is found to be dependent on shear rate, temperature, and blowing agent concentration. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 89: 1132–1138, 2003

Rheological Behavior of Ethylene-Octene Copolymer Vulcanizates: Effect of Blowing Agent and Precipitated Aluminium Silicate Filler

An experimental study of the rheological behaviour of aluminium silicate filled ethylene-octene copolymer vulcanizates in extrusion containing blowing agent has been carried out. The cell morphology development has been studied through scanning electron microscope. Rheological properties of unfilled and aluminium silicate filled systems with variation of blowing agent, extrusion temperature and shear rate have been studied using Monsanto processability tester (MPT). The total extrusion pressure (ΔPT), apparent shear stress (x), apparent viscosity (ηa), and die swell (%) of the unfilled and aluminium silicate filled compounds have been determined by using MPT. It is found that there is a reduction in stress and viscosity with blowing agent loading. It is observed that incorporation of blowing agent led to decreased shear thinning behaviour resulting in an increase in power law index. The viscosity reduction factor (VRF) of unfilled vulcanizates is found to be dependent on concentration of blowing agent, shear rate and temperature, whereas VRF of aluminium silicate filled vulcanizates is found to be independent of shear rate and temperature but dependent on blowing agent concentration.

Modelling the flow of power law fluids in a packed bed using a volume-averaged equation of motion

The development of a theoretical model for the prediction of velocity and pressure drop for the flow of a viscous power law fluid through a bed packed with uniform spherical particles is presented. The model is developed by volume averaging the equation of motion. A porous microstructure model based on a cell model is used. Numerical solution of the resulting equation is effected using a penalty Galerkin finite element method. Experimental pressure drop values for dilute solutions of carboxymethylcellulose flowing in narrow tubes packed with uniformly sized spherical particles are compared to theoretical predictions over a range of operating conditions. Overall agreement between experimental and theoretical values is within 15%. The extra pressure drop due to the presence of the wall is incorporated directly into the model through the application of the no-slip boundary condition at the container wall. The extra pressure drop reaches a maximum of about 10% of the bed pressure drop without wall effect. The wall effect increases as the ratio of tube diameter to particle diameter decreases, as the Reynolds number decreases and as the power law index increases.

The effects of viscous dissipation on heat transfer to power law fluids in arbitrary cross-sectional ducts

In this paper, the analytical study of forced convection heat transfer to power-law fluids in arbitrary cross-sectional ducts with finite viscous dissipation is undertaken. Both the flow and heat transfer develop simultaneously from the entrance of the duct the walls of which are maintained at a constant temperature different from the entering fluid temperature. The governing conservation equations written in curvilinear coordinates are solved using the Line-Successive-Over relaxation (LSOR) method. Numerical results of dimensionless heat transfer coefficients and temperature profiles are presented for the trapezoidal, triangular, circular and square ducts. For cooling, viscous dissipation generally augments heat transfer. At low values of Brinkman number (Br∼0.1), the cooling effect dominates over viscous heating in the entrance region. AsBr is increased, the location where viscous dissipation becomes important shifts closer to the entrance until a value is reached for which the effect of viscous dissipation is always predominant irrespective of the axial location. When the walls are heated, for a non-zero Brinkman number, theNu X* distribution exhibits a singularity from the negative side of theNu X* axis. As the power-law index increases, the position of this singularity shifts closer to the entrance of the duct. Far downstream of the duct, for a fixedn, Nu X* attains an asymptotic value which is independent ofBr and is at least thrice that for forced convection without viscous dissipation.

Viscous dissipation effects on thermal entrance heat transfer to power-law fluids in arbitrary cross-sectional ducts

The effect of viscous dissipation on forced convection heat transfer to power-law fluids in arbitrary cross-sectional ducts is examined. Both the flow and heat transfer develop simultaneously from the entrance of the duct, the walls of which are maintained at a constant temperature which is different from the entering fluid temperature. The governing conservation equations, written in curvilinear coordinates, are solved using the line successive overrelaxation (LSOR) method. Numerical results of dimensionless heat transfer coefficients are presented for the triangular, circular, trapezoidal and square ducts for several values of the Brinkman number Br and the power-law index n. For cooling (Br > 0.0), viscous dissipation generally augments heat transfer. At low values of the Brinkman number (Br ≈ 0.1), the cooling effect dominates over viscous heating in the entrance region. As Br is increased the location where viscous dissipation becomes important shifts closer to the entrance, until a value (which is geometry dependent) is reached for which the effect of viscous dissipation is always predominant irrespective of the axial location. For heating (Br < 0.0), the Nu x* distribution exhibits a singularity from the negative side of the Nu x* axis. As the power-law index increases, the position of this singularity shifts closer to the entrance of the duct. Before the singularity, Nu x* decreases with increasing magnitude of Br, while the opposite behaviour is observed beyond the singular point. At any X*, an increase in the power-law index results in a decrease in Nu x*. Far downstream of the duct, for a fixed n, Nu x* attains an asymptotic value which is independent of Br and is at least three times that for forced convection without viscous dissipation.

Temperature and Velocity Profiles in Drag Flow of a Temperature Dependent Power Law Fluid

Abstract This paper analyzes non-isothermal drag flow between parallel plates. First, non-isothermal drag flow is considered of a Newtonian fluid with a non-zero pressure gradient, assuming viscous dissipation is negligible. Second, non-isothermal drag flow is considered of a power law fluid with zero pressure gradient, again assuming viscous dissipation is negligible. Finally, pure drag flow of a temperature dependent power law fluid is analyzed, including the effect of viscous dissipation. When the temperature of the stationary plate is increased, the flow rate increases. Applying this analysis to screw extrusion, this means that increasing the screw temperature relative to the barrel temperature will increase the flow rate. Also, the temperature sensitivity of the flow rate reduces with increasing screw temperature and with increasing power law index. This points to useful benefits of screw heating. With shear thinning fluids increased channel depth raises the stock temperatures, even though the shear rates in the fluid will be reduced. The increased temperatures are due to longer heat transfer distances; the viscous heat generated is less easily conducted away to the walls. At high values of the Brinkman number, the heat flux at the lower plate reduces when the upper plate temperature increases, while at low Brinkman numbers the opposite is true. This is important in the analysis of melting in extruders. It indicates that while high barrel temperatures in the melting zone may be beneficial at low screw speeds, it can be detrimental at high screw speeds. At high screw speeds, therefore, lower barrel temperatures may be more appropriate to obtain efficient melting.