Hydrodynamics of power-law fluids in trickle-flow reactors: Mechanistic model, experimental verification and simulations

A fully predictive one-dimensional mechanistic model was developed for describing the hydrodynamics of power-law fluids in trickle-bed reactors. The model is a generalization of the slit approach to the case of non-Newtonian fluids obeying Ostwald-deWaele rheological behavior. Without recourse to adjustable parameters, the proposed model enabled prediction of the experimental values of (i) total two-phase total pressure drop and total liquid holdup in the trickle flow regime, (ii) frictional pressure drop in single-phase flows through packed beds, and (iii) total liquid holdup in gravity driven liquid downflow and stagnant gas through packed beds. Parametric simulations guided by knowledge of the behavior of highly viscous Newtonian liquids in trickle beds highlighted the capability of the model in the simulation and design of trickle flow operation using power-law fluids.     

The influence of non-Newtonian flow on effective viscosity and channel efficiency in screw pumps

A method is given for selecting the “effective viscosity” for isothermal flow of non-Newtonian liquids in screw pumps or melt extruders. Effective viscosity is the Newtonian viscosity that would give the same screw-pump performance with non-Newtonian liquids at the same flow rate and speed. When effective viscosity is known, it can be inserted in performance equations for simple Newtonian flow. The analysis is restricted to shallow screw-pump channels with large aspect ratios and to shear stress/shear rate curves with modest curvatures when shown in a double logarithmic plot. The shear stress/shear rate curve is replaced by a power-law tangent to that curve in the domain of prevailing shear rates, but the analysis could be extended to more complex behavior. Curves are also included for calculating the efficiency of the screw-pump channel, which can be used to estimate the energy dissipated in screw-pumps. It is shown that efficiency decreases with decreasing power-law exponent.     

MOTION OF AND MASS TRANSFER FROM AN ASSEMBLAGE OF SOLID SPHERES MOVING IN A NON-NEWTONIAN FLUID AT HIGH REYNOLDS NUMBERS

An approximate solution for the motion of an assemblage of solid spheres moving in a power-law fluid in the high Reynolds number region is obtained using a combination of Happel's free-surface cell model and the boundary layer theory. It is theoretically predicted that the drag coefficient will decrease with the increase of the shear-thinning anomaly. The results of the present analysis are in reasonably good agreement with the available experimental data for fixed and fluidized beds. The influence of the non-Newtonian behavior on the mass transfer rate from an assemblage of solid spheres is also discussed.     

Residence time distribution of the liquid in two-phase cocurrent downflow in packed beds: Air/newtonian and non-newtonian liquid systems

New data on the liquid residence time distribution for two-phase downflow of air-Newtonian and non-Newtonian liquids through packed beds of porous and non-porous particles are presented. The piston-dispersion-exchange model is used to describe the liquid flow. With porous particles the dynamic evolution of the tracer concentration in the particles is described in terms of diffusion phenomena. The axial dispersion is very important in the case of two-phase downflow of air-water (trickle flow regime) and air-CMC systems through fixed beds with porous particles, and is negligible in the case of non-porous particles. With the porous particles, a key value is the effective diffusion coefficient of the tracer in the pores of the particles.     

A macroscopic model for shear-thinning flow in packed beds based on network modeling

The flow of non-Newtonian fluids in packed beds and other porous media is important in several applications such as polymer processing, filtration, and enhanced oil recovery. Expressions for flowrate versus pressure gradient are desirable for a-priori prediction and for substitution into continuum models. In this work, physically representative network models are used to model the flow of shear-thinning fluids, including power-law and Ellis fluids. The networks are used to investigate the effects of fluid rheology and bed morphology on flow.A simple macroscopic model is developed for the flow of power-law and Ellis fluids in packed beds using results from the network model. The model has the same general functionality as those developed using the popular bundle-of-tubes approach. The constant β, which appears in these models, is often directly derived from the tortuosity and a simple representation of the porous media. It is shown here that this can lead to incorrect and ambiguous values of the constant. Furthermore, the constant is a weak function of the shear-thinning index, indicating that no single bundle-of-tubes could ever properly model flow for a wide variety of shear-thinning fluids.The macroscopic model is compared to experimental data for shear-thinning fluids available in the literature. The model fits the data well when β is treated as an experimental parameter. The best-fit values of β vary, which is expected because even the constant C in the Blake-Kozeny equation varies depending on the source consulted. Additionally, physical effects, such as adsorption and filtration, as well as rheological effects such as viscoelasticity may affect the value of β. We believe that in the absence of these effects, β equals approximately 1.46 for packed beds of uniform spheres at relatively moderate values of the shear-thinning index (>0.3).     

PRESSURE DROPS OF NON-NEWTONIAN PURELY VISCOUS FLUID FLOW THROUGH SYNTHETIC FOAMS

This work aims at giving a first insight of non-Newtonian fluid flow through synthetic foams.

At first, a review of experimental pressure drops measured with Newtonian fluids through various foams is proposed as well as a recall of a capillary-type flow model used to determine structural parameters. In this particular case of Newtonian fluid flow, a single equation is shown to correlate experimental data whatever the grade of the foam. Results of an image analysis study are also given; they allow to give a physical sense to the value of the equivalent diameter of pore given by the flow model.

In a second part, results of pressure drops measured with a non-Newtonian fluid are reported. A model allowing the determination of the pressure drops for non-Newtonian purely viscous fluid flow through packed beds of particles, based on the same capillary representation of porous media, is tested in the case of fluid flow through synthetic foams. The model predictions are acceptable for foams of high grades, but a discrepancy is observed for foams of low grades. An attempt is made to explain the underevaluation of pressure drops with the aid of results of image analysis.     

Application of a three-layer modeling approach for solids transport in horizontal and inclined channels

The three-layer model concept developed previously for solid-liquid flow has been adapted to model solids transport in inclined channels. The present model predicts the pressure loss and transport rate of solids in Newtonian and power-law fluid suspensions by assuming stratified flow conditions. Sets of stationary sand bed transport rate tests were performed to verify the predictions of the model. A 70-mm flow loop was constructed to measure the average transport rates and critical flow rates, which are required to initiate the motion of solids bed particles. The tests were carried out by eroding stationary sand beds with water and an aqueous solution of poly anionic cellulose (PAC) in a transparent pipe. Four sand beds with different particle size ranges were used. The average transport rates of the beds were predicted using the model. The model predictions show a satisfactory agreement with experimentally measured results when the grain Reynolds number is between 15 and 400 and the flow rate is sufficiently higher than the critical flow rate. Therefore, with some degree of limitation, the three-layer model can be applicable for predicting the transport rates of stationary solids beds in inclined channels for both Newtonian and power-law fluids.     

Rheology of high solid coatings. II. Analysis of combined sagging and leveling

The rheology of combined sagging and leveling of high solid coatings is analyzed in terms of non-Newtonian power-law model. The results indicate that, in order to have good leveling, good sag control, and good sprayability at the same time, high solid coatings should have pseudoplastic rheology with power constant of about 0.5 and viscosity at sec^-1 of about 50 poises. This theoretical prediction is confirmed experimentally.     

Rheology of high solid coatings. I. Analysis of sagging and slumping

The rheology of sagging and slumping of high solid coatings on vertical surfaces is analyzed in terms of a non-Newtonian power-law model. The results indicate that, in order to have good sag control and good sprayability at the same time, high solid coatings should be pseudoplastics with a power constant of about 0.6 and a viscosity at 1 sec^-1 of about 25 poise at the temperature of interest. This theoretical prediction is confirmed experimentally.     

Filtration of non-Newtonian fluids

Equations of motion characterizing the flow of incompressible, time-independent non-Newtonian fluids, exhibiting the anomalous surface effect, through compressible porous media are developed. Approximations are made to arrive at workable filtration equations for slurries of non-Newtonian (power-law) fluids. The constant-pressure and constant-rate filtration relationships developed are verified experimentally using slurries of calcium carbonate in water and dilute CMC solutions. The anomalous surface effect is found to exist in the filtration of the non-Newtonian fluids. The specific cake resistance in the case of the non-Newtonian sludge and the ratio of the effective slip velocity to the pore velocity are found to be functions of both the CMC concentration and the pressure drop across the filter bed.     

Forced convection heat transfer from axisymmetric bodies to non-Newtonian fluids

A theoretical analysis for forced convection heat transfer from axisymmetric bodies immersed in non-Newtonian power-law fluids has been performed. Results for the velocity shape factor, local friction coefficient, and the Nusselt number are presented for different values of Prandtl number and the power-law index. The local friction coefficient results are compared with available experimental data and it is shown that asymptotic expressions suffice to get accurate predictions of heat transfer at low and high Prandtl numbers.     

Trickle bed hydrodynamics and flow regime transition at elevated temperature for a Newtonian and a non-Newtonian liquid

Despite the hydrodynamics of trickle beds experiencing high pressures has become largely documented in the recent literature, trickle bed hydrodynamic behavior at elevated temperatures, on the contrary, largely remains terra incognita. This study's aim was to demonstrate experimentally the temperature shift of trickle-to-pulse flow regime transition, pulse velocity, two-phase pressure drop, liquid holdup and liquid axial dispersion coefficient. These parameters were determined for Newtonian (air-water) and non-Newtonian (air-0.25% Carboxymethylcellulose (CMC)) liquids, and the various experimental results were compared to available literature models and correlations for confrontation and recommendations. The trickle-to-pulse flow transition boundary shifted towards higher gas and liquid superficial velocities with increasingly temperatures, aligning with the findings on pressure effects which likewise were confirmed to broaden the trickle flow domain. The Larachi-Charpentier-Favier diagram [Larachi et al., 1993, The Canadian Journal of Chemical Engineering 71, 319-321] provided good predictions of the transition locus at elevated temperature for Newtonian liquids. Conversely, everything else being kept identical, increasingly temperatures occasioned a decrease in both two-phase pressure drop and liquid holdup; whereas pulse velocity was observed to increase with temperature. The Iliuta and Larachi slit model for non-Newtonian fluids [Iliuta and Larachi, 2002, Chemical Engineering Science 46, 1233-1246] predicted with very good accuracy both the pressure drops and the liquid holdups regardless of pressure and temperature without requiring any adjustable parameter. The Burghardt et al. [2004, Industrial and Engineering Chemistry Research 43, 4511-4521] pulse velocity correlation can be recommended for preliminary engineering calculations of pulse velocity at elevated temperature, pressure, Newtonian and non-Newtonian liquids. The liquid axial dispersion coefficient (D_ax) extracted from the axial dispersion RTD model revealed that temperatures did not affect in a substantial manner this parameter. Both Newtonian and power-law non-Newtonian fluids behaved qualitatively similarly regarding the effect of temperature.     

PRESSURE DROPS FOR PURELY VISCOUS NON-NEWTONIAN FLUID FLOW THROUGH BEDS PACKED WITH MIXED-SIZE SPHERES

Extensive new measurements on pressure drop for the flow of purely viscous non-Newtonian fluids through packed beds made up of binary-and-quaternary size spheres are reported herein. These results have been interpreted using the previously available Sabiri and Comiti capillary model (1995), which has been quite successful in correlating the pressure drop data for the beds of uniform size spherical and nonspherical particles. The resulting predictions are also very good for mixed beds if some results are available for a Newtonian fluid, thereby enabling the evaluation of the tortuosity factor. However, since such data is always not available, an empirical scheme for estimating the tortuosity that allows the prediction of pressure drop with a mean relative error of about 10% is also presented. This sort of accuracy is quite acceptable for process engineering design calculations.     

Pressure drops for purely viscous non-newtonian fluid flow through beds packed with mixed-size spheres

Extensive new measurements on pressure drop for the flow of purely viscous non-Newtonian fluids through packed beds made up of binary-and-quaternary size spheres are reported herein. These results have been interpreted using the previously available Sabiri and Comiti capillary model (1995), which has been quite successful in correlating the pressure drop data for the beds of uniform size spherical and nonspherical particles. The resulting predictions are also very good for mixed beds if some results are available for a Newtonian fluid, thereby enabling the evaluation of the tortuosity factor. However, since such data is always not available, an empirical scheme for estimating the tortuosity that allows the prediction of pressure drop with a mean relative error of about 10% is also presented. This sort of accuracy is quite acceptable for process engineering design calculations.     

A simplified couette flow solution of non-newtonian power-law fluids in eccentric annuli

Flow of non-Newtonian fluids in both the concentric and eccentric annuli is of great importance in extruders for molten plastics and wellbore fluid circulation for the removal of drilling cuttings. The steady laminar couette flow of non-Newtonian power-law fluids in eccentric annulus is employed in this study to analyze the problems of surge or swab pressures encountered when running or pulling tubular goods (pipes) in a liquid filled borehole. This is similar to the annular space created by two long co-axial cylinders with the inner cylinder in motion at a steady velocity, and a stationary outer cylinder. The solutions of the equations of motion are presented in both dimensionless form and as a family of curves for different pipe/borehole eccentricity ratios and power-law fluid index values for a more general application. The expected error in surge computation for concentric annulus as a result of eccentricity is evaluated.     

MASS AND MOMENTUM TRANSFER WITH NON-NEWTONIAN FLUIDS IN FLUIDIZED BEDS

A new model for non-Newtonian fluid flows through fluidized beds is developed by extending the Richardson-Zaki concept. In the proposed model, the existing correlations for an isolated single sphere are simply extended to those for a fluidized bed by incorporating a voidage function. The model is found to be in a good agreement with other correlations for flows of a non-Newtonian fluid through multi-particle systems. The model is also applied to interpret mass transfer in fluidized beds.     

非牛顿流体薄膜流中质量传递的实验研究

本文通过实验,研究非牛顿流体层流降膜流中质量传递过程.实验系采用温壁塔测定二氧化碳在高分子水溶液中吸收速率.这些溶液符合幂律模型.实验证明非牛顿幂律流体降膜流中考虑速度分布的微分方程精确解是正确的;对拟塑性流体,用无因次长度Z<0.1作为渗透论适用范围的判据是合适的,而精确解则不受此范围的限制.     

Free coating of non-newtonian liquids onto a vertical surface

The coating of non-Newtonian liquids onto a vertical surface continuously withdrawn from the liquid bath is considered. An analytic treatment is presented for purely viscous non-Newtonial liquids using the Ellis and generalised Bingham models both of which may be reduced to a new theory for power-law fluids. The theories give a relationship between the dimensionless film thickness, T₁, and the Capillary number, C₁, as a function of the fluid physical properties and the parameters of the viscous model. The dimensionless groups have been generalised to allow for non-Newtonian behaviour. The power-law and Ellis model predictions are compared with previous theoretical studies and shown to be consistent with known limits. Experimental data are also presented for a wide range of non-Newtonian fluids and compared with the new theories.     

MASS AND MOMENTUM TRANSFER WITH NON-NEWTONIAN FLUIDS IN FLUIDIZED BEDS

A new model for non-Newtonian fluid flows through fluidized beds is developed by extending the Richardson-Zaki concept. In the proposed model, the existing correlations for an isolated single sphere are simply extended to those for a fluidized bed by incorporating a voidage function. The model is found to be in a good agreement with other correlations for flows of a non-Newtonian fluid through multi-particle systems. The model is also applied to interpret mass transfer in fluidized beds.     

MODELING OF CONCENTRATED LIQUID-SOLIDS FLOW IN PIPES DISPLAYING SHEAR-THINNING PHENOMENA

This paper describes our work in modeling concentrated liquid-solids flows in pipes. Based on our previous analyses, some concentrated liquid-solids suspension flows display shear-thinning rather than Newtonian phenomena. Therefore, we developed a new two-phase non-Newtonian power-law model that includes the effect of solids concentration on solids viscosity. With this new two-phase power-law solids-viscosity model, and with constitutive relationships for interfacial drag, virtual mass and shear lift forces, and solids partial-slip boundary conditions at the pipe walls, COMMIX-M is capable of analyzing concentrated three-dimensional liquid-solids flows.