Pressure drop during the flow of stokesian fluids through granular beds

The resistance to flow of Stokesian fluids (i.e. time—independent fluids with no yield stress) through granular beds is discussed. A definition of friction factor λ and generalized Reynolds number Re′_BK is proposed for fluids obeying the “power-law” shear stress—shear rate relation.The generalized Ergun equation, derived in this paper, gives the dependence of the friction factor on the generalized Reynolds number and flow behaviour index n. The validity of the generalized Ergun equation was proved experimentally. In the case of Newtonian fluid (for n = 1·0) a more exact form of the classical Ergun equation is obtained.     

RHEOLOGICAL CHARACTERIZATION OF NON-NEWTONIAN FLUIDS WITH A NOVEL VISCOMETER

A hydrostatic head viscometer and its novel viscosity equation were developed to determine flow characteristics of Newtonian and non-Newtonian fluids. The objective of this research is to test capabilities of the hydrostatic head viscometer and its novel non-Newtonian viscosity equation by characterizing rheological behaviors of well-known polyethylene oxide aqueous solutions as non-Newtonian fluids with 60 wt.% sucrose aqueous solution as a reference/calibration fluid. Non-Newtonian characteristics of 0.3–0.7 wt.% polyethylene oxide aqueous solutions were extensively investigated with the hydrostatic head viscometer and its non-Newtonian viscosity equation over a 294–306 K temperature range, a 0.14–40 Reynolds number range, and a 55–784 s^?1 shear rate range at atmospheric pressure. Dynamic viscosity values of 60 wt.% sucrose aqueous solution were determined with the calibrated hydrostatic head viscometer and its Newtonian viscosity equation over a 3–5 Reynolds number range at 299.15 K and atmospheric pressure and compared with the literature dynamic viscosity value.     

RHEOLOGICAL CHARACTERIZATION OF NON-NEWTONIAN FLUIDS WITH A NOVEL VISCOMETER

A hydrostatic head viscometer and its novel viscosity equation were developed to determine flow characteristics of Newtonian and non-Newtonian fluids. The objective of this research is to test capabilities of the hydrostatic head viscometer and its novel non-Newtonian viscosity equation by characterizing rheological behaviors of well-known polyethylene oxide aqueous solutions as non-Newtonian fluids with 60 wt.% sucrose aqueous solution as a reference/calibration fluid. Non-Newtonian characteristics of 0.3-0.7 wt.% polyethylene oxide aqueous solutions were extensively investigated with the hydrostatic head viscometer and its non-Newtonian viscosity equation over a 294-306 K temperature range, a 0.14-40 Reynolds number range, and a 55-784 s^-1 shear rate range at atmospheric pressure. Dynamic viscosity values of 60 wt.% sucrose aqueous solution were determined with the calibrated hydrostatic head viscometer and its Newtonian viscosity equation over a 3-5 Reynolds number range at 299.15 K and atmospheric pressure and compared with the literature dynamic viscosity value.     

Orifice Flowmeter for Measuring Extensional Rheological Properties

Extensional rheological properties play an important role in processes in which the fluid is subjected to highly decelerated or accelerated flows. This paper describes an orifice flowmeter used to measure extensional properties of rheologically complex fluids at high strain rates. The operating principle of the flowmeter is based on the pressure drop due to the flow through a small size orifice. The flowmeter was first calibrated, by plotting the pressure drop‐flow rate curve of the orifice, in terms of a dimensionless Euler number versus Reynolds number. Newtonian fluids consisting of aqueous solutions of corn syrup were used as calibration fluids. The calibration curve was then used to determine the apparent extensional viscosity of three different paper coating colors. The apparent extensional viscosity is compared to the shear viscosity in terms of the Trouton ratio. The Trouton ratio for one coating color is shown to exceed considerably the theoretical value of 3 expected for Newtonian fluids.     

Mass Transfer in a Flat Gas/Liquid Interface using non‐Newtonian Media

Gas/liquid mass transfer has been investigated using a stirred vessel gas/liquid contactor using non‐Newtonian media and carbon dioxide as absorbent and gas phase, respectively. The volumetric mass transfer coefficients at different operational variables have been determined. Non‐Newtonian media (liquid phase) were prepared as aqueous solutions of sodium carboxymethyl cellulose salt. The influence of the rheological properties, polymer concentration, stirring rate, and gas flow rate on mass transfer was studied for these liquid phases. Kinematic viscosity and density experimental data were used to calculate the average molecular weight corresponding to the polymer employed. The Ostwald model has been used to fit the rheological behavior of aqueous solutions of the polymer employed as absorbent phase. Reasonably good agreement was found between the predictions of the proposed models and the experimental data of mass transfer coefficients.     

Lattice‐Boltzmann simulation of sago (Metroxylon sagu) starch solutions in porous media

From rheological experiments in gelatinized sago starch solution already reported in the literature and a Lattice‐Boltzmann simulation, we provide some insight into the understanding of the non‐Newtonian fluid dynamics of sago‐starch‐type solutions in porous media. In this paper, permeability and wall shear stress in arbitrarily generated and randomly generated porous media are predicted in the range of the modified Darcy's law. Additional results on flow paths, velocity, shear‐stress tensor, and pressure fields are provided. We prove that our LBE model for sago starch solutions reproduces Blake‐Kozeny and Ergun laws. The model presented in this paper is intended to be used for simulating packed beds.     

Interfacial stress in non-Newtonian flow through packed bed

This study investigates the pressure drop characteristics, shear stress in packed bed with shear thinning power law type non-Newtonian liquid. A mechanistic model has also been developed to analyze the pressure drop and interfacial stress in packed bed with non-Newtonian liquid by considering the loss of energy due to wettability. The Ergun's and Foscolo's equations were used for comparison with the experimental data. The Ergun equation was modified to account for the effect of flow behavior index of non-Newtonian fluid in the column. The intensity factor of shear stress and the friction factor were analyzed based on energy loss due to wettability effect of liquid on the solid surface.     

Wetting kinetics of polymer solutions and force‐based contact angles

The effect of the shear thinning behavior and elasticity of polymer solutions on the dynamic contact angles are investigated. Under dynamic conditions, the contact angle of a liquid on a solid surface changes significantly with the substrate velocity from its equilibrium value. The dynamic contact angles for polyethylene oxide (PEO) solutions of two molecular weights 3 × 10⁵ and 4 × 10⁶ have been measured using a polyethylene terephthalate (PET) plate. The three‐parameter Ellis model to fit the rheological data to obtain shear thinning power n, characteristic shear stress, and the zero‐shear viscosity is used. The theory indicates that dynamic contact angles follow power law in this instance instead of showing Newtonian behavior with zero‐shear viscosity when the shear thinning effects are considered. The elastic effect becomes important at larger polymer concentrations that reduces the dependence on capillary number, that is, reduces n keeping with the experiments. © 2016 American Institute of Chemical Engineers AIChE J, 62: 2533–2541, 2016     

Comportement des lits fluidisés en milieu viscoélastique

The flow of various fluids of different rheological behaviour has been studied in both fixed and fluidized beds. These fluids were water and aqueous solutions of glycerin, Carbopol and Separan AP-30. For small Deborah numbers, the flow of Carbopol and Separan AP-30 solutions thorough fixed beds is well described by the Ergun relation using the generalized Reynolds number of Christopher and Middleman (1965). Deviations from this behaviour are observed in fluidized beds as preferential flow paths easily occur near the distributor. These deviations are small for Newtonian fluids but take more importance with non-Newtonian solutions and are increased by viscoelasticity.     

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.     

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     

Pressure drop in a split‐and‐recombine caterpillar micromixer in case of newtonian and non‐newtonian fluids

Microreactors are very promising tools for the design of future chemical processes. For example, emulsions of very narrow size distribution are obtained at much lower energy consumption than the one spent with usual processes. Micromixers play thereby an eminent role. The goal of this study is to better understand the hydrodynamic properties of a split‐and‐recombine Caterpillar micromixer (CPMM) specially with regard to handling viscoelastic fluids, a topic hardly addressed so far in the context of micromixers in general, although industrial fluids like detergent, cosmetic, or food emulsions are non‐Newtonian. Friction factor was measured in a CPMM for both Newtonian and non‐Newtonian fluids. For Newtonian fluids, the friction factor in the laminar regime is f/2 = 24/Re. The laminar regime exists up to Reynolds numbers of 15. For shear‐thinning fluids like Carbopol 940 or viscoelastic fluids like Poly Acryl Amide (PAAm) aqueous solutions, the friction factor scales identically within statistical errors up to a generalized Reynolds number of 10 and 0.01, respectively. Above that limit, there is an excess pressure drop for the viscoelastic PAAm solution. This excess pressure drop multiplies the friction factor by more than a decade over a decade of Reynolds numbers. The origin of this excess pressure drop is the high elongational flow present in the Caterpillar static mixer applied to a highly viscoelastic fluid. This result can be extended to almost all static mixers, because their flows are generally highly elongational. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2679–2685, 2013     

Modeling rheological behavior of bentonite suspensions as Casson and Robertson-Stiff fluids using Newtonian and true shear rates in Couette viscometry

The Casson model and the Robertson-Stiff model have been used to determine whether they can describe the rheology of aqueous bentonite suspensions. The assessment utilized a total of twelve sets of experimental viscometric data from literature and from this work. Equations have been presented which allowed the determination of the true shear rates experienced by the fluids within the gap of the rotational viscometer for both rheological models. Non-linear regression has been applied to determine the two rheological parameters for the Casson model and the three rheological parameters for the Robertson-Stiff model using true shear rates and Newtonian shear rates, which are used most often in the analysis of rheometric data. The results showed that both models describe well the experimental data of these bentonite suspensions with good statistical indicators. Furthermore, analysis showed that true shear rates are always higher than Newtonian shear rates for both models. The differences depend on the particular suspension and are larger at low shear rates while they become smaller at higher shear rates indicating that the fluid behavior approaches Newtonian behavior at higher shear rates. The shapes of the rheograms remained essentially unchanged indicating that the rheological parameters determined with the use of true shear rates are very similar to the rheological parameters determined with the use of Newtonian shear rates. This was further confirmed with the computation of the rheological parameters for both models and both approaches. For the Casson model differences in the yield value computed with true shear rates were at most at 7% while for the plastic viscosity at 3%. For the Robertson-Stiff model, differences of the order of 2 to 5% were observed for the K-values, of 7% for γ˙₀-values while no differences were observed for the n-values. These small differences, however, do not justify use of Newtonian shear rates when analytical solutions exist which allow use of true shear rates without any compromise.     

Rheological properties of hydroxypropyl- and sodium carboxymethyl-substituted guar gums in aqueous solution

The aqueous solutions of guar gum and its hydroxypropyl (HP) and sodium carboxymethyl (sod CM) derivatives are pseudoplastic, and the transition from Newtonian to pseudoplastic occurs in the low-shear-rate range at the concentrations of interest to industries. The flow properties of these polysaccharide solutions were studied in the range of low to moderately high shear rates, using a very simple technique and instrument. The flow of these polysaccharide solutions can be described by equation of state based on Cross model, and the basic rheological parameters, like zero shear rate viscosity (η₀), elasticity modulus (G₀), and relaxation time (γ₀), were calculated from simple and established relations. Master viscosity curves indicate that the molecular weight distribution of native guar gum has not been changed by derivatization. The effect of concentration and temperature on rheological parameters (η₀ and γ₀) has been studied, and the relations among these were established by simple equations.     

An analysis of the viscous behaviour of polymeric solutions

Viscosity data of an aluminum soap solution, a polyisobutylene solution, a polyacrylamide solution, and ten polystyrene solutions have been analysed using two rheological models: the Carreau model A and the Ellis model. The model parameters were obtained by non-linear regression. The importance of the zero-shear viscosity for design purposes is illustrated by observing the appearance of a master curve in the representation of the dimensionless viscosity versus the product of the zero-shear viscosity by the shear rate. A linear relation is shown to exist between the logarithm of the zero-shear viscosity and the polymer concentration. It is asserted that either model should replace advantageously the commonly used power-law expression.     

THE EFFECT OF SURFACE ROUGHNESS ON PRESSURE DROP IN A PACKED BED

Particle surface roughness is shown to have a significant effect on the pressure drop in a packed bed of adsorbent particles. The packed bed friction factor is determined using three spherical adsorbents of differing degree of surface roughness in the Reynolds number range 1-62. The results were successfully correlated using a correlation of the Ergun type. It is shown that surface roughness significantly increases the friction factor.     

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.     

High shear rheology and shear degradation of aqueous polymer solutions

We have investigated the possibility of replacing hydraulic oils with aqueous polymer solutions having the same rheological properties. We measured the low and high shear rate viscosities of polymer solutions at several moderate concentrations and compared the results to the predictions of a molecular model. We found that viscosities measured at shear rates near 1 million reciprocal seconds are in good agreement with those calculated using the model of Mochimaru.⁶ Shear degradation studies were also conducted using higher molecular weight versions of some of the polymers. Exposure of these to very high shear rate caused a permanent decrease in viscosity and a corresponding change in the molecular weight and molecular weight distribution. Taken together, these results show that very low molecular weight polymers at moderate concentrations are necessary to formulate an aqueous hydraulic fluid with approximately Newtonian behavior at shear rates near 1 million reciprocal seconds, and without long-term polymer degradation.     

Physical Properties of Aqueous Solutions of Pectin Containing Sunflower Wax

The aim of this study is to investigate the physical properties of aqueous solutions of pectin (PA) containing sunflower wax (SFW), which are used as a basis for producing edible films. The stability and the rheological and microstructural characteristics of SFW/PA mixtures were evaluated. SFW/PA mixtures formed oil-in-water emulsions that were milky and opaque in appearance and were stable towards phase separation. Polarized micrographs revealed the presence of wax crystals, whose size decreased as pectin concentration increased. The rheological behavior of the aqueous solutions of pectin containing different amounts of SFW were best described by the generalized power law model of Herschel–Bulkley (H–B), which gave the best fit in all the range of shear rate values. Apparent viscosities and yield stress were determined using this model, and both properties increased with increasing pectin content. The apparent viscosity values were between 0.0095 and 0.1031 Pa s. SFW addition resulted in a small decrease in viscosity for emulsions formulated with 1 and 2 % PA, but the opposite effect was observed for emulsions formulated with 3 % PA. In addition, shear stress values were higher for emulsions with higher PA content, but were not affected by SFW addition.     

Nonlinear rheological behavior of a novel oil displacing agent

In this work, the shear and elongational rheologies have been investigated for a newly developed oil displacing agent, polymeric surfactant‐PSf. It was found that the PSf solutions exhibited Newtonian, shear‐thinning, and shear‐thickening behavior, respectively, depending on the polymer concentration and shear rate, and Cox–Merz rule was not applicable to these systems. The first normal stress difference (N₁) versus shear rate plots for PSf were complicated, which varied with the composition of the solutions. The uniaxial elongation in capillary breakup experimental results indicated that Exponential model could be used to fit the experimental data of the PSf solutions at lower polymer concentrations. In addition, it was found that PSf was more effective in improving shear viscosity than partially hydrolyzed polyacrylamide, but not in the case of elongational viscosity. The experimental results indicated that the microstructural mechanisms are responsible for the rheological behavior of the polymers. © 2014 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2014 , 131, 40813.