Power Consumption of a Rushton Turbine Mixing Viscous Newtonian and Shear-thinning Fluids: Comparison between Experimental and Numerical Results

A fluid dynamics software package (Fluent) was used to calculate the local velocities in a standard tank fitted with a Rushton turbine. We defined the local average shear rate γ˙¯ as a user variable. The power was computed from the values of γ˙ with a user subroutine. The influence of the rheological behaviour of the fluid was investigated: the shape factor K_p for a Newtonian fluid, and in the case of a shear-thinning fluid Metzner and Otto′s concept was used to determine the effective shear rate γ˙_e · γ˙_e verified the classical relation: γ˙_e = K_s·N. The influence of position and stirring speed on the local average shear rate γ˙¯ was also studied. As expected, higher values were obtained near the turbine. At a fixed position, γ˙¯ was found to be proportional to the stirring speed N. The power values obtained by numerical procedures were compared to experimental results: the Fluent software yielded results in excellent agreement with experimental findings. The calculation of local parameters such as the local average shear rate γ˙¯ as well as global characteristic shape factors demonstrate the validity of Fluent software in the laminar range.     

A new investigation of the metzner-otto concept for anchor mixing impellers

Anchor impellers are commonly used for the homogenization of non-Newtonian fluids, often in association with a set of coaxial turbines. The optimal design of such mixers relies on the knowledge of power drawn by the individual impellers. In non-Newtonian mixing, this can be readily obtained using the Metzner-Otto (1957) concept. In this article, the Metzner-Otto concept and the determination of the constant K_s for anchor impellers have been revisited using numerical and experimental techniques for the case of shear-thinning and shear-thickening fluids. Contrary to literature findings, it is shown that the constant K_s does not vary strongly with the power law index and that, for mixer design purposes, the use of a constant value of K_s for each of the rheological behaviors considered is adequate.     

Hydrodynamic Performance of a Ring-Style High-Shear Impeller in Newtonian and Shear-Thinning Fluids

Laminar mixing of Newtonian and shear-thinning fluids induced by a Hockmeyer®-type impeller was investigated. Two unbaffled tanks at three impellers off-bottom clearances (c) were studied. Six geometric combinations, i.e., two d/T and three c/T, were examined where d and T are the impeller and tank diameters, respectively. Determination of the Metzner-Otto constant (K_s) was undertaken. The effects of d/T, c/T, and fluid rheology on K_s, power demand, pumping, shear and viscous dissipation were analyzed. The evaluated geometric ratios and rheology do not significantly affect K_s and power demand, only the rheology had an impact on the remaining hydrodynamic parameters. Pumping was favored with the Newtonian fluid, and shear and viscous dissipation increased with the shear-thinning fluid.     

Power Requirement when Mixing a Shear‐Thickening Fluid with a Helical Ribbon Impeller Type

The optimal design of close clearance impellers requires the knowledge of the power demand of the mixing equipment. In non‐Newtonian mixing, this can be readily obtained using the Metzner and Otto concept [1]. In this work, this concept and the determination of the K_s value for an atypical helical agitator (PARAVISC system from Ekato firm) have been revised in the case of shear‐thinning fluids and a shear‐thickening fluid. For poor shear‐thinning fluids, it has been shown that for our mixing system the K_s value does not vary strongly with the flow behavior index, and may be regarded as a constant for the mixing purpose design. By contrast, for the shear‐thickening fluid, power consumption measurements indicate that the relationship between the K_s values and the flow behavior index is much more complex due to a partial solidification of the product around the impeller.     

Power-draw analysis of a coaxial mixer with Newtonian and non-Newtonian fluids in the laminar regime

The power consumption of a new coaxial mixer composed of a wall scraping arm and a series of rods and a pitched-blade turbine mounted on the same axis of revolution and operated in a contra-rotating mode has been characterized. The work is based on experimental measurements and 3D numerical simulations in the case of homogeneous Newtonian and non-Newtonian fluids in the laminar regime. Very good agreements between experimental and numerical results have been obtained. It has been shown that the Metzner-Otto concept can be extended to account for the speed ratio between the impellers, which allows to represent the power consumption results of the coaxial mixer on a single power master curve like with a single agitator mixer.     

半弧面斜叶桨的流体力学性能与氧传质特性

氧传质系数是气液搅拌反应器设计的关键参数,研究新型搅拌桨的氧传质性能对气液两相搅拌反应器的强化有着重要的意义。本文实验研究了气体分布器、搅拌转速、气量对氧传质系数、搅拌功耗及气含率的影响,结果表明,氧传质系数随搅拌转速和气量的增加而增加;并建立了氧传质系数与搅拌功耗和表观气速的经验公式,为进一步放大应用提供了基础。采用欧拉-欧拉多相流模型及群体平衡模型对半弧面新型斜叶桨进行了计算流体力学（CFD）数值模拟研究,模拟研究了不同结构、搅拌转速、气量下的流体力学性能和氧传质系数,模拟计算结果与实验值的相对偏差在20%以内;这为研究这一半弧面新型斜叶桨提供了一种可靠的数值模拟方法;优化了半弧面新型斜叶桨的结构,提高了搅拌釜的氧传质效率。     

Viscoelastic Liquids in Stirred Vessels – Part I: Power Consumption in Unaerated Vessels

Different shear‐thinning and elastic fluids (STE fluids) have been stirred under unaerated conditions, in vessels equipped with Rushton disc turbines. Their power consumption has been evaluated over a wide range of stirring rates and their Metzner‐Otto constant (k_s) has been measured. A correlation has then been proposed to predict k_s values for a Rushton turbine operating in non‐Newtonian solutions. Power curves of STE fluids have been drawn and compared with reference curves (Newtonian, shear‐thinning inelastic and elastic with constant shear viscosity fluids). The STE fluids have thus been divided into two categories. The STE fluids of the first category (STE I fluids), which are concentrated viscous solutions of polymers (guar, CMC) reducing the power consumption at the beginning of the transitional region and connecting with the Newtonian reference at higher Reynolds numbers. In contrast, STE solutions of the second category (STE II fluids), which are solutions of drag reducing polymers (PAA), are less viscous and more elastic. They reduce the power consumption at the end of the transitional region and do not connect with the Newtonian reference, at least until Re = 6000. A general correlation has finally been proposed to model the power curve of STE fluids stirred by a Rushton turbine from the laminar to the turbulent regions, as a function of their elasticity.     

Control strategy of pH,dissolved oxygen concentration and stirring speed for enhancing β‐poly (malic acid) production by Aureobasidium pullulans ipe‐1

BACKGROUND: β‐poly(malic acid) (PMLA) can be used as a pro‐drug or for a drug‐delivery system. Effects of pH, dissolved oxygen concentration (DO) and stirring speed were investigated to improve PMLA production by A. pullulans ipe‐1. RESULTS: The strain produced a high PMLA concentration when pH and DO remained at about 6.0 and above 70%, respectively, and the yeast‐like cells were the main PMLA producers. To further promote PMLA production, the cultivation could be divided into three phases. In phase I, cell growth was accelerated by maintaining high DO (>70%) with a constant stirring speed of 800 rpm. In phase II, PMLA production was increased by controlling DO at 70% using the automatically controlled stirring speed. In phase III, PMLA production on per gram of glucose (Y_p/s) was enhanced by keeping DO at 70%, and using a low stirring speed to decrease cell growth. Compared with batch cultures, a higher PMLA yield was obtained with this strategy, i.e. PMLA production and Y_p/s increased by 15% and 18%, respectively. CONCLUSION: Control strategies for pH, DO and stirring speed provide a good reference for process development and optimization of PMLA production. © 2012 Society of Chemical Industry     

Relaxation of stresses in crazes at crack tips and rate of craze extension

The Barenblatt theory of cohesive stresses at crack tips is used to investigate the effect of the relaxation of craze stresses at crack tips on the rate of craze extension. The craze stresses are equated to the cohesive stresses of the Barenblatt theory. The cancellation by the cohesive/craze stress of the singularity that would exist at the crack tip in their absence is assumed to hold for an extending craze. With this assumption, relaxation of the craze stresses produces craze extension, an effect which has been called ‘relaxation controlled growth’ by Williams and Marshall. A general equation relating the rate of change of craze length to the rate of change of stress intensity factor (K₁) and the rate of change of the craze stress is derived. It is argued from this equation that uniform crack growth with a constant craze length can occur only at constant K₁. Using plausibility arguments for the behaviour of the craze stress with time and position in the craze, and assuming a generalized Dugdale model, differential equations for the rate of craze extension with no crack growth are derived for the constant load and constant K₁ cases. These equations relate the rate of change of craze length to the craze stress at the tip of the crack. Assuming a specific form for the time dependence of this stress, the equation for the constant K₁ case is solved to yield an expression for the craze length as a function of time.     

Gas transfer in an external loop bioreactor with built-in ultrafiltration

The SGI Ultrafermenter is an external loop bioreactor with circulation through an ultrafiltration module allowing removal of medium and soluble products during fermentation. The contents are continually circulated during operation and the vessel is also equipped with a stirring turbine. The interaction of these two mixing agents on gas transfer was investigated. Both mechanisms produced similar increments in K_La over their working ranges, with values from approximately 0–700 h^?1 at 200 dm³ h^?1 airflow. The effects of these two mechanisms on K_La were approximately additive at low values but the combined mixing produced a maximum K_La value also in the region of 700 h^?1. The power-draw of mixing using the two agents was calculated and stirring was found to be 10–20 times more efficient than circulation.     

Power Requirement for Mixing Shear‐Thinning Fluids with a Planetary Mixer

The optimal control of processes dealing with non‐Newtonian liquids requires the knowledge and control of the power demand of the mixing equipment. In this context, an extension of the Metzner and Otto concept to planetary mixers is proposed to adapt this concept to planetary mixers. The theoretical part of this work defines modified expressions of Reynolds and power numbers. These definitions introduce a characteristic velocity u_ch that is used to define the parameter K_s. A planetary mixer is employed to experimentally ascertain this guideline. Power consumption measurements carried out by mixing shear‐thinning fluids permit to determine the K_s factor. This factor varies only slightly with the flow behavior index and may be regarded as a defined constant for this geometry. Finally, experiments with an additional shear‐thickening fluid confirm the validity of this approach.     

Performance of an RTL contactor for gas-liquid systems: Effective interfacial area and volumetric mass transfer coefficient by oxidation of sodium sulphite solution

Effective interfacial area a and volumetric liquid-side mass transfer coefficient k_La of an RTL contactor were obtained at different stirring speeds by absorption of oxygen from air into 0.8 kmol/m³ sodium sulphite solution, in the presence of Co⁺⁺ ions. The values of a and k_La ranged from 80 to 150 m²/m³ and 0.0003 to 0.00053 s^?1, respectively, when stirrer speed was increased from 8 to 40 rpm. When k_L alone was evaluated, it was found to be practically constant, irrespective of stirring speed.     

Drag of a cylinder moving near a wall in a yield stress fluid

The drag of a cylindrical obstacle moving at a constant velocity in a yield stress fluid close to a wall is studied experimentally and numerically. The wall influence has been explored for gap values between the cylinder of diameter D and the wall ranging from 0.01D to 100D, which corresponds, respectively, to hydrodynamic lubrication and to unconfined domain conditions. A model yield stress fluid (Carbopol gel) is used in the experiments. The viscous and plastic drag coefficients have been calculated and measured as depending on the Oldroyd number, in conditions where the yield stress effects are more important than those of viscosity and the inertia negligible. We have performed experimental and numerical validations in the Newtonian case and provided more specifically comparisons of our measured data on yield stress materials with those resulting from viscoplastic flow simulations. © 2018 American Institute of Chemical Engineers AIChE J, 64: 4118–4130, 2018     

Sedimentation in a dilute emulsion

In the present work the method of Batchelor[1] is adopted to calculate the statistical mean speed of sedimentation in a dilute dispersion of identical spherical droplets. The mean settling speed is shown to be

where U₀ is the settling velocity of a single fluid sphere falling under gravity through an unbounded quiescent fluid, σ is the ratio of the viscosities μ′/μ where the upper prime indicates the disperse phase, c is the volume fraction of the spheres, and K (σ) is a numerical factor whose values are displayed in Table 1.An exact evaluation of K (σ) requires knowledge of the solutions for the drag force components acting on two fluid spheres in the directions of their line of centre and perpendicular to it. Such a solution for the latter case is not yet available, hence, an approximate solution based on the ‘method of reflections’ is used. This causes some error in the numerical results for K (σ), the magnitude of which is examined and discussed.     

Impact of thixotropy on flow patterns induced in a stirred tank: Numerical and experimental studies

Agitation of a thixotropic shear-thinning fluid exhibiting a yield stress is investigated both experimentally and via simulations. Steady-state experiments are conducted at three impeller rotation rates (1, 2 and 8 s⁻¹) for a tank stirred with an axial-impeller and flow-field measurements are made using particle image velocimetry (PIV) measurements. Three-dimensional numerical simulations are also performed using the commercial CFD code ANSYS CFX10.0. The viscosity of the suspension is determined experimentally and is modelled using two shear-dependant laws, one of which takes into account the flow instabilities of such fluids at low shear rates. At the highest impeller speed, the flow exhibits the familiar outward pumping action associated with axial-flow impellers. However, as the impeller speed decreases, a cavern is formed around the impeller, the flow generated in the vicinity of the agitator reorganizes and its pumping capacity vanishes. An unusual flow pattern, where the radial velocity dominates, is observed experimentally at the lowest stirring speed. It is found to result from wall slip effects. Using blades with rough surfaces prevents this peculiar behaviour and mainly resolves the discrepancies between the experimental and computational results.     

The role of mass transfer in the electrolytic reduction of hexavalent chromium at gas evolving rotating cylinder electrodes

The rate of electrolytic reduction of hexavalent chromium from acidic solution at a hydrogen-evolving rotating cylinder lead cathode was studied under conditions of different current densities, Cr⁶⁺ concentrations and rotation speeds. The rate of the reaction was found to follow a first order rate equation. The specific reaction rate constant was found to increase with increasing rotation speed until a limiting value was reached with further increase in rotation speed. Mechanistic study of the reaction has shown that at relatively low rotation speeds the reduction of Cr⁶⁺ is partially diffusion controlled, at higher speeds the reaction becomes chemically controlled. The limiting specific reaction rate constant was related to the operating current density by the equationK=0.044i ^1.385. The current efficiency of Cr⁶⁺-reduction was measured as a function of current density, initial Cr⁶⁺ concentration and rotation speed. Possible practical applications are discussed.Nomenclature A electrode area (cm²) - a, b constants in Equations 5 and 13, respectively - C bulk concentration of Cr⁶⁺ at timet(M) - C _o initial concentration of Cr⁶⁺ (M) - C _i interfacial concentration of Cr⁶⁺ (M) - d cylinder diameter (cm) - D diffusivity of Cr⁶⁺ (cm² s^–1) - e _o standard electrode potential (V) - F Faraday's constant (96 487 C) - current consumed in hydrogen discharge (A) - i current density (A cm^–2) - I cell current (A) - K _l mass transfer coefficient (cm s^–1) - K _r mass transfer coefficient due to cylinder rotation (cm s^–1) - K _o natural convection mass transfer coefficient (cm s^–1) - K _g mass transfer coefficient due to hydrogen stirring (cm s^–1) - K ₂ specific reaction rate constant (cm s^–1) - K overall rate constant (cm s^–1) - m theoretical amount of Cr⁶⁺ reduced during electrolysis (g) - P gas pressure (atm) - R gas constant (atm cm³ mol^–1 K^–1) - T temperature (K) - t time (s) - V linear speed of the rotating cylinder (cm s^–1) - hydrogen discharge rate (cm³ cm^–2 s^–1) - V _s solution volume (cm³) - z electrochemical equivalent (g C^–1) - Z number of electrons involved in the reaction - Re Reynolds number (Vd/v) - Sh Sherwood number (K _r d/D) - Sc Schmidt number (v/D) - rotation speed (r.p.m.) - kinematic viscosity (cm² s^–1)     

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.     

Maxwell Fluid Model for Generation of Stress–Strain Curves of Viscoelastic Solid Rocket Propellants

Solid rocket propellants are modeled as Maxwell Fluid with single spring and single dashpot in series. Complete stress–strain curve is generated for case‐bonded composite propellant formulations by taking suitable values of spring constant and damping coefficient. Propellants from same lot are tested at different strain rate. It is observed that change in spring constant, representing elastic part is very small with strain rate but damping constant varies significantly with variation in strain rate. For a typical propellant formulation, when strain rate is varied from 0.00037 to 0.185 per second, spring constant (K) changed from 5.5 to 7.9 MPa, but damping coefficient (D) varied from 1400 to 4 MPas. For all strain rates, stress–strain curve is generated using developed Maxwell model and close matching with actual test curve is observed. This indicates validity of Maxwell fluid model for case‐bonded solid propellant formulations. It is observed that with increases in strain rate, spring constant increases but damping coefficient decreases representing solid rocket propellant as a true viscoelastic material. It is also established that at higher strain rate, damping coefficient becomes negligible as compared to spring constant. It is also observed that variation of spring constant is logarithmic with strain rate and that of damping coefficient follows a power law. The correlation coefficients are introduced to ascertain spring constants and damping coefficients at any strain rate from that at a reference strain rate. Correlation for spring constant needs a coefficient “H,” which is function of propellant formulation alone and not of test conditions and the equation developed is K₂=(K₁‐H)×{ln(dε₂/dt)/ln(dε₁/dt)}+H. Similarly for damping coefficient (D) also another constant “S” is introduced and prediction formula is given by D₂=D₁×{(dε₂/dt)/(dε₁/dt)}^S. Evaluating constants “H” and “S” at different strain rates validate this mathematical formulation for different propellant formulations. Close matching of test and predicted stress–strain curve indicates propellant behavior as viscoelastic Maxwell Fluid. Uniqueness of approach is to predict complete stress–strain curves, which are not attempted by any other researchers.     

Power consumption of helical ribbon mixers in viscous newtonian and non-newtonian fluids

Power input measurements are reported for helical ribbon impellers for two scales; a 0.15 m diameter and a 0.4 m diameter tank. Data for viscous Newtonian and non-Newtonian fluids are brought together by use of the average apparent viscosity concept and the following equation:

where k_s is the shear rate constant, c is the clearance between vessel wall and impeller with diameter D.Power measurements from this work combined with relevant information extracted from the published literature indicate that impeller geometry has a profound effect upon power requirement, particularly in the laminar region, where the reported data can be described by:

where K_p is a geometric constant and all the other symbols have their usual significance. Theoretical models which fail to allow for system geometry and fluid properties give values which may be seriously in error.     

Power consumption and mass transfer in agitated gas-liquid columns: A comparative study

Power consumption, gas holdup and oxygen mass transfer in agitated gas-liquid columns have been studied for an air-water system. Measurements have been carried out in a reciprocating plate reactor using five different types of perforated plates and in a stirred tank reactor with one, two and three Rushton turbines, a helical ribbon impeller with and without surface baffles. Each mixing vessel had an identical geometry with a working volume of 17 L. For reciprocating plate stacks, the gas holdup is a complex function of the perforation diameter, the frequency of agitation and the gas superficial velocity. For radial-type mixing devices, the gas holdup increases more rapidly with the speed of rotation for the helical ribbon. The power imparted to the fluid by the mixing device is independent of the gas superficial velocity for the plate stacks and the helical ribbon impeller for a given frequency or speed of agitation whereas it decreases for Rushton turbines. The correlation of the power consumption obtained for all mixing devices plotted against the reciprocating frequency or speed of rotation to the third power shows a linear fit. K_La values were correlated very well with the power input per unit volume and superficial gas velocity for all mixing devices. At lower power input per unit volume, K_La is a function of only the gas superficial velocity. At higher input power per unit volume, K_La increases rapidly with an increase in the intensity of agitation. Reciprocating plates with larger diameter perforations led to higher K_La values whereas the lowest K_La were obtained with the helical ribbon impeller. Correlations for one and three Rushton impeller assemblies were almost identical whereas measured K_La were much higher for the two-impeller assembly due to the presence of a highly mixed zone in the vicinity of the dissolved oxygen probe.