Local Energy Dissipation Rate Research Articles

Simulation of Flow in Stirred Vessel with Axial Flow Impeller: Zonal Modeling and Optimization of Parameters

The usefulness of the turbulent k-ε model for simulation of flow in a stirred vessel was studied in detail. For optimization, the effects of model parameters on flow characteristics were analyzed. It was observed that, any single set of model parameters could not yield even reasonable agreement between the model predictions and the experimental observations throughout the vessel. Therefore, the concept of zonal modeling was introduced for the r-z plane. It was observed that with the introduction of zonal modeling, the predicted values of flow variables, except the energy dissipation rate, were in good agreement with experimental data even close to the top surface of the tank. The same results could not be obtained with the standard set of parameters and a single zone. Local turbulent energy dissipation rate (ε) was calculated using the expression (ε = k3/2/Lres) proposed by Wu and Patterson (1989). The zonal modeling prediction was better than the prediction by a standard set of parameters. However, above the impeller, a wide discrepancy between the predicted and experimental value for energy dissipation rate was observed.

Use of CFD in prediction of particle size distribution in suspension polymer reactors

A two-compartment population balance model has been developed for taking into account the large spatial variations of the local turbulent kinetic energy, in order to predict the evolution of droplet sizes in a high holdup (i.e., 47–50 vol%) suspension polymerization system as a function of the most important process conditions. Phenomenological expressions were applied for describing the breakage and coalescence rates as a function of the local energy dissipation rate and physical properties of the system. Computational fluid dynamics (CFD) simulations were used for estimating the volume ratio of the impeller and circulation regions, the ratio of turbulent dissipation rates and the exchange flow rate of the two compartments at different agitation rates and continuous phase viscosities. A satisfactory agreement was obtained between theoretically and experimentally determined drop size distributions.

Hydrodynamics of suspensions agitated by pitched‐blade turbine

AbstractThe complete local fluid velocity vector was determined by three‐dimensional phase‐Doppler anemometry, while the velocity vector of suspended process particles was being measured. Three‐dimensional mean and fluctuating velocities, complete Reynolds stresses, 3‐D length macroscales, and turbulent energy dissipation rates are presented. The difference in mean and fluctuating velocities of the process particles and the fluid is also examined. Tangential mean flow opposite to the impeller rotation is observed at certain locations just outside and above the impeller, and impeller fluctuations are found also in the inflow to the impeller. The outflow of the impeller exhibits strong spatial variations. Length macroscales are essentially independent of impeller speed, but vary significantly in the tank. Local energy dissipation rates increase with agitation rate raised to power three. Differences in mean and fluctuating velocities of the two phases are normally small, but increase in the discharge flow, where differences in the direction are also observed.

Kinematic effects on local energy dissipation rate and local enstrophy in fluid turbulence

A simple analysis shows that for incompressible isotropic turbulence there exists a generic inequality between the energy dissipation rate and enstrophy whenever the moment correlations of the velocity field is quasi-normal. This indicates that the cause of the difference of spatial fluctuations between the two quantities is at least in part kinematic.

3‐D hydrodynamics generated in a stirred vessel by a multiple‐propeller system

AbstractThe aim of this study is to optimize the position and the number of propellers in a non‐standard tall vessel. Laser sheet flow visualization experiments were carried out for selected geometrical arrangements which produced stable flow patterns and good transport between the propellers. Four double‐propeller arrangements corresponding to frequent industrial cases and a three‐propeller system have been chosen. Comparison of LDA measurements in the r‐z plane, dimensionless global parameters NQp, Ntm, Np, Ep and spatial distribution of local energy dissipation rate ϵ shows that a three‐propeller system is the most efficient.

Turbulence in flocculators: Effects of tank size and impeller type

AbstractPast research shows that for the same power per unit volume, flocculation performance varies with tank size and impeller type. This study was performed to characterize effects of scale and impeller design on turbulence produced in the flocculation process. The study was performed with a Rushton turbine and an A310 foil impeller in three square tanks of 5‐, 28‐ and 560‐L volume. Fluid velocities were measured using a dual‐channel laser Doppler velocimeter with an enhanced burst spectrum analyzer. Flocculation tanks were operated at a constant average unit‐mass energy‐dissipation rate of 0.0016 m2/s3. The results show that the turbulence intensity and local turbulent energy‐dissipation rate were higher for the Rushton turbine than for the A310 impeller. The turbulence intensity was found to increase with increasing tank size regardless of impeller type. The local turbulent energy dissipation rate decreased for the Rushton turbine and remained constant for the A310 impeller with increasing tank size.

Impact of tank geometry on the maximum turbulence energy dissipation rate for impellers

AbstractThe maximum turbulence energy dissipation rate per unit mass, εmax, is an important variable in dispersion systems, particularly for drop breakup and coalescence, and for gas dispersion. The effect of tank geometry (number of baffles, impeller diameter, and off‐bottom clearance) on εmax for four impellers (the Rushton turbine, RT; the pitched blade turbine, PBT; the fluidfoil turbine, A310; and the high‐efficiency turbine, HE3) is examined. Mean and fluctuating velocity profiles close to the impellers were measured in a cylindrical baffled tank using laser doppler velocimetry. Local and maximum turbulence energy dissipation rates in the impeller region were estimated using ε = Av3/L with A = 1 and L = D/10 for all four impellers. Factorial designs were used to test for the effects of single geometric variables under widely varying conditions and interactions between variables. Several factorial designs were used to ensure that real effects were separated from effects that appeared as an artifact of the experimental design. Results show that the tank geometry has a significant effect on εmax, primarily with respect to variations in impeller diameter and interactions between the off‐bottom clearance and impeller diameter. For the same power input and tank geometry, the RT consistently produces the largest εmax and/or εmax scaled with N3D2.

Cellular response to agitation characterized by energy dissipation at the impeller tip

Sf-9 insect cells in suspension show an increase in intracellular calcium concentration as a function of local energy dissipation rate in two fluid systems. A significant calcium response (more than a 10% increase) resulted at a dissipation rate above 10 7 cm 2s −3 in 15 min capillary flow experiments. Similar results were obtained after 4 h with a dissipation rate above 10 5 cm 2s −3 or in a bioreactor stirred at ≥280 rpm for 4 h with a 6-bladed Rushton impeller. This impeller speed produces a local energy dissipation rate of 10 5 cm 2s −3 if the dissipation rate is estimated using the volume of the trailing vortices as the dissipation volume ( V = 0.075 d i 3). Our results show the capillary flow system to be a good model for prediction of cell damage in a stirred bioreactor.

Study of hydrodynamics in microcarrier culture spinner vessels: A particle tracking velocimetry approach

Three-dimensional particle tracking velocimetry (3-D PTV), a modern, quantitative, visualization tool, has been applied to the characterization of the flow field in the impeller region of cell culture reactor vessels. The experimental system used here is a 250-mL microcarrier spinner vessel. The studies were conducted at three different agitation rates, 90, 150, and 210 rpm, corresponding to healthy, mildly damaging, and severely damaging shear intensities, respectively. The flow can be classified into three regions: a predominantly tangential (azimuthal) flow generated by the impeller; a trailing vortex region coming off the impeller tip; and a converging flow region close to the center of the vessel. The latter two are the regions of highest velocity gradients. Energy dissipation rates due to mean velocity gradients were also calculated to characterize the impeller stream. Local specific energy dissipation rates > 10,000 erg/(cm3sec) · have been measured. It is proposed that the critical regions for microcarrier culture damage due to impeller hydrodynamics are the trailing vortex region and the high energy converging flow region. Graphical representation of the mean velocity flow fields and the distribution of energy dissipation rates in the impeller region are also presented here. The merits of using the dissipation function (measure of specific energy dissipation rate) as a possible scale-up parameter are also discussed. © 1996 John Wiley & Sons, Inc.

EDDY NUMBER DISTRIBUTION IN ISOTROPIC TURBULENCE AND ITS APPLICATION FOR ESTIMATING MASS TRANSFER COEFFICIENTS

The eddy number distribution for isotropic turbulent flows has been derived by manipulating well-established formulae for describing turbulence. The distribution has been used to derive an expression relating the fractional rate of surface renewal, and hence mass transfer coefficients across gas-liquid interfaces, to key process variables such as the local energy dissipation rate and the Kolmogorov scale. The expression has been compared with the previously published roll-cell and eddy-cell models, which were applied to correlating local mass transfer coefficients in a pipe and an open absorption channel. Excellent agreement has been found. The expression has also been applied to correlating the gas-side mass transfer coefficients for liquid evaporation in wetted wall pipe flows, and the mass transfer coefficient for solid dissolution adjacent to a solid-liquid interface in a stirred tank. The theoretical correlations have been validated by published experimental data.

Refined similarity hypotheses for turbulent velocity and temperature fields

The refined similarity hypothesis of Kolmogorov [J. Fluid Mech. 13, 82 (1962)] is extended to a scalar field. These hypotheses are tested using measurements in a circular jet and the atmospheric surface layer. Over a significant part of the inertial range, statistics of the normalized stochastic variables for velocity and temperature indicate a dependence on the separation r. This dependence is also quantified through the probability density functions of the stochastic variables and the correlation between the velocity (or temperature) increment and the local energy (or temperature) dissipation rates. Probability density functions of the stochastic variables are conditioned on the local Reynolds number Rer based on r and the local energy dissipation rate. These functions depend on Rer when the latter is small and are approximately universal when Rer is very large. This behaviour is consistent with the refined similarity hypothesis. There is however a slight difference between the shapes of the conditional probability density functions in the two flows, implying a weak dependence on the turbulence Reynolds number Rλ and flow conditions.

The multifractal lagrangian nature of turbulence

The multifractal formalism for the eulerian statistics of small-scale dynamics in turbulent flows is reviewed. Theoretical extensions of these results (the statistics of small volume averages of the energy dissipation rate) are used to predict properties of the probability distribution of the local energy dissipation rate at a fixed point. The improved parametrization of the eulerian statistics allows the lagrangian statistics (those for a fixed fluid particle in contrast to the eulerian statistics at a fixed point) to be determined exactly by using results derived as a consequence of incompressibility. Several properties of particle trajectories in a turbulent flow can be predicted with these new lagrangian statistics. In particular, a trajectory is typically smooth and generally unremarkable in its features. This contrasts the often suggested description: that of a highly convoluted and intricately structured ‘fractal’ curve. Some of the traditional dispersion results, which depend on the lagrangian statistics, are shown to be only weakly influenced by the intermittency inherent in the multifractal character of turbulence.

Statistical properties of two-dimensional magnetohydrodynamic turbulence

The statistical properties of two-dimensional (2-D) magnetohydrodynamic (MHD) turbulence are studied by means of high-resolution numerical simulations. As a theoretical point of reference, the β model of intermittent turbulence is adapted to the MHD case. Comparison of simulation results for energy spectra with the β-model predictions shows intermittency corrections to be small, δ<0.2, while fourth-order correlation functions exhibit a stronger effect δ≂0.35, consistent with the numerically observed Reynolds-number dependence of the flatness factor F∝R1/2λ. An argument is given that this scaling valid for Rλ∼102 is, however, not characteristic of the asymptotic regime Rλ→∞, where a constant value of F is to be expected. The probability distributions of the field difference δv(x,t), δB(x,t) are Gaussian for large separation x or t, approaching an approximately exponential distribution for x, t→0. This behavior can be understood by a simple probabilistic argument. The probability distribution of the local energy dissipation rate ε is roughly consistent with a log-normal distribution at larger ε but shows a different behavior at small ε.

Multifractal structure of fully developed hydrodynamic turbulence. I. Kolmogorov's third hypothesis revisited

We discuss intermittency effects in fully developed hydrodynamic turbulence. It is shown that the application of the bounded log-normal distribution to the fluctuations of the local energy dissipation rate resolves some basic difficulties related to Kolmogorov's third hypothesis and gives a good agreement with experiment. The nonlinear interaction of the large-scale and inertial-range turbulent pulsations of the velocities may explain the observable characteristics of the intermittency. We give also a detailed comparison of the results obtained with the use of the bounded log-normal distribution with that obtained in the framework of the homogeneous and random /~-models, a two-scale Cantor set approximation, and the original unbounded log-normal distribution suggested by Kolmogorov and Obukhov.

Laser-Doppler measurements of turbulent-flow parameters in a stirred mixer

It is recognized that a detailed knowledge of turbulence parameters, as well as velocities, can aid in understanding and modelling mixing rate dominated phenomena in stirred vessels. Turbulent-flow parameters were measured in a baffled, Rushton turbine agitated vessel with a laser-Doppler velocimeter. The necessary corrections for the periodic, nondissipative velocity fluctuations in the near impeller region were made by an autocorrelation method. Two components of periodic fluctuation, one corresponding to impeller blade frequency, the other corresponding to twice that frequency, were found to be significant. With the periodicity removed, meaningful turbulence intensities, autocorrelation functions, turbulence scales, energy spectra, and turbulence energy dissipation reates were obtained. Integral scales and turbulence energy dissipation rates were a particular objective in this work because of their usefulness in modelling local mixing rates in turbulence flows. From an energy balance around the impeller and impeller stream, it was found that 60% of the energy transmitted into the tank via impeller was dissipated in that region, and 40% was dissipated in the bulk of the tank. An equation for calculating local energy dissipation rates from resultant fluctuation velocities and resultant turbulence macroscales, ▪, appeared adequate. Constant A was found to be 0.85.

Random walk models for particle displacements in inhomogeneous unsteady turbulent flows

First a small time analysis is developed for the first and second moments of the velocity (W) and displacement (Z) in one direction of particles marked at a given point in an inhomogeneous unsteady turbulent flow, in terms of the local energy dissipation rate, and the local derivatives of the second and third moments of the vertical component of the velocity field, ∂∼(u23)/∂z and ∼(∂u33)/∂z. Then the appropriate form of a Langevin equation in inhomogeneous turbulence is suggested, namely, dW=(−W/TL+a1)dt+a1/22 dωt where a1, a2, and TL are functions of the particle position and time, and dωt is a random Gaussian velocity increment with ∼(dωt)=0 and ∼((dωt))2=dt. For simplicity, only one component of the particle motion, W(t), is considered. The functions a1 and a2 are determined by relating the random walk model to the Eulerian conservation equations for the mass of the contaminant and volume of the flow (i.e., the continuity equation), using the Fokker–Planck equation and the Eulerian equations for the moments of a vertical velocity. The coefficients a1 and a2 reduce to the same form as that obtained by the statistical analysis, namely a1=∂∼(u23)/∂z, (≊dW̄/dt, when t → 0) and a2=2∼(u23)/TL +d∼(u23)/dt (≊2∼(W′2)/TL+d∼(W′2)/dt, when t → 0). It is shown that the random walk model has the correct behavior as t/TL → 0. The theory is shown to agree reasonably well with the measurements of mean height and mean vertical displacement of particles released in a convective boundary layer [Q. J. R. Meteorol. Soc. 102, 427 (1976)]. Yaglom’s [Isv. Atmos. Oceanic Phys. 8, 333 (1972)] surface similarity result is recovered as a special case. For t ≫ TL, and in a zero-skewness steady turbulence, the random walk model reduces to the familiar K-diffusion equation. Some examples are presented to show that mean and mean square particle displacements from the random walk model are virtually identical to those obtained from the analytical solution of the corresponding Eulerian moment equations. Careful analysis is still required when concentration distributions in turbulent flows near a boundary are evaluated using random walk models.

Information theory and the spectrum of isotropic turbulence

The problem of determining the spectrum of isotropic turbulence can be thought of as one of finding the most appropriate joint probability distribution for the flow taken as a whole. From the point of view of information theory, what one means by the most appropriate distribution is clearly defined and easily justified; it is the probability distribution that maximises the information theory entropy, subject to whatever constraints one can impose on the flow. In this work, the relevant constraints are taken to be the Reynolds number and energy dissipation rate of the flow, energy balance (on average) at every point in wavenumber space, and adherence to the Navier-Stokes equations. Using these constraints, it is shown that the maximum entropy formalism leads to a pair of coupled equations describing the distribution of energy in the turbulent spectrum, and the correlations between the amplitudes of velocity components with nearly identical wavenumbers. Although solutions to these equations are not presented, it develops that if a power-law solution exists, it can only be the Kolmogorov law E(k) varies as k-53/. In arriving at this result, a useful concept is that of the 'turbulent temperature', defined as the reciprocal of the derivative of the entropy with respect to the local energy dissipation rate. This quantity plays a role directly analogous to the thermodynamic temperature, governing the rate of energy exchange between different wavenumbers. It is found that, within the spectrum's inertial subrange, the turbulent temperature is virtually constant, with only a minute temperature gradient required to drive the energy cascade.

Contrasting Fracture Models For Granite Under Dynamic Loading

A “symmetric impact” experiment was used as the basis for comparing different models of the fracture mechanism in granite. Two different tensile fracture models were considered, but better results were obtained when both shear and tensile fracture were admitted. The combined model employs a fracture criterion based on static measurements of the tensile and triaxial compressive strength and emphasizes the role of bulking, which is related directly to the local rate of energy dissipation. It is shown that the behavior subsequent to incipient fracture has a major influence on the overall material response.

TURBULENCE IN NON-BAFFLED MIXING VESSEL

Time-mean liquid velocity, RMS velocity, various scales of turbulence, and energy dissipation rate in non-baffled mixing vessel are measured. As the lateral RMS velocity is one-third to one-fourth the longitudinal RMS velocity, liquid in the vessel is in the non-isotropic turbulence field. However, the theory of local isotropy of Kolmogoroff can be applied because existence of the inertial subrange is observed. As the integrated value of local energy dissipation rate agrees with the power per unit mass of liquid from the impeller, almost all energy from the impeller is viscous dissipated in eddies of micro scale. At low impeller speed, there exist poorly turbulent portions at parts distant from the impeller or in motion of micro scale eddies in the high frequency range in spite of the fact that Re number indicates full turbulence. Turbulent diffusivities in radial direction are from 0.1 to 10 cm2/sec in this experiment and are almost proportional to impeller speed. Turbulent diffusivities in tangential direction are three to five times larger than those of radial or axial diffusivities and mixing between cylindrically rotating zone and outer quasi-free vortex zone is poor.

Vertical Profiling of Velocity Microstructure

Abstract A free-fall oceanographic instrument capable of producing direct estimates of the local rate of energy dissipation has been developed. The velocity sensor in an adaptation of the two-component airfoil probe to the oceanic environment. Spectra of the velocity data show that the instrument achieves complete spatial resolution of the vertical current shear and there is no apparent contamination of the signal from vibrations or body motions in the frequency range contributing to the variance of the shear. The energy dissipation is proportional to the variance of the vertical shear. Thus, it is possible to estimate the energy dissipation without comparing the velocity spectra to any “universal curve.”