Mixing Time Results Research Articles

Design and scale up of a mixing process for a novel automotive coating

The study reported here was performed as part of a larger project aiming to develop a high performance automotive coating and the associated manufacturing process to enable large scale manufacture of the new product. We report from the process development aspect of the project which aimed at establishing the comparative blending performance of two types of impeller- one currently in use and an alternative- at different scales. Mixing time measurements compared on the basis of volumetric power input were comparable for the sawtooth impeller and pitched blade turbine over the turbulent and transitional regimes at formulation (T= 0.13 m, ∼2 l) and pilot scales (T= 0.30 m, ∼ 20 l). At the largest scale of ∼ 160 l, it took a lot longer to blend when using the sawtooth impeller. Measured mixing time results were in agreement with predicted values using the well-established correlation for both impellers in T= 0.30 m and also for the PBT at large scale, For the sawtooth impeller the correlation underpredicted at large scale unless a higher value for the constant in the mixing time correlation (about twice) was used. This deviation in the performance at large scale has highlighted the complexity of design and scale up using a sawtooth impeller.

Investigation of mixing miscible liquids with high viscosity contrasts in turbulently stirred vessels using electrical resistance tomography

The time required to attain a sufficient degree of homogeneity i.e., mixing time, is an important parameter in mixing processes. This paper presents results from a study employing an experimental approach to estimate mixing times for a miscible Newtonian liquid mixture system with high viscosity contrasts in a turbulent stirred vessel. An Electrical Resistance Tomography (ERT) based technique has been adopted to monitor dimensionless mixing time across a range of additive viscosities, impeller designs, sizes, and speed. Dimensional analysis has been used to interpret these mixing time results in terms of the magnitude of the bulk inertia forces relative to the initial viscous forces within the added fluid. Critical non-dimensional numbers, uniting the properties of the two liquids, have been proposed as the criterion for avoiding undesirable operating conditions under which the mixing time is much longer than that required for mixing fluids with similar properties. The work proposes novel correlations for mixing time and incorporates a new dimensionless group, thereby enabling a more nuanced and accurate characterisation of mixing behaviours. This research stands to contribute to energy saving and waste minimisation efforts in the industry. It provides insights for process designers and simulation engineers, propelling a leap forward in the design and operation of mixing processes when dealing with liquid systems that have significant viscosity differences.

Efficient sampling and counting algorithms for the Potts model on ℤd at all temperatures

AbstractFor and all we give an efficient algorithm to approximately sample from the ‐state ferromagnetic Potts and random cluster models on finite tori for any inverse temperature . This shows that the physical phase transition of the Potts model presents no algorithmic barrier to efficient sampling, and stands in contrast to Markov chain mixing time results: the Glauber dynamics mix slowly at and below the critical temperature, and the Swendsen–Wang dynamics mix slowly at the critical temperature. We also provide an efficient algorithm (an FPRAS) for approximating the partition functions of these models at all temperatures. Our algorithms are based on representing the random cluster model as a contour model using Pirogov–Sinai theory. The main innovation of our approach is an algorithmic treatment of unstable ground states, which is essential for our algorithms to apply to all inverse temperatures .

Coalescing and branching simple symmetric exclusion process

Motivated by kinetically constrained interacting particle systems (KCM), we consider a reversible coalescing and branching simple exclusion process on a general finite graph G=(V,E) dual to the biased voter model on G. Our main goal is tight bounds on its logarithmic Sobolev constant and relaxation time, with particular focus on the delicate slightly supercritical regime in which the equilibrium density of particles tends to zero as |V|→∞. Our results allow us to recover very directly and improve to ℓp-mixing, p≥2, and to more general graphs, the mixing time results of Pillai and Smith for the Fredrickson–Andersen one spin facilitated (FA-1f) KCM on the discrete d-dimensional torus. In view of applications to the more complex FA-jf KCM, j>1, we also extend part of the analysis to an analogous process with a more general product state space.

Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications.

Ribonucleic acid (RNA) secondary structures and branching properties are important for determining functional ramifications in biology. While energy minimization of the Nearest Neighbor Thermodynamic Model (NNTM) is commonly used to identify such properties (number of hairpins, maximum ladder distance, etc.), it is difficult to know whether the resultant values fall within expected dispersion thresholds for a given energy function. The goal of this study was to construct a Markov chain capable of examining the dispersion of RNA secondary structures and branching properties obtained from NNTM energy function minimization independent of a specific nucleotide sequence. Plane trees are studied as a model for RNA secondary structure, with energy assigned to each tree based on the NNTM, and a corresponding Gibbs distribution is defined on the trees. Through a bijection between plane trees and 2-Motzkin paths, a Markov chain converging to the Gibbs distribution is constructed, and fast mixing time is established by estimating the spectral gap of the chain. The spectral gap estimate is obtained through a series of decompositions of the chain and also by building on known mixing time results for other chains on Dyck paths. The resulting algorithm can be used as a tool for exploring the branching structure of RNA, especially for long sequences, and to examine branching structure dependence on energy model parameters. Full exposition is provided for the mathematical techniques used with the expectation that these techniques will prove useful in bioinformatics, computational biology, and additional extended applications.

Macromixing study for various designs of impellers in a stirred vessel

The effect of the impeller designs and impeller clearance level (C/T) on power consumption, mixing time and air entrainment point in a single liquid phase under turbulent conditions (Re > 104) were investigated. Different impeller designs including conventional and new designs, were used to consider both axial and radial flow impellers. The electric conductivity method, suspended motor system and observation method were employed to determine the mixing time, the power consumption and the air entrainment point, respectively. The reduction in the impeller clearance level form T/3 to T/6 resulted in a decrease in power number values for up-flow pumping impellers while it was increased for down-flow pumping. The same trend was observed for the mixing time results. Moreover, axial flow impellers and specially HE3 are preferable for higher agitation speeds due to the less air entrainment. The results verified that the axial flow impellers and specifically down-flow impellers are more efficient than the radial flow impellers. ANFIS-Fuzzy C–means (ANFIS–FcM) and nonlinear regression were used to develop models to predict the mixing time based on the energy dissipation rate and clearance. The results verified that the model predictions successfully fit the experimental mixing time data.

Online Network Optimization Using Product-Form Markov Processes

We develop a gradient algorithm for optimizing the performance of product-form networks through online adjustment of control parameters. The use of standard algorithms for finding optimal parameter settings is hampered by the prohibitive computational burden of calculating the gradient in terms of the stationary probabilities. The proposed approach instead relies on measuring empirical frequencies of the various states through simulation or online operation so as to obtain estimates for the gradient. Besides the reduction in computational effort, a further benefit of the online operation lies in the natural adaptation to slow variations in ambient parameters as commonly occurring in dynamic environments. On the downside, the measurements result in inherently noisy and biased estimates. We exploit mixing time results in order to overcome the impact of the bias and establish sufficient conditions for convergence to a globally optimal solution. We discuss our algorithm in the context of different systems, including queueing networks, loss networks, and wireless networks. We also illustrate how the algorithm can be used in such systems to optimize a service/cost trade-off, to map parameter regions that lead to systems meeting specified constraints, and to achieve target performance measures. For the latter application, we first identify which performance measures can be controlled depending on the set of configurable parameters. We then characterize the achievable region of performance measures in product-form networks, and finally we describe how our algorithm can be used to achieve the target performance in an online, distributed fashion, depending on the application context.

Characterisation of the mixing of non‐newtonian fluids with a scaba 6SRGT impeller through ert and CFD

AbstractAn attempt has been made to study the mixing of yield‐pseudoplastic fluids with a Scaba 6SRGT impeller using electrical resistance tomography (ERT) and computational fluid dynamics (CFD). The ERT system with four sensor planes, each containing 16 equispaced stainless steel electrodes, was used to measure the mixing time. The multiple reference frames (MRF) technique and the modified Herschel–Bulkley model were applied to simulate the impeller rotation and the rheological behaviour of the non‐Newtonian fluids, respectively. To validate the model, the CFD results for the power consumption were compared to the experimental data. The validated model was then employed to obtain further information regarding the averaged impeller shear rate, impeller circulation, and pumping capacities. The CFD and ERT data were utilised to investigate the effect of the impeller power, fluid rheology, and impeller size on the mixing time. The mixing time results obtained in this study were in good agreement with those reported in the literature. © 2011 Canadian Society for Chemical Engineering

CFD Modelling of Global Mixing Parameters in a Peirce-Smith Converter with Comparison to Physical Modelling

The flow pattern and mixing in an industrial Peirce-Smith converter (PSC) has been experimentally and numerically studied using cold model simulations. The effects of air volumetric flow rate and presence of overlaying slag phase on matte on the flow structure and mixing were investigated. The 2-D and 3-D simulations of the three phase system were carried out using volume of fluid (VOF) and realizable k - ɛ turbulence model to account for the multiphase and turbulence nature of the flow respectively. These models were implemented using commercial Computational Fluid Dynamics (CFD) numerical code FLUENT. The cold model for physical simulations was a 1:5 horizontal cylindrical container made of Perspex with seven tuyeres on one side of the cylinder typifying a Peirce-Smith converter. Compressed air was blown into the cylinder through the tuyeres, simulating air or oxygen enriched air injection into the PSC. The matte and slag phases were simulated with water and kerosene respectively in this study. The influence of varying blowing conditions and simulated slag quantities on the bulk mixing was studied with five different air volumetric flow rates and five levels of simulated slag thickness. Mixing time results were evaluated in terms of total specific mixing power and two mixing time correlations were proposed for estimating mixing times in the model of PSC for low slag and high slag volumes. Both numerical and experimental simulations were in good agreement to predict the variation characteristics of the system in relation to global flow field variables set up in the converter through mathematical calculation of relevant integrated quantities of turbulence, Volume Fraction (VF) and velocity magnitudes. The findings revealed that both air volumetric flow rate and presence of the overlaying slag layer have profound effects on the mixing efficiency of the converter.

Using CFD and Ultrasonic velocimetry to Study the Mixing of Pseudoplastic Fluids with a Helical Ribbon Impeller

AbstractA commercial CFD package was used to simulate the 3D flow field generated in a cylindrical tank by a helical ribbon impeller. The study was carried out using a pseudoplastic fluid with yield stress in the laminar mixing region. Ultrasonic Doppler velocimetry (UDV), a noninvasive fluid flow measurement technique for opaque systems, was used to measure xanthan gum velocity. From flow field calculations and tracer homogenization simulations, power consumption and mixing time results were obtained. The torque and power characteristics remain the same for upward and downward pumping of the impeller, but the mixing times are considerably longer for the downward pumping mode. Overall, the numerical results showed good agreement with experimental results and correlations developed by other researchers. From the power and mixing time results, two efficiency criteria were utilized to determine the best pumping mode of the impeller.

Abrasion of tablets during scale-up: The influence of different crushing forces in laboratory and production perforated pan coaters

The purpose of this study was to investigate the influence of batch size during scale-up on the abrasion of biconvex tablets. Labelled tracer tablets of six different crushing forces (23–116 N) were mixed at different times and peripheral speeds in a laboratory (4 kg) and production (360 kg) perforated pan coater. The weight loss of these tracer tablets was determined. The main factor affecting the abrasion in both scales is the tablet crushing force as a nonlinear decrease in the abrasion was observed with increasing crushing force of the tablets. An increase in mixing time results in an increase in abrasion for the laboratory scale. In contrast to the production scale an influence of the peripheral speed on the abrasion could not be observed in laboratory scale. There is no difference in total abrasion for the laboratory scale and production scale for low peripheral speed. At higher peripheral speeds the abrasion in the production scale is slightly higher than in the laboratory scale.

Mixing and solid-liquid mass-transfer rates in a creusot-loire uddeholm vessel: A water model case study

The process of mixing and solid-liquid mass transfer in a one-fifth scale water model of a 100-ton Creusot-Loire Uddeholm (CLU) converter was investigated. The modified Froude number was used to relate gas flow rates between the model and its protoype. The influences of gas flow rate between 0.010 and 0.018 m3/s and bath height from 0.50 to 0.70 m on mixing time were examined. The results indicated that mixing time decreased with increasing gas flow rate and increased with increasing bath height. The mixing time results were evaluated in terms of specific energy input and the following correlation was proposed for estimating mixing times in the model CLU converter: Tmix=1.08Q−1.05W0.35, where Q (m3/s) is the gas flow rate and W (tons) is the model bath weight. Solid-liquid mass-transfer rates from benzoic acid specimens immersed in the gas-agitated liquid phase were assessed by a weight loss measurement technique. The calculated mass-transfer coefficients were highest at the bath surface reaching a value of 6.40 × 10−5 m/s in the sprout region. Mass-transfer coefficients and turbulence parameters decreased with depth, reaching minimum values at the bottom of the vessel.