Academic literature on the topic 'Hindered settling function'

The Vesilind settling velocity function forms the basis of flux theory used both in state point analysis (for design and capacity rating) and one-dimensional dynamic models (for dynamic process modelling). This paper proposes new methods to address known shortcomings of these methods, based on an extensive set of batch settling tests conducted at different scales. The experimental method to determine the Vesilind parameters from a series of bench scale settling tests is reviewed. It is confirmed that settling cylinders must be slowly stirred in order to represent settling performance of full scale plants for the whole range of solids concentrations. Two new methods to extract the Vesilind parameters from settling test series are proposed and tested against the traditional manual method. Finally, the same data set is used to propose an extension to one-dimensional (1-D) dynamic settler models to account for compression settling. Using the modified empirical function, the model is able to describe the batch settling interface independently of the number of layers.

The settling velocity of a sediment particle is an important parameter needed for modelling the vertical flux in rivers, estuaries, deltas and the marine environment. It has been observed that a particle settles more slowly in the presence of other particles in the fluid than in a clear fluid, and this phenomenon has been termed ‘hindered settling’. The Richardson and Zaki equation has been a widely used expression for relating the hindered settling velocity of a particle with that in a clear fluid in terms of a concentration function and the power of the concentration function, and the power index is known as the exponent of reduction of the settling velocity. This study attempts to formulate the model for the exponent of reduction of the settling velocity by using the probability method based on the Tsallis entropy theory. The derived expression is a function of the volumetric concentration of the suspended particle, the relative mass density of the particle and the particle’s Reynolds number. This model is tested against experimental data collected from the literature and against five existing deterministic models, and this model shows good agreement with the experimental data and gives better prediction accuracy than the other deterministic models. The derived Tsallis entropy-based model is also compared with the existing Shannon entropy-based model for experimental data, and the Tsallis entropy-based model is comparable to the Shannon entropy-based model for predicting the hindered settling velocity of a falling particle in a particle-fluid mixture. This study shows the potential of using the Tsallis entropy together with the principle of maximum entropy to predict the hindered settling velocity of a falling particle in a particle-fluid mixture.

Numerical models for the prediction of turbulent flow field and suspended solid distribution in sedimentation tanks are characterized by refined modeling of hydrodynamics, but apparently weak modeling of settling properties of suspensions. It is known that sedimentation tanks typically treat highly heterodisperse suspensions, whose concentrations range from relatively high to low values. However, settling is modeled either by considering one or more particle classes of different settling velocity, without accounting for hindered settling conditions, or by treating the suspension as monodisperse, even in regions of low concentration. A new generalized settling model is proposed to account for both discrete settling conditions in low concentration regions of the tanks and hindered settling conditions in high concentration regions. Settling velocities of heterodisperse suspensions are then determined as a function of particle velocities in isolation and their total concentration. The settling model is used in the framework of a transport model for the simulation of hydrodynamics and solid distribution in a rectangular sedimentation tank. Results show that solid distribution is mainly affected by particle interactions in the inlet region and by settling properties of individual particles in the outlet region. Comparison of the proposed settling model with other settling models suggests that a generalized approach of the modeling of settling properties of suspensions is a primary concern to obtain reliable predictions of the removal rate.

The consistent modelling methodology for secondary settling tanks (SSTs) leads to a partial differential equation (PDE) of nonlinear convection–diffusion type as a one-dimensional model for the solids concentration as a function of depth and time. This PDE includes a flux that depends discontinuously on spatial position modelling hindered settling and bulk flows, a singular source term describing the feed mechanism, a degenerating term accounting for sediment compressibility, and a dispersion term for turbulence. In addition, the solution itself is discontinuous. A consistent, reliable and robust numerical method that properly handles these difficulties is presented. Many constitutive relations for hindered settling, compression and dispersion can be used within the model, allowing the user to switch on and off effects of interest depending on the modelling goal as well as investigate the suitability of certain constitutive expressions. Simulations show the effect of the dispersion term on effluent suspended solids and total sludge mass in the SST. The focus is on correct implementation whereas calibration and validation are not pursued.

A systematic analysis of the state of the art in the methods for enhancing processes of thermochemical treatment of oil is carried out. A new method and a new system for controlling the process of dynamic settling of oil emulsion (OE) is developed, which allows increasing the efficiency of managing the process of dynamic settling by more accurately measuring the degree of phase separation, while avoiding the process of “flooding”. The mechanism of formation of an electrical double layer around emulsified water droplets (EWD) and the interaction energy of these droplets as a distance function is shown. An adequate mathematical model of hindered settling of EWD is proposed. It is shown that OE and intermediate emulsion layer (IEL) can be broken down by using microwave radiation. By virtue of this, the authors develop a new method, algorithm and system for automatic measurement of the water cushion level and the thickness of the İEL in settlers based on measuring the optical density of oil.

Abstract Most models of sedimentation contain the nonlinear hindered-settling flux function. If one assumes ideal conditions and no compression, then there exist several theoretically possible ways of identifying a large portion of the flux function from only one experiment by means of formulas derived from the theory of solutions of partial differential equations. Previously used identification methods and recently published such, which are based on utilizing conical vessels or centrifuges, are reviewed and compared with synthetic data (simulated experiments). This means that the identification methods are evaluated from a theoretical viewpoint without experimental errors or difficulties. The main contribution of the recent methods reviewed is that they, in theory, can identify a large portion of the flux function from a single experiment, in contrast to the traditional method that provides one point on the flux curve from each test. The new methods lay the foundation of rapid flux identification; however, experimental procedures need to be elaborated.

In this paper we present direct numerical simulations of monodisperse and polydisperse suspensions of non-Brownian particles sedimenting at low Reynolds number. We describe a scheme to generate ergodic ensembles of random particulate systems and a numerical procedure for computing interactions among spherical particles based on Ewald summation technique for hydrodynamic mobility tensors. From the generation process truly random both monodisperse and multimodal size distributions of particles were obtained for dilute and moderate densities based on a minimum energy criterion. Concerned with computations of the Ewald sums our numerical procedure drastically reduces the CPU simulation time providing results of the hindered settling function in good agreement with available experimental data and asymptotic results for ordered and random periodic arrays of particles. We show new computer simulations with no flux boundary perpendicular to gravity and periodic boundary conditions in horizontal direction. The simulations reproduce the experimental correlation-time and anisotropy of the velocity fluctuations, but have the magnitude of these fluctuations increasing proportional to the size of the system.

Batch sedimentation is a method that enables us to understand the mechanism of compaction and compression of sedimenting slurry. However, batch settling behaviour is a very complex phenomenon that is not easily described fully by a mathematical model. This causes unrealistically large empirical calculations when the thickener size estimations are required. Channelling, reverse concentration gradients and the initial concentration of the slurry have large effects on batch settling. Existing procedures do not provide clear relationships involving these three significant variables. In this study, batch sedimentation phenomena are examined in detail and possible explanations are given to clarify the complex behaviour using recent theories. Modern research has shown that channelling is an unwanted formation because channels can change the concentration at the bottom and top of the bed by carrying a great amount of flocs upwards. Batch sedimentation tests were performed using flocculated slurry of Calcium Carbonate at various initial concentrations such as 250 g/l, 500 g/l, 750 g/l and 1000 g/l to observe channelling and reverse concentration gradients. Flux plots for the batch system reveal behaviour which can be attributed to the upward flow of solids. In addition, photographic methods were used to observe settling processes, channelling mechanisms and flocs in the channels. One of the purposes of this work was to examine the phenomenological solid-liquid separation theory of Buscall and White (1987), which employs the material properties of the local volume fraction, compressive yield stress Py ()ö and hindered settling function R()ö to identify the material behaviour in batch sedimentation. Stepped-pressure filtration and batch settling tests were used to measure the material characteristics for the flocculated CaCO3 suspension. Experimental data were demonstrated using Height versus Time and Height versus Concentration graphs and displayed the possible region of reverse concentration gradients and channelling in the settling bed. Mathematical predictions adopted from Usher (2002) were performed employing material characteristics of the material and graphical documentations were presented. The results of mathematical predictions were compared to the experimental results and the modes of sedimentation explained by Lester et al. (2005). Fundamental theoretical models and experimental observations highlight that the main driving force for channelling is the high-pressure gradient at the bottom of the bed and the most important factors that cause channelling are high initial concentration of slurry and settling time. The predictions also show that the material and flocculant used for the batch settling tests demonstrate important effect on the settling process. The knowledge and information gained from this study is valuable to maximize the thickening process.

This paper describes the physical properties for defining the operation of a pulse jet mixing system. Pulse jet mixing systems operate with no moving parts located in the vessel or in the fluid to be mixed. Pulse tubes submerged in the vessel provide a pulsating flow that mixes the fluid due to a controlled combination of applied pressure to expel the fluid from the pulse tube nozzle followed by suction to refill the pulse tube through the same nozzle. For mixing slurries nondimensional parameters to define mixing operation include slurry properties, geometric properties and operational parameters. Primary parameters include jet Reynolds number and Froude number; alternate parameters may include particle Galileo number, particle Reynolds number, settling velocity ratio, and hindered settling velocity ratio. Rating metrics for system performance include just suspended velocity, concentration distribution as a function of elevation, and blend time.

Abstract A large fraction of HLW glass readily precipitates spinel. The presence of solid particles, including spinel crystals, is undesirable in a high-level waste (HLW) glass melter because the settling of solids can disrupt melter operation and shorten melter lifetime. Spinel formation in the melter can be reduced by lowering waste loading. When formulating HLW glass to maximize waste loading (thus minimizing the cost), the settling of insolubles must be considered. The rate of nucleation, growth, and dissolution of spinel crystals in a molten HLW glass was measured as a function of temperature and the presence of nucleation agents, such as noble metals. The mass transfer coefficient was calculated using the Hixson-Crowell model and expressed as an Arrhenius function of temperature that was identical for both crystal growth and crystal dissolution. The Stokes law for hindered settling and the Hixson-Crowell equation for crystal growth and dissolution were chosen as a convenient representation of spinel behavior for mathematical models of HLW glass melters. Levich’s analysis of the dissolution or growth of falling particles was used to estimate the effective diffusion coefficient (D) and the concentration-boundary-layer thickness (δ) around growing and dissolving crystals. The estimated values were in reasonable agreement with the measured concentration profile of Fe at a dissolving spinel crystal at 1200°C, determined with energy dispersive spectroscopy. The D value is comparable to that obtained by Borom and Pask for a sodium disilicate-magnetite couple. The calculated δ value agreed with the concentration distribution of Fe around falling crystals of spinel as imaged by optical microscopy. This methodology, in conjunction with mathematical modeling, provides a basis for developing optimized technology and glass formulation for HLW vitrification.

Here we observe the spatial and temporal patterns that erosion fronts driven by pulsed radial wall jets develop in double ring arrays of pulse tubes within slurry mixing vessels with curved bottoms. Although erosion of unbounded particle beds driven by individual steady jets has been studied for decades, the patterns developed within mixing vessels as neighboring transient erosion fronts collide and the subsequent relaxation of the particle bed towards the vessel center when the jets stop (i.e., as the pulse tubes refill within mixing vessels) remain incompletely understood. Relaxation here refers to motion of fluidized particle beds that were driven toward the vessel seam by radial wall jets that subsequently return or relax from the seam toward the center of the vessel when the jets turn off. Relaxation does not refer to downward individual or hindered particle settling. Spatial variations in the particle bed due to these relaxing particle beds comprise an important “initial” condition to the mathematical description of the evolution of the jet driven erosion front, and erosion fronts other than the one that expands radially from the pulse tube axis have only recently been described. For example, Bamberger, et al. (2017) [9], recently evaluated five selected cases of erosion patterns found in vessels 15 and 70 inches in diameter with 2:1 semi-elliptical bottoms. A highlight of that study was the discovery of a second type of erosion front that forms at the plane of symmetry between two adjacent pulse tubes. As neighboring radial wall jets collide they form an upwelling sheet of fluid; this second type of erosion front forms immediately beneath this upwelling flow. However, variations in this type of planar erosion front have not been cataloged previously. In this study, we systematically probe the erosion fronts driven by these upwelling sheets in greater detail and evaluate the relaxation of the particle bed to its “initial” condition after the pulse ceases. Variations in the erosion patterns and particle bed relaxation are evaluated as a function of particle concentration, density, and size. This study specifically focusses on video images collected from the 15 inch vessel because it provides distinctive visualization of erosion pattern behavior. We find the upwelling sheets to be more influential on the erosion patterns at lower particle concentrations, making these findings particularly important to low solids concentration vessels. At lower particle concentrations, flow at the base of the plane of symmetry readily erodes particle beds. At higher particle concentrations, piles of unmobilized solids accumulate beneath colliding jets either because the erosion mechanism vanishes or because erosion at the plane of symmetry is slow compared to radial erosion. We also find that the upwelling sheets introduce a flow that drives erosion patterns from outer ring jets toward the vessel center along the curved vessel floor along the plane of symmetry between nozzles. We further find that the rate of particle bed relaxation back toward the vessel center after the pulse ceases may correlate with concentration, particle density, and size. Higher concentrations and particle densities relax faster. The rate at which the entire bed relaxes toward the vessel center is faster near the vessel seam but slows as the relaxing front approaches the vessel center. This paper discusses competing mechanisms to explain these observations, including particle rolling, bed avalanches, gravity driven fluidized bed motion, and suspended particle sedimentation.