Modelling the Hindered Settling Velocity of a Falling Particle in a Particle-Fluid Mixture by the Tsallis Entropy Theory

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

Evaluation of a data-driven primary sedimentation tank model using settleometer data

Data-driven primary sedimentation (or settling) tank (DDPST) model was recently developed to improve the existing primary sedimentation (or settling) tank (PST) models which are largely total suspended solids (TSS) based. The purpose of the DDPST model is to realistically simulate the full-scale PST (FS-PST) underflow and overflow outputs in terms of biodegradable particulate organics (BPO), unbiodegradable particulate organics (UPO), and inorganic suspended solids (ISS) compositions. This characterization is fundamental to the planning of the downstream resource recovery systems such as anaerobic digestion (AD) and activated sludge (AS) systems through accurate process predictions. The DDPST model was previously subjected to rigorous mass balance verifications and has undergone a general sensitivity analysis (GSA) to identify the most important parameters required to allow the accurate predictions. The settling velocities and settling proportions of the five particle settling velocity groups (SVGs) were identified as significant. These five SVGs are characterized by different proportions of settleable BPO (BPOset), settleable UPO (UPOset) and settleable ISS (ISetS). Due to variability in wastewater (WW) characteristics from plant to plant, the DDPST model requires settleability data that is specific to a plant to make correct predictions. A five-column settleometer (5C-settleometer) has been proposed as a tool that can provide this necessary data. Essentially, the 5C-settleometer is regarded a labscale PST that allows detailed study of the FS-PST critical parameters through segregation of settleable TSS into five different SVGs. Along with the Augment Biomethane Potential (AugBMP) test procedure and parameter estimation tools, the particles from the SVGs can be fractionated into BPOset, UPOset, and ISetS. To this date, however, the accurateness of the 5C-settleometer to provide accurate FS-PST settleability characteristics has not been confirmed. This implies that the 5C-settleometer, as a suggested tool to provide the necessary data, cannot be used with confidence, and so is the DDPST model. The purpose of this study was to evaluate the DDPST model through comprehensive characterization of the primary sewage sludge (PSS). To achieve this, the ability of the 5Csettleometer to accurately characterize the PSS in terms of settling velocities and settling proportions was verified. This was done by collecting the FS-PST influent, underflow, and overflow diurnal data in parallel with the 5C-settleometer test runs at the Bellville Wastewater Treatment Works (WWTWs). Using the FS-PST diurnal data, the related percentage removals for settleable solids (in terms of BPOset, UPOset, and ISetS) were determined. Using the 5Csettleometer characterized PSS outputs, the parameters required to run the DDPST model (i.e., five settling velocity groups and their corresponding BPOset, UPOset, and ISetS settling proportions) were determined to allow the PST underflow and overflow model predictions. Following this, the PST percentage removals (underflow) and overflow characteristics from (i) the DDPST model, and (ii) the FS-PST, were compared to confirm whether the model predictions fully replicate the FS-PST. Important to note is that both the FS-PST and the 5Csettleometer received the same influent characteristics to ensure that the settleometer overflow and underflow are reasonably compared with the FS-PST to check the representativeness. While the 5C-settleometer has proven (i) the ability to segregate the particles into distinguishable particle sizes, and (ii) the ability to somewhat account for the overall removal proportions that are comparable to the FS-PST at a flow rate fluctuating between 0.6 and 0.8 l/min, its relationship with the DDPST model needs to be reassessed. The settleometer upflow velocities appeared to be too high to allow accurate model predictions. Unfortunately, it is impossible to achieve very low velocities to fulfil the DDPST model requirements with the current settleometer design. This shortfall on the model to represent the 5C-settleometer makes it impossible to yield reasonable predictions. This gap requires to be addressed to ensure that the 5C-settleometer, as a lab-scale version of the FS-PST, is correctly translated into the DDPST model, not only in terms of the upflow velocities which govern the particle settling, but also, the accompanying settleometer features which affects the particle or flow movements, i.e., (i) the occurrence of sharp bends or curvatures in a 5C-settleometer, or (ii) the increased probability of inter-particle forces especially when operating the settleometer at low flow rates. While it may be a challenge to model these interfering instances, it is essential that they are accounted for in the model to ensure accurate DDPST model predictions. It is further recommended to explore ways to improve the operation of the current settleometer. For instance, investing in a higher capacity pump and a settleometer flow meter will, with no doubt, add a benefit to the quality of the settleometer output data. In addition, the use of multiple settleometer runs with a combination of high and low flow rates can be explored, to achieve exceptionally representative FS-PST data. In terms of the elemental compositions (EMs), the five SVGs showed a marginally varied BPOset and UPOset compositions. This could be the result of particulate material differences between the SVGs. Nonetheless, the average BPO and UPO compositions recorded in terms of CXHYOZNAPB are C1H5.99O1.13N0.05P0.00 and C1H1.79O0.45N0.22P0.02, respectively. The subscript “Y” (i.e., 5.99) of the BPO composition marginally differs from what has already been reported on literature. This difference could be linked to (i) the discrepancies related to the carbon measurements experimentally (carbon is calculated from biogas measurement and alkalinity), and (ii) the source of Bellville WW, i.e., Bellville WWTWs receives both domestic and industrial WW. Overall, the ISetS (76.7%) and UPOset (78.1%) were found to be removed in greater proportions than BPOset (37.7%) in the FS-PST