Fluid flow through a bed of solid particles is an important process that occurs in full-scale water treatment operations. The Carman–Kozeny model remains highly popular for estimating the resistance across the bed. It is common practice to use particle shape factors in fixed bed state to match the predicted drag coefficient with experimentally obtained drag coefficients. In fluidised state, however, where the same particles are considered, this particle shape factor is usually simply omitted from the model without providing appropriate reasoning. In this research, it is shown that a shape factor is not a constant particle property but is dependent on the fluid properties as well. This dynamic shape factor for irregularly shaped grains increases from approximately 0.6 to 1.0 in fluidised state.

We found that unstable packed beds in moderate up-flow conditions are pseudo-fixed and in a setting state. This results in a decreasing bed voidage and simultaneously in a decreasing drag coefficient, which seems quite contradictory. This can be explained by the collapse of local channels in the bed, leading to a more uniform flow distribution through the bed and improving the available surface for flow-through. Our experimental measurements show that the drag coefficient decreases considerably in the laminar and transition regions. This is most likely caused by particle orientation, realignment and rearrangement in particles’ packing position.

A thorough hydraulic analysis shows that up-flow filtration in rapid sand filters under backwash conditions causes the particle bed to collapse almost imperceptibly. In addition, an improved expression of the drag coefficient demonstrated that the Carman–Kozeny model constant, however often assumed to be constant, is in fact not constant for increasing flow rates. Furthermore, we propose a new pseudo-3D image analysis for particles with an irregular shape. In this way, we can explain the successful method using optimisation of the extended terminal sub-fluidisation wash (ETSW) filter backwashing procedure, in which turbidity and peaks in the number of particles are reduced with a positive effect on water quality.@en

Granular activated carbon (GAC) filtration is an important unit operation in drinking water treatment. GAC filtration is widely used for its filtration and adsorption capabilities as a barrier for undesired organic macro- and micro-pollutants. GAC filtration consists of two successive phases: adsorption and filtration, capturing the impurities from the water in conjunction with a backwash procedure in which the suspended particles are flushed out of the system. Available literature predominantly focusses on adsorption. A less frequently discussed but nevertheless equally crucial aspect of this operation is the backwash procedure of GAC beds. To prevent accumulation of suspended particles and to avoid additional operation costs, optimal backwashing is required. Another factor is sustainability: water utilities are showing increasing interest in exploring new sustainable GAC media. As these have different bed expansion tendencies due to different GAC characteristics with varying geometries, operational developments are needed for prediction models to estimate the expansion degree during backwashing. The prediction of the bed expansion of GAC is complex as the particles are non-spherical, porous and polydisperse. Through a combination of advanced particle laboratory and fluidisation experiments, we demonstrate a new approach which leads to an improved expansion prediction model for the backwashing of GAC filters.@en

Liquid-solid fluidisation is frequently encountered in drinking water treatment processes, often to obtain a large liquid-solid interfacial surface area. A large surface area is crucial for optimal seeded crystallisation in full-scale softening reactors. Due to crystallisation, particles grow and migrate to a lower zone in the reactor which leads to a stratified bed. Larger particles adversely affect the surface area. To maintain optimal process conditions in the fluidised beds, information is needed about the distribution of particle size, local voidage and available surface area, over the reactor height.

In this work, a sensor is developed to obtain the hydraulic state gradient, based on Archimedes’ principle. A cylindrical heavy object is submerged in the fluidised bed and lowered gradually while its weight is measured at various heights using a sensitive force measuring device.

Based on accurate fluidisation experiments with calcite grains, the voidage is determined and a straightforward empirical model is developed to estimate the particle size as a function of superficial fluid velocity, kinematic viscosity, suspension density, voidage and particle density. The surface area and specific space velocity can be estimated accordingly, which represent key performance indicators regarding the hydraulic state of the fluidised bed reactor. The prediction error for voidage is 5 ± 2 % and for particle size 9 ± 4 %.

The newly developed soft sensor is a more time-effective method for obtaining the hydraulic state in full-scale liquid-solid fluidised bed reactors.@en

A novel effective drag relation for liquid-solid fluidisation is proposed, suitable for application in full-scale installations. This is achieved by presenting new insights related to the influence of the temporal-spatial heterogeneity on the effective hydrodynamic drag for large fluidised systems. While heterogeneous flow behaviour can be predicted increasingly accurately in CFD simulations that explicitly model the heterogeneous solids distribution, for the operation of many large-scale applications it is infeasible to perform such computationally intensive simulations. Therefore, there is a clear need for full-scale drag relations that effectively take into account the heterogeneous behaviour and irregular spatial particle distributions. Our new drag relation is based on a large set of experiments, which shows that the degree of overall expansion is not only dependent on the ratio of laminar-turbulent flow, but also on the amount of homogenous versus heterogeneous flow, which is not included in current full-scale drag relations. To include the effect of heterogeneity, the standard drag relation, based on the Reynolds number, is extended with a specific type of Froude number. Because fully turbulent flow regimes are rare in applications of liquid-solid fluidisation, our focus is not on the turbulent flow regime but instead on laminar and transitional flow regimes. In these regimes, three types of models are investigated. The first type is based on a theoretical similarity with terminal settling, the second is based on the semi-empirical Carman-Kozeny model, and the third is based on empirical equations using symbolic regression techniques. For all three types of models, coefficients are calibrated on experimental data with monodisperse and almost spherical glass beads. The models are validated with a series of calcium carbonate grains applied in drinking water treatment processes as well as data obtained from the literature. Using these models, we show that the voidage prediction average relative error decreases from approximately 5% (according to the best literature equations which use Reynolds number only) to 1-2% (using both Reynolds and Froude number). This implies that our new models are more suitable for operational control in full-scale fluidised bed applications, such as pellet softening in drinking water treatment processes.@en

In full-scale drinking water production plants in the Netherlands, central softening is widely used for reasons related to public health, client comfort, and economic and environmental benefits. Almost 500 million cubic meters of water is softened annually through seeded crystallisation in fluidised bed reactors. The societal call for a circular economy has put pressure on this treatment process to become more sustainable. By optimising relevant process conditions, the consumption of chemicals can be reduced, and raw materials reused. Optimal process conditions are feasible if the specific crystallisation surface area in the fluidised bed is large enough to support the performance of the seeded crystallisation process. To determine the specific surface area, crucial variables including voidage and particle size must be known. Numerous models can be found in the literature to estimate the voidage in liquid-solid fluidisation processes. Many of these models are based on semi-empirical porous-media-based drag relations like Ergun or semi-empirical terminal-settling based models such as Richardson-Zaki and fitted for monodisperse, almost perfectly round particles. In this study, we present new voidage prediction models based on accurate data obtained from elaborate pilot plant experiments and non-linear symbolic regression methods. The models were compared with the most popular voidage prediction models using different statistical methods. An explicit model for voidage estimation based on the dimensionless Reynolds and Froude numbers is presented here that can be used for a wide range of particle sizes, fluid velocities and temperatures and that can therefore be directly used in water treatment processes such as drinking water pellet softening. The advantage of this model is that there is no need for applying numerical solutions; therefore, it can be explicitly implemented. The prediction errors for classical models from the literature lie between 2.7 % and 11.4 %. With our new model, the voidage prediction error is reduced to 1.9 %.@en

One of the most popular and frequently used models for describing homogeneous liquid-solid fluidised suspensions is the model developed by Richardson & Zaki in 1954. The superficial fluid velocity and terminal settling velocity together with an index makes it possible to determine the fluid porosity in a straightforward way. The reference point for the Richardson-Zaki model is the terminal settling velocity at maximum porosity conditions. To be able to predict porosity in the proximity of minimum fluidisation conditions, either the minimum fluidisation velocity must be known or the Richardson-Zaki index must be very accurate. To maintain optimal process and control conditions in multiphase drinking water treatment processes, the porosity is kept relatively low. Unfortunately, the Richardson-Zaki index models tends to overestimate the minimum fluidisation velocity and therefore also results in less accurate predictions with respect to porosity values. We extended the Richardson-Zaki model with proven hydraulics-based models. The minimum fluidisation velocity is acquired using the model proposed by Kozeny (1927), Ergun (1952) and Carman (1937). The terminal settling velocity is obtained through the model developed by Brown & Lawler (2003), which is an improved version of the well-known model developed by Schiller & Naumann (1933). The proposed models are compared with data from expansion experiments with calcium carbonate grains, crushed calcite and garnet grains applied in drinking water softening using the fluidised bed process. With respect to porosity, prediction accuracy is improved, with the average relative error decreasing from 15% to 3% when the classic Richardson-Zaki model is extended with these hydraulics-based models. With respect to minimum fluidisation velocity, the average relative error decreases from 100% to 12%. In addition, simplified analytical equations are given for a straightforward estimation of the index n.

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For an accurate prediction of the porosity of a liquid-solid homogenous fluidized bed, various empirical prediction models have been developed. Symbolic regression machine learning techniques are suitable for analyzing experimental fluidization data to produce empirical expressions for porosity as a function not only of fluid velocity and viscosity but also of particle size and shape. On the basis of this porosity, it becomes possible to calculate the specific surface area for reactions for seeded crystallization in a fluidized bed.@en

The literature provides a comprehensive collection of equations to estimate the terminal settling velocity of single solid particles in a liquid system. The settling velocity for perfectly round spheres can accurately be calculated. In the contrary for natural imperfect particles the experimentally measured settling velocity deviates considerably compared the calculated value. In drinking water treatment processes natural particles are frequently applied in up flow fluidisation processes and in addition sedimentation processes are applied to clarify water and to concentrate solids.@en

Natural particles are frequently applied in drinking water treatment in up-flow fluidisation processes. Additionally, sedimentation processes are applied to clarify water and to concentrate solids. To estimate the terminal settling velocity of single solid particles in a liquid system, a comprehensive collection of equations is available. For perfectly round spheres, settling velocity can be calculated accurately. For naturally imperfect particles, however, experimentally measured settling velocity shows considerable deviation compared to calculated values. This article discusses a number of experiments demonstrating this deviation and the applicability of commonly used drag-coefficient equation by Brown-Lawler.@en