Drinking water softening has primarily prioritized public health, environmental benefits, social costs and enhanced client comfort. Annually, over 35 billion cubic meters of water is softened worldwide, often utilizing three main techniques: nanofiltration, ion exchange and seeded crystallization by pellet softening. However, recent modifications in pellet softening, including changes in seeding materials and acid conditioning used post-softening, have not fully achieved desired flexibility and optimization. This highlights the need of an integral approach, as drinking water softening is just one step in the drinking water treatment chain, which includes ozonation, softening, biological active carbon filtration (BACF) and sand filtration among others. In addition, pellet softening is often practiced based on operator knowledge, lacking practical key reactor performance indicators (KPIs) for efficient control. For that reason, we propose a newly and improved integral mechanistic model designed to accurately predict (1) calcite removal rates in drinking water through seeded crystallization in pellet softening reactors, (2) the saturation of the filter bed in the subsequent treatment step, (3) values for the KPIs steering the softening efficiency. Our new mechanistic model integrates insights from hydrodynamics, thermodynamics, mass transfer kinetics, nucleation and reactor engineering, focussing on critical variables such as temperature, linear velocity, pellet particle size and saturation index with respect to calcite. Our model was validated with data from the Waternet Weesperkarspel drinking water treatment plant in Amsterdam, The Netherlands, but implies universal applicability for addressing industrial challenges beyond drinking water softening. The implementation of our model proposes five effective KPIs to optimize the softening process, chemical usage, and reactor design. The advantage of this model is that it eliminates the application of numerical methods and fills a significant gap in the field by providing predictions of the carry-over (i.e., the produced CaCO₃ fines leaving the fluidized bed) from water softening practices. With our model, the calcium removal rate is predicted with an average standard deviation (SD) of 40 % and the consequential clogging prediction of the BACF bed with an average SD of 130 %. Ultimately, our model provides crucial insights for operational management and decision-making in drinking water treatment plants, steering towards a more circular and environmentally sustainable process.

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One of the main targets of Waternet, the water-cycle company of Amsterdam, is to increase the sustainability of water treatment. To achieve this goal, it is necessary to improve the efficiency and to decrease the use of chemicals during pellet softening process. However, the model of Van Schagen et al. (2008 b,c) that is currently used to optimize the pellet softening process is not predicting accurately enough the pH and calcium profile over the height of the pellet softening reactor. The calcium carbonate crystallization is calculated, in this model, using a linear relationship between the rate of crystallization and supersaturation with an additional diffusion parameter to take into account the flow conditions inside the reactor.
To determine more accurately the rate of calcium carbonate during the pellet softening process, two types of experiments were conducted during this research: STR batch and PFR fluidized bed experiments. Firstly, the experimental results were compared with the predictions of two linear models: the model of Wiechers et al. (1975) from literature and the one-rate-constant model developed in this research. Based on the results, it was concluded that it is not possible to improve the prediction of calcium carbonate crystallization kinetics if a linear model with one-rate-constant is used as proposed by Van Schagen. When the rate of crystallization is plotted against supersaturation a bending of the curve is observed at low supersaturation due to a sharp decrease in the rate of crystallization. Other researchers, such as Dreybrodt et al. (1997) has also observed that the rate of calcium carbonate crystallization is not linearly related to supersaturation when water or seeding material with inhibiting compounds is used. To describe this bending of the curve, two models were considered: the exponential model of Lasaga (1998) and the two-rate-constants model that consists of two linear equations. In this research, the two-rate-constants model was chosen instead of the exponential Lasaga model because it is easier to fit to the experimental results and gives a better overview of the dependence of the rate of crystallization from supersaturation. The two-rate-constants model significantly improves the prediction of calcium carbonate crystallization in a pellet softening fluidized bed reactor. The average relative error of this model, for the prediction of the calcium profile in a full-scale reactor, is only 2-5% while the average relative error of the one-rate-constant model is approximately 15-30%. Therefore, the two-rate-constants model predicts better the calcium carbonate crystallization and can be used to describe much more accurately the pellet softening process compared to the models found in literature.
Based on the results of the research, it can be concluded that the performance of a pellet softening fluidized bed reactor cannot be significantly improved by increasing the height of the reactor. On the other hand, it is possible that performance is enhanced by removing inhibitors such as organic carbon from the water. Nevertheless, further research is necessary to determine the effect of inhibitors, such as organic carbon, on water softening. Also, in order to determine more accurately the model parameters, the experimental set up should be adjusted in order to represent better the conditions inside a pellet softening fluidized bed reactor. In particular, increasing the height of the reactor and mixing the caustic soda at the bottom of the column is necessary.

In this study a simplified model of the rivers Tigris and Euphrates during the Uruk period (3500-3100 B.C.) was created in order to define the hydraulic behaviour of the rivers in ancient Mesopotamia. The model was used to test the effect of a number of factors on the water-level of the river system. Especially, the role of the development of settlements in the Mesopotamian valley and the way the irrigation system was operated was examined. Also, physical processes such as the avulsion of the secondary branches of the two rivers was taken into account.
According to the results of the simulation Euphrates is much more sensitive to an increase in water demand than Tigris. However, when two branches in the upper part of the river are closed the problem of low water level in Euphrates is solved. Furthermore, the development of settlements and irrigation nodes in ancient Mesopotamia, at the lower part of the valley, has as a result a lack of sufficient water even at the early stages of societal development. On the other hand, Euphrates does not seem to be so sensitive to an increase in water demand in the upper part of the valley.