Mathematical modelling and analysis of dynamic behaviour for seeded batch potash alum crystallization process

Crystal size distribution (CSD) is an essential criterion for determining the production of high-quality crystals since it influences the efficiency of the crystallization process. Producing specified CSD in the crystallization process represents a main challenge as it depends on temperature control, which indirectly regulates the solution’s concentration and affects the crystal’s evolution. Different temperature profiles may influence the distribution of crystal products, and a suitable optimization algorithm is required to produce an optimum temperature trajectory that produces the desired CSD. Thus, this study aims to maximize the CSD of the grown seed crystals while minimizing the nucleus-grown crystals by employing the best optimization algorithm for the potash alum crystallization process. The crystallization process was developed and simulated in Matlab software using a potash alum in the water system. Four optimization algorithms were proposed with different objective functions, such as maximizing mean crystal size (I), minimizing coefficient of variation (II), minimizing nucleus-grown crystals (III), and maximizing CSD (IV). Based on the simulation results, optimization IV, which maximizes CSD, performs best with a large mean crystal size of 490 µm. Furthermore, the number of fine crystals was among the lowest at a volume distribution of 0.00071 m3/m compared to the linear profile at 0.00191 m3/m. Optimization IV employs a dissolution strategy, which manipulates two quality specifications in one algorithm (size of crystals and number of fines), which is considered the best optimal cooling profile for seeded batch crystallization by maximizing CSD and minimizing the generation of nucleus-grown crystals.

The ideal requirements for highly efficient evaporative crystallization in batch or semi-batch type operations are a high crystal growth rate, a high density of suspended crystalline particles in a slurry and a sharp crystal size distribution. On the other hand, the initial seed crystal conditions are also very important because they affect the crystal size and distribution during the entire period of operation. In the present study, the influence of seeding conditions (average diameter and mass) and heating rate on both the crystal growth rate of sodium chloride (NaCl) and the crystal size distribution were investigated in an optimal design draft-tube stirred vessel containing high suspension density slurries of sodium chloride particles up to 30 vol%. In order to determine the optimal seeding conditions to simultaneously achieve a high crystal growth rate and narrow size distribution, an optimal operation index Iop has been defined based on a comparison of the experimental data and a model calculation with the assumption of ideal crystal growth. The results show that Iop correlate with the particle number of seed crystals and have a maximum value. With Iop correlation, the appropriate seeding conditions for achieving a high crystal growth rate and sharp size distribution have been determined.

The temperature profile is a critical parameter in industrial batch cooling crystallization processes which directly impacts the crystal size distribution (CSD) and subsequent down streaming operations. Conventional cooling approaches, including natural and linear cooling, often fail to achieve optimal results. Linear cooling may trigger excessive primary nucleation by crossing metastable limits, while natural cooling leads to inconsistent nucleation and growth, resulting in smaller and non-uniform crystals. Hence, optimized cooling profiles are used to achieve the desired target CSD. This study employs reinforcement learning (RL) to optimize the temperature profile for batch cooling crystallization for two systems: paracetamol in water and potassium dihydrogen phosphate (KDP) in aqueous solution. RL dynamically adjusts the cooling trajectory by iteratively interacting with a process model, aiming to maximize mean crystal size while minimizing the coefficient of variation. A comprehensive mathematical model incorporating population balance equations, mass balance, and energy balance is developed to simulate the crystallization process. For the paracetamol system, the RL-optimization strategy resulted in a 107.5 % increase in mean crystal size compared to the natural cooling profile. In the case of the KDP system, a 12 % increase in mean crystal length is achieved relative to a linear cooling profile, along with a significant reduction in the coefficient of variation, indicating improved crystal size uniformity. The optimization results obtained using RL are also compared with that from a genetic algorithm for both cases, and RL demonstrated superior performance. This work underscores the potential of RL in advancing broader applications in chemical process optimization.

Study of batch maltitol (4- O-α- d-glucopyranosyl- d-glucitol) crystallization by cooling and water evaporation

The textures of plagioclase crystals within large olivine oikocrysts preserve the sequence of the formation of anorthosite. Such textures have been quantified in a troctolite‐anorthosite from the Proterozoic Lac-St-Jean anorthosite complex, Canada. Crystal size distributions (CSDs) indicate that initially plagioclase nucleated and grew in an INTRODUCTION environment of linearly increasing undercooling, producing a straight- The textures of plutonic rocks are one of their most line CSD. During this phase, latent heat of crystallization was striking aspects, but they have been little used to elucidate largely removed by circulation of magma through the porous crystal petrological processes as compared with chemical and mush. By about 25% solidification the crystallinity was such that isotopic methods. Where used, discussion has tended it reduced, but did not eliminate, the circulation of magma, resulting to be qualitative, in contrast to quantitative chemical in the retention of more latent heat within the crystal pile. The methods, hence reducing the weight of this approach. temperature rose until it was buVered by the solution of plagioclase Indeed, it is not possible to verify physical models if close to the liquidus temperature of plagioclase. Nucleation of quantitative data are not available. Quantitative textural plagioclase was inhibited and conditions were suitable for textural studies of volcanic rocks are becoming more common coarsening of both plagioclase and olivine to occur (Ostwald and these methods are applied here to the development ripening). In this process small crystals were resorbed, whereas of anorthosite. larger crystals grew from both material recycled by the resorption of Quantitative textural studies of plutonic rocks really crystals smaller than the critical size and new material brought in started with Jackson’s studies of the Stillwater complex by the circulating magmatic fluid. The results of this process resemble in 1961 ( Jackson, 1961). He measured crystal size disthose of high-temperature metasomatism but there is no necessity for tributions, but did not record the area that he measured, a magmatic fluid with a diVerent composition. The shapes of the hence his data cannot be completely compared with plagioclase CSDs fit better the communicating neighbours equation of modern data. His pioneering work was not really followed textural coarsening, rather than the classic Lifshitz‐Slyozov‐Wagner up, and since then most studies have looked only at equation, as do other examples drawn from the literature. If olivine ‘average’ or ‘typical’ crystal sizes. These data are much started to nucleate at a higher temperature than plagioclase, then less revealing of petrological processes than crystal size during the textural coarsening phase olivine would have been more distributions, partly because they only give one parameter undercooled than plagioclase, and would have had a higher maximum for each sample. The few existing studies of crystal size growth rate. In these conditions olivine would coarsen more rapidly distributions in plutonic rocks have not treated than plagioclase and engulf it. Hence the order of crystallization plagioclase. determined from the textures would be the reverse of the order of One problem especially pertinent to quantitative texfirst nucleation of the two phases, from equilibrium phase diagrams. tural studies is the way that many igneous petrologists Maintenance of the temperature near the plagioclase liquidus may look at rocks: they commonly try to reconstruct how a rock has formed by examining the final, fully crystallized also inhibit the nucleation and growth of other phases.

New perspectives for the on-line monitoring of pharmaceutical crystallization processes using in situ infrared spectroscopy

Characterization of tectono-metamorphic events using crystal size distribution (CSD) diagrams. A case study from the Acebuches metabasites (SW Spain)

Influence of calcium ions on the crystallization of sodium bicarbonate

Crystalline products with a narrow and uniform distribution of crystals by size (CSD), characterized by a desired average size, are necessary in many practices. Therefore, extensive, but mostly experimental, research is devoted to the problem of obtaining such CSDs. Alternatively, this manuscript presents a theoretical approach for calculating CSD resulting from crystallization in unstirred solutions. First, classical equations for the rates of diffusion-controlled and kinetically controlled growth of crystals are used to discuss the size-dependent growth of the nucleated crystals and the initial CSD (which arises from the non-simultaneous nucleation of crystals). Then, applying the law of conservation of matter, it is proved that the CSD continues to expand during the growth stage. Furthermore, it is substantiated that, due to their uneven spatial distribution, crystals of the same size can grow at different rates. This depends on whether the crystals are outside the diffusion fields of other crystals or are clustered together in “nests”. Moreover, by calculating the growth rates of crystals in “nests”, an explanation is given for the observation that closely spaced crystals are smaller in size than the separately growing crystals. Finally, the CSD established during the Ostwald ripening is discussed quantitatively, step-by-step.

In this paper, a data-driven modeling and optimization method is proposed for batch operation of the cooling crystallization process based on the design of experiments (DoE) in the operation ranges of cooling rate and initial solution supersaturation. The proposed model reflecting a mapping relationship between the manipulated variables of operating conditions and the product crystal size distribution (CSD) is established by two classes of basis functions, one is the wavelet basis function for reshaping the CSD and the other is the polynomial basis function for weighting the chosen wavelet basis functions to reflect the nonlinear relationship between the manipulated variables and the size distribution of product crystals. Meanwhile, a fractional factorial design of DoE is presented to reduce the number of experiments for the above modeling, in comparison with the classical full factorial design. The optimal operating conditions are designed by introducing a comprehensive objective function that combines the information entropy of the product CSD and the ratio of desired product yield with respect to the target crystal size. A particle swarm optimization algorithm with guaranteed convergence is given to solve the nonconvex optimization problem based on the established CSD prediction model. Simulation studies and experiments on the batch seeded β-form L-glutamic acid (LGA) crystallization process are conducted to demonstrate the effectiveness and advantage of the proposed method.

Sodium bicarbonate is a substance which is produced in the middle stages of the soda ash production process. In this precipitation process, carbon dioxide gas is continuously injected into the bubble column reactor which contains carbonate and bicarbonate solutions. To elucidate the key parameters affecting precipitation kinetics, an experimental study was conducted to understand nucleation and growth in conditions of industrial reactor. The composition of the solution is followed during the crystallization process by titration. Magma density is also monitored and crystal size distribution (CSD) is obtained by sieving. The method of moments was used to determine nucleation and growth rates of crystals. The nucleation and growth rate correlations for sodium bicarbonate precipitation in industrial scale were correlated by empirical power laws as B = 26.685MT0.42Δw1.31 and G = 1.381×10–4Δw1.53. The nucleation and growth rate correlations obtained in this study can be used to simulate the crystallization of sodium bicarbonate plants. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In this work, the effect of crystallization parameters, i.e., supersaturation, seeding, and temperature, on the polymorphism and crystal size of a nonsteroidal anti-inflammatory drug, indomethacin (IMC), was investigated. First, several crystallization solvents (ethanol, methanol, ethyl acetate, acetone, acetonitrile, and dichloromethane) were screened through the measurement of IMC solubility at different temperatures. This was followed by the investigation of IMC nucleation through measurement of induction times in selected solvents at two supersaturations. Finally, seeded cooling crystallization of IMC in ethanol was performed with different process parameters to investigate the influence on the polymorphism and crystal size distribution. Remarkably long induction times were observed for IMC in ethanol and ethyl acetate solutions, while a shorter induction time was observed in acetone. Cooling crystallization of IMC from ethanol confirmed that supersaturation, operating temperature, and seeding do affect the polymorphism as well as crystal size distribution of IMC. Fine needle-shaped crystals of metastable α-IMC were obtained at 5 °C with high supersaturation even in the presence of γ-IMC seeds, while rhombic platelike crystals of thermodynamically stable γ-IMC were obtained in the remaining experiments. The seed loading only marginally influenced the crystal growth rate and median particle diameter. Particle size analysis of the crystals obtained showed a bimodal distribution in all experiments, and a larger median particle diameter was observed at 15 °C with high supersaturation.

Integrated membrane emulsification and solution cooling crystallization to obtain a narrow and predictable crystal size distribution

Owing to the rise in population growth and demands for the current living standards, mineral and water resources are depleting at the alarming rate. Also, the mineral–water-energy nexus raises a concern of high-quality water and mineral production at low costs. Membrane distillation crystallization (MDC) has emerged as a promising technology capable of simultaneous water and mineral recovery from wastewater or industrially discharged brines. Herein, the MDC was evaluated for recovery of water and precious minerals from acid mine drainage discharged from the mine tailings. High quality fresh water with 99.9 % salt rejection and permeate flux of 2.32 kg·m−2·h−1 was recovered at 52 °C feed inlet temperature. Also, feed temperatures were varied to determine their impact on mineral crystal morphology, size distribution and growth rate. Predominantly recovered minerals were gypsum, manganese fluoride, potassium iron oxide, nickel oxide hydroxide, sodium sulphate and dihydrogen sulphate. Orthorhombic crystal structure of mineral salts was attained at low feed inlet temperature (34 °C). Higher feed inlet temperatures (i.e. 41° C and 52° C) promoted co-crystallization giving rise to orthorhombic polymorph structures. Narrow crystal size distribution was reported at low water recovery factor (initial growth) for all temperatures. However, the particle size distribution broadened as the recovery factor increased. The crystal sizes ranged from 3.91 ± 0.97 µm to 14.04 ± 3.31. Remarkably, crystal growth rate increased with an increase in feed inlet temperature. These findings open a research direction towards economic recovery of mineral and water to supplement the existing shortages. With a myriad of crystals obtained from the MDC process, an additional operational unit is required to selectively recover high economic minerals.

Introducing uncertainty analysis of nucleation and crystal growth models in Process Analytical Technology (PAT) system design of crystallization processes