Evolution Of Crystal Size Distribution Research Articles

Computationally Efficient Algorithm for Solving Population Balances with Size-Dependent Growth, Nucleation, and Growth-Dissolution Cycles

A computationally efficient algorithm for solving population balance equations (PBEs) for describing the evolution of crystal size distribution (CSD) in seeded batch crystallization processes is developed. The algorithm is particularly suitable for solving problems with size-dependent growth and dissolution kinetics where temperature cycling is applied to modify the CSD. It is much faster than the quadrature method of moments (QMOM) which has been widely applied to solve this type of problem and has the added benefit that it provides the complete crystal size distribution rather than only the moments. These features make it especially well suited for solving problems in optimization and determination of feasible regions, where many batch simulations are required. The algorithm is applied to determine attainable regions for seed-grown product crystals (characterized by the mean and variance of the seed-grown product CSD) and analyze the trade-off between the objectives of minimizing batch time and nucleated crystal volume in a batch crystallization process with temperature cycling. Two main conclusions are drawn from the results. First, the attainable region expands with an increasing number of growth-dissolution cycles as expected, but the extent of the increase or decrease in the attainable variance depends on the sensitivity of the growth and dissolution rates to the crystal size. Second, the trade-off between the abovementioned objectives subject to constraints on the seed-grown crystal mean size and variance is significant and an outcome representing a good compromise can be achieved by specifying proper supersaturation and undersaturation set points.

Correction to: CrystalMom: a new model for the evolution of crystal size distributions in magmas with the quadrature-based method of moments

The order first name–family name of the authors (Colucci Simone, de’ Michieli Vitturi Mattia, Landi Patrizia) is published incorrectly.

CrystalMom: a new model for the evolution of crystal size distributions in magmas with the quadrature-based method of moments

Nucleation and growth of crystals, and the resulting crystal size distribution, play a fundamental role in controlling the physical properties of magmas and consequently the dynamics of the eruptions. In the past decades, laboratory experiments demonstrated that size and shape of crystals strongly control the physical properties of magma and lava. Additionally, natural and experimental samples are usually characterized in terms of their crystal size distribution to link it with physical processes that are not directly observable, such as cooling or decompression mechanisms. In this paper, we present CrystalMoM, a new predictive model, based on the quadrature-based method of moments, developed for studying the kinetic of crystallization in volcanic systems. The quadrature-based method of moments, well established in the field of chemical engineering, represents a mesoscale modelling approach that rigorously simulates the space–time evolution of a distribution of particles, by considering its moments. The method is applied here, for the first time, for studying the equilibrium/disequilibrium crystallization in magma, modelling the temporal evolution of the moments of a crystal size distribution. The model, verified against numerical and experimental data, represents a valuable tool to infer the cooling and decompression rates from the crystal size distribution observed in natural samples.

Quantifying the effect of fillers on the breakage behaviour of needle-shaped particles

Understanding the relationship between the applied stress and the behaviour of packings of breakable granular particles is required in many chemical engineering applications. A pharmaceutical tablet is an example where a composite packing (a packing containing a mixture of several kinds of particles) is compacted in the process of tablet formation. In this work, the unidirectional compaction and the breakage processes in composite packings formed from a mixture of breakable needle-shaped crystals and elastic spherical filler particles were studied using a Discrete Element Method (DEM) simulation. The evolution of crystal size distribution and contacts topology with the increasing stress was computed for different relative size, volume fraction and elasticity of filler particles inside the packing. We show that crystals breakability in a packing is directly related to the number of contacts among particles and that it can be significantly influenced by an appropriate choice of added filler particles. The results can be used for the development of mathematical models that describe the pharmaceutical tablet production and other processes involving composite packings of fragile particles.

Crystal Shape Evolution Using Polyhedral Population Balance

Research on crystal morphology covers shape evolution of single crystal and evolution of the size and shape distribution of populations of crystals. The population balance models have been used for the single or multiple dimensions for predicting the evolution of crystal size distributions (CSDs). More often than not these models do not consider the change in solute concentration due to disappearance/appearance of the crystal faces and therefore compromise rigorous mass balance. The present work focuses on the methodology for the morphological evolution of crystal shape using a polyhedral population balance model. This methodology is based on the morphodrome which illustrates the shapes of crystals at various operating conditions. gPROMS software (PSE, UK) along with Microsoft Excel is used to devise the framework for predicting the crystal shape evolution. The developed approach and models can be implemented for various crystallization systems and will be useful for simulating crystal shape evolutions.

Crystal growth rate dispersion modeling using morphological population balance

AbstractCrystal growth in solution is a surface‐controlled process. The variation of growth rates of different crystal faces is considered to be due to the molecular arrangement in the crystal unit cell as well as the crystal surface structures of different faces. As a result, for some crystals, the growth rate for a specific facet is not only a function of supersaturation, but also dependent on some other factors such as its size and the lattice spread angle. This phenomenon of growth rate dispersion (GRD) or fluctuation has been described in literature to have attributed to the formation of some interesting and sophisticated crystal structures observed in experimental studies. In this article, GRD is introduced to a recently proposed morphological population balance model to simulate the dynamic evolution of crystal size distribution in each face direction for the crystallization of potash alum, a chemical that has been reported to show GRD phenomenon and sophisticated crystal structures. The GRD is modeled as a function of the effective relative supersaturation, which is directly related to crystal size, lattice spread angle, relative supersaturation, and solution temperature. The predicted results clearly demonstrated the significant effect of GRD on the shape evolution of the crystals. © 2008 American Institute of Chemical Engineers AIChE J, 2008

Aging of Crude Terephthalic Acid Crystals at High Temperatures

By using a specially contrived experimental technique, crude terephthalic acid (TPA) crystals have been aged in their own saturated 90 mass % acetic acid + 10 mass % solution at 467, 477, 487, 497, and 507 K. The time evolution of crystal size distribution, as well as contents of the contaminant 4-carboxybenzaldehyde (4-CBA) in both solid and liquid phases, has been experimentally measured. Experimental results reveal the following: (1) The effect of stirring rate on the aging of crude TPA crystals is negligible under the conditions of this study. (2) The crystal size distribution is observed to broaden with time, accompanied by a decrease in number and an increase in averaged crystal size during aging. The aging rate increases with the increasing of temperature. (3) Under the conditions of this study, substantial purification occurs during aging. The rate of purification increases as the temperature increases. The Ostwald ripening mechanism is proposed to model the effect of aging on crystal size distri...

Determination of growth and breakage kinetics of l-threonine crystals

The crystal growth and breakage kinetics of l-threonine by evaporation crystallization were studied at the laboratory scale using a 1200 ml crystallizer with draft tube. Evaporation experiments were carried out at three different temperatures and three different stirrer speeds. Crystal breakage phenomena were demonstrated by image analysis and evolution of crystal size distribution. Breakage of binary uniform distribution and volume-independent growth kinetics were extracted by moment method. Moreover, the breakage and growth kinetics were proved to be appropriate for l-threonine/water evaporation process using numeric simulation. The growth kinetic parameters indicated that diffusion should be the rate-controlling step of crystal growth, and higher temperature higher growth rate. The higher exponent of agitation speed for breakage kinetics indicated that the removal of fragments stimulated by stirring should be the main breakage mechanism.

Evolution of crystal size distributions in a CMSMPR—the effect of temperature, flow rate, and mixing speed with time

A continuous mixed-suspension, mixed-product removal (CMSMPR) crystallizer with automatic computer control and data acquisition was constructed. Operating this CMSMPR at rpm’s greater than 100 rpm assures well-mixed conditions. The crystallization of NaCl, an impurity in a concentrated CaCl2 solution, was performed by cooling under well-mixed conditions. Atomic absorption spectrometry (AAS) and inductively coupled plasma-atomic emission spectrometry (ICP-AES) have been used to measure solution concentrations. Scanning electron microscopy (SEM) with energy dispersive spectroscopy (EDS), light microscopy and particle size analyzer with advanced laser light scattering have been used to characterize the crystals produced by the CMSMPR crystallizer. Octahedral NaCl crystals are observed. Steady state crystal size distribution (CSD) is observed at nine to ten times the mean residence time for the crystallizer. The CSD has been analyzed to look at the details of nucleation, growth, and aggregation rate as a function of crystallization temperature, rpm, and feed flow rate with the application of a Hounslow’s constant aggregation rate equation.

Precipitation of Piperine from Hydrotropic Solutions: Study of Crystal Nucleation and Growth Kinetics from Batch Experiments

ABSTRACTPrecipitation of piperine from aqueous hydrotropic solutions of sodium cumene sulfonate was investigated in a laboratory-scale agitated vessel by cooling aqueous sodium cumene sulfonate solution saturated with piperine from 353K to 303K. This temperature swing is a more economical approach than usual dilution by water for recovery of piperine from hydrotrope solutions. The time evolution of crystal size distribution was used to deduce simultaneously crystal nucleation and growth kinetics. The results indicate higher nucleation and slower crystal growth rates and thus formation of very fine crystals of piperine in the range of 1–400 μm.

On the Interpretation of Crystal Size Distributions in Magmatic Systems

holocrystalline under a wide spectrum of cooling regimes implies batch system. Instead, the CSDs of each system reflect a combination that cooling and crystallization can be uncoupled and considered of kinetic and dynamic influences on crystallization. Heterogeneous separately. This is tantamount to realizing that the Avrami number nucleation and annexation of small crystals by larger ones, enis large in most igneous systems. Crystallization automatically trainment of earlier grown and ripened crystals, rate of solidification adjusts through nucleation and growth to the cooling regime, and front advance, and protracted transit of a well-established mush all aspects of the ensuing crystal population reflect the relative roles column are some of the eVects revealed in the observed CSDs. There of nucleation and growth, which reflect the cooling regime. The may be an overall CSD evolution, reflecting the maturity of characteristic scales of crystal size, crystal number, and crys- the magmatic system, from simple straight nonkinked CSDs in tallization time are intimately tied to the characteristic rates of monogenetic systems to multiply kinked, piecewise continuous CSDs nucleation and growth, but it is the crystal size distributions (CSDs) in well-established systems such as Hawaii and Mount Etna. This that provide fundamental insight on the time variations of nucleation is not unlike the evolution of CSDs in some industrial systems. and growth and also on the dynamics of magmatic systems. Crystal Finally, the fact that comagmatic CSDs are not often captured size distributions for batch systems are calculated by employing evolving systematically through large changes in nucleation rates, the Johnson‐Mehl‐Avrami equation for crystallinity related to even in low crystallinity systems, may suggest that magma is always exponential variations in time of both nucleation and growth. The laced with high population densities of nuclei, supernuclei, and slope of the CSD is set by the diVerence a ‐ b, where a and b crystallites or clusters that together set the initial CSD at high are exponential constants describing, respectively, nucleation and characteristic population densities. Further evolution of the CSD growth. The batch CSD has constant slope and systematically occurs through sustained heterogeneous nucleation and rapid anmigrates to larger crystal size (L) with increasing crystallinity. The nealing at all crystallinities beginning at the liquidus itself and diminution in nucleation with loss of melt is reflected in the CSD operating under more or less steady (not exponentially increasing) at late times by a strong decrease in population density at small rates of nucleation. crystal sizes, which is rarely seen in igneous rocks themselves. Observed CSDs suggest that a ‐ b ~6‐10 and that b ~0. That is, growth rate is approximately constant and nucleation rate apparently increases exponentially with time. Correlations among CSD slope, intercept, and maximum crystal size for both batch and open systems suggest that certain diagnostic relations may be useful in interpreting the CSD of comagmatic sequences. These systematics are explored heuristically and through the detailed