Steady-state Crystal Size Distribution Research Articles

Calcium sulphate crystallisation in the presence of mesoporous silica particles: Experiments and population balance modelling

A population balance model is used to investigate the effect of mesoporous silica particles on calcium sulphate crystallisation in a stirred batch crystalliser. The model accounts for nucleation, growth, agglomeration, breakage, and particle-assisted nucleation, and the model equations are solved numerically using the method of classes over a logarithmic, non-uniform mesh. The crystallisation process is characterized experimentally using electrical conductivity to track the ion concentration and laser diffraction to measure the steady-state crystal size distribution obtained at the end of the experiments. The experiments are carried out over a range of temperatures, initial supersaturations, particle pore diameters, and particle loadings. The model is first fitted to experimental data obtained in the absence of particles to determine kinetic parameters of the nucleation, growth, agglomeration, and breakage for pure calcium sulphate crystallisation. Varying pore diameter did not influence the catalytic effect of the particles, however, particle loading was found to significantly decrease the nucleation induction time. The model was extended to account for the presence of particles by fitting two additional mechanisms. The first proposed a particle-assisted nucleation where nuclei are produced via heterogeneous crystallisation, then detach by particle-particle collision that is second-order with respect to particle loading. The second proposed that the crystal breakage frequency increases linearly with particle loading. Good agreement with the experimental data is demonstrated over the range of conditions examined.

Frazil-ice growth rate and dynamics in mixed layers and sub-ice-shelf plumes

Abstract. The growth of frazil or granular ice is an important mode of ice formation in the cryosphere. Recent advances have improved our understanding of the microphysical processes that control the rate of ice-crystal growth when water is cooled beneath its freezing temperature. These advances suggest that crystals grow much faster than previously thought. In this paper, we consider models of a population of ice crystals with different sizes to provide insight into the treatment of frazil ice in large-scale models. We consider the role of crystal growth alongside the other physical processes that determine the dynamics of frazil ice. We apply our model to a simple mixed layer (such as at the surface of the ocean) and to a buoyant plume under a floating ice shelf. We provide numerical calculations and scaling arguments to predict the occurrence of frazil-ice explosions, which we show are controlled by crystal growth, nucleation, and gravitational removal. Faster crystal growth, higher secondary nucleation, and slower gravitational removal make frazil-ice explosions more likely. We identify steady-state crystal size distributions, which are largely insensitive to crystal growth rate but are affected by the relative importance of secondary nucleation to gravitational removal. Finally, we show that the fate of plumes underneath ice shelves is dramatically affected by frazil-ice dynamics. Differences in the parameterization of crystal growth and nucleation give rise to radically different predictions of basal accretion and plume dynamics, and can even impact whether a plume reaches the end of the ice shelf or intrudes at depth.

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.

Use of focused beam reflectance measurement (FBRM) and process video imaging (PVI) in a modified mixed suspension mixed product removal (MSMPR) cooling crystallizer

The FBRM instrument is a ‘powerful’ tool developed by Lasentec as an ‘in situ’ particle monitoring technique for in-line real-time measurement of particle size. This technique was successfully used to monitor steady-state operation in a modified MSMPR crystallizer. The FBRM technique was also used to estimate crystallization kinetics. The FBRM particle size measurements were complimented by an in-line process video imaging (PVI) system developed ‘ in-house’ (Microscopy and Microanalysis, 6 (Suppl. 2) (2000) 996–997), to visualize habit and crystal behaviour within an MSMPR crystallizer. A comparison of the steady-state crystal size distributions measured by low angle light scattering (LALLS) and FBRM was made, showing poor sensitivity of the FBRM technique to particles of less than 1 μm hence the technique was not suitable for the measurement of crystallization kinetics for this organic system.

Simulation and analysis of industrial crystallization processes through multidimensional population balance equations. Part 1: a resolution algorithm based on the method of classes

In order to obtain constant solid properties with particles exhibiting a low order of symmetry, it is necessary to monitor and to control several distributed parameters characterising the crystal shape and size. A bi-dimensional population balance model was developed to simulate the time variations of two characteristic sizes of crystals. The nonlinear population balance equations were solved numerically over the bi-dimensional size domain using the so-called method of classes. An effort was made to improve usual simulation studies through the introduction of physical knowledge in the kinetic laws involved during nucleation and growth phenomena of complex organic products. The performances of the simulation algorithm were successfully assessed through the reproduction of two well-known theoretical and experimental features of ideal continuous crystallization processes: the computation of size-independent growth rates from the plot of the steady-state crystal size distribution and the possibility for MSMPR crystallizers to exhibit low-frequency oscillatory behaviours in the case of insufficient secondary nucleation.

CFD Study of Local Crystal Growth Rate in a Continuous Suspension Crystallizer.

The objective of the present work is to determine the local crystal growth rate in an imperfectly mixed suspension crystallizer. Experiments were carried out in pure potassium sulphate-water solution. The steady-state Crystal Size Distributions (CSDs) were measured at four different heights of the crystallizer using an in-situ particle image analyzer. Experimentally obtained CSD was used to determine the overall growth rate. Then, Computational Fluid Dynamics (CFD) was used to calculate the Particle Size Distributions (PSDs) of the same kind of suspension which was used in the experiments. The computational study was focused to simulate the mass transfer coefficient between particles and solution, and supersaturation. These quantities were determined from hydrodynamics of suspension in a crystallizer. The multi-fluid model was applied to describe suspension flow in the crystallizer. One continuous phase (water) and six dispersed phases, which were taken to be particles in six size classes, were used to approximate PSD. The crystal growth rate model was based on the diffusion process and was implemented into a CFD environment. The local growth rates were calculated from the slip velocity or from energy dissipation rate at the local temperature and supersaturation.

Solution of population balance equations with a new combined Lax-Wendroff/Crank-Nicholson method

A population balance equation was solved by combining the Lax-Wendroff and Crank-Nicholson methods. The population balance was used in the simulation of an evaporative cooling crystallizer with fines dissolution. The stability of the method, along with that of the Lax-Wendroff and Crank-Nicholson methods alone, was compared under different operating and initial conditions. When tested under conditions of uninterrupted growth to steady state, it was found that the three methods produced virtually identical final crystal size distributions (CSDs), though the Lax-Wendroff and Crank-Nicholson methods showed some oscillatory behaviour early on. Under dissolution conditions, only the combined Lax-Wendroff/Crank-Nicholson method proposed in this paper produced a population density that steadily reduced to zero. The Lax-Wendroff method became unstable, while the Crank-Nicholson method became extremely oscillatory. Under dynamic conditions, both the Lax-Wendroff and Crank-Nicholson methods showed some oscillations that did not affect the ultimate CSD. The combined Lax-Wendroff/Crank-Nicholson method showed no oscillations in any case, producing smooth dynamic and steady state CSDs.

Monte Carlo Simulation of Transient Crystal Size Distribution in a Continuous Crystallizer Using the ASL Model

A Monte Carlo simulation scheme is proposed for transient crystal size distribution in a continuous crystallizer to account for size-dependent growth rate. Crystal growth rates are described by Abegg, Stevens and Larson (ASL) model. The proposed model is used to predict the transient and steady state crystal size distribution from potassium carbonate crystallizer. The agreement between theory and available data confirms the validity of the model.

Monte Carlo Simulation of Transient CSD in a Continuous Crystallizer under Stochastic Dispersion Effects

AbstractMonte Carlo simulation is a very powerful tool for simulation of transient and steady state crystal size distribution (CSD) in a continuous crystallizer under stochastic dispersion effects. In the present work, transient CSD in a continuous crystallizer has been reported when shape factor and growth rate dispersions conform to normal distribution. For the steady state run, the algorithm reported by Sen Gupta and Dutta elsewhere has been used to validate the results obtained in the present work when the steady state is reached.

The calculation of growth and nucleation kinetics from MSMPR crystallizer data including growth rate dispersion

One of the most important applications of the population balance approach to crystallizer modeling is the recovery of crystal nucleation and growth rate data from steady state crystal size distributions as described by Randolph and Larson in 1971. Recent crystallization studies have revealed that growth rate dispersion, wherein crystals of the same size do not have the same growth rate, causes nonidealities that limit the usefulness of the Randolph and Larson approach. The present work presents and shows application of a procedure that allows calculation of unambiguous nucleation and growth rate kinetics in the presence of growth rate dispersion. It requires the appropriate choice of growth rate distribution for a fit to the experimental data. The method is applied to analysis of pilot scale crystallization experiments with the sucrose-water system and indicates that unimodal distributions may be incapable of describing such data.

TRANSIENT ANALYSIS OF CRYSTALLIZATION: EFFECT OF THE INITIAL SIZE DISTRIBUTION

The effect of seed size distribution on the transient and steady-state crystal size distribution in a continuous mixed suspension, mixed product removal crystallizer has been analyzed and simulated from a stochastic point of view. The resultant stochastic model provides an easier way of obtaining the dynamic information on the performance of the crystallizer under consideration than the conventional population balance techniques. Moreover, the moments of the number of crystals at each size can be evaluated. Simulation results indicate that the seed size distribution gives rise to a maximum in the crystal size distribution. The steady-state representation can be recovered from the resultant dynamic model; it reduces to the conventional deterministic population balance model under the condition of no random variation.

Numerical modelling of steady state continuous crystallization processes using piecewise cubic spline functions

An efficient technique for computing the steady state crystal size distributions from continuous crystallization units is developed using piecewise cub The technique is simple, explicit and offers considerable computation savings over other available techniques. Solution techniques are developed for cr dependent growth and size dispersion. The numerical results are very satisfactory when compared with other techniques or exact analytical results.

Stochastic analysis of the Transient Behavior of an msmpr Crystallizer

The transient crystal size distribution (CSD) in a continuous mixed suspension, mixed product removal crystallizer has been modeled through a stochastic approach. Effects of the seed size distribution and size-dependent growth rate on both the transient and steady-state CSD have been investigated. It has been found that the seed size distribution causes a maximum in the steady-state CSD and the size-dependent growth rate results in an upward curvature at the lower end in a semilogarithmic plot of steady-state CSD against crystal size. Under appropriate conditions, the mean CSD of the present stochastic model reduces to the results predicted by the corresponding deterministic population balance model.

Crystallization kinetics of calcium nitrate tetrahydrate from MSMPR crystallizer

The steady-state crystal size distribution (CSD) of calcium nitrate tetrahydrate in both Ca(NO 3) 2H 2O and Ca(NO 3) 2H 3PO 4HNO 3H 2O systems were measured in a continuous mixed suspension mixed product removal (MSMPR) crystallizer at 19°C. The secondary nucleation rate ( B°) and the growth rate ( G) were correlated by a generalized semiempirical power law model. Freon-113 was found to be satisfactory for washing wet crystals of Ca(NO 3) 2·4H 2O separated from mother liquor. The influence of some impurities usually present in phosphate rock on B° and G were examined. The results showed that impurity ions increased generally the nucleation rate and inhibited, to some degree, growth rate. The sequence of effectiveness is: Al +3 > Mg +2 > SiF 6 −2 > Fe +3.

Crystallization and agglomeration kinetics of nickel ammonium sulphate in an MSMPR crystallizer

Precipitation of nickel ammonium sulphate achieved by mixing solutions of nickel sulphate and ammonium sulphate in a 5 1 MSMPR crystallizer is used to investigate both crystallization and agglomeration kinetics from the steady-state crystal size distribution. An iterative non-linear parameter estimation procedure is used to deduce the kinetic rate parameters in the solution of the agglomeration model suggested by Liao and Hulburt from the data-set obtained by size analysis of the product crystals. The results obtained are correlated in terms of conventional power law kinetic expressions.

The effect of temperature on the precipitation of calcium oxalate

Nucleation and growth kinetics have been determined for the precipitation of calcium oxalate from aqueous solution using the steady state crystal size distribution in an MSMPR crystallizer. Experiments were performed over the temperature range 9–45°C. Above about 30°C the thermodynamically stable calcium oxalate monohydrate was formed whereas below about 25°C only the trihydrate was detected. Throughout the temperature range the steady state solution concentration was greater than the saturation values of both hydrates and the particular hydrate found in the product appears to be governed by kinetic rather than thermodynamic factors.

On the measurement of crystal size distributions in the micrometer size range

The steady state crystal size distribution in a continuous mixed suspension mixed product removal crystallizer has been measured during the precipitati of calcium oxalate trihydrate from aqueous solution. Two different instruments, both working over the size range 1.4–25 μm, were used, a Coulter Counter and an instrument based on the principle of forward scattering of laser light. The form of size distribution obtained with both instruments was very similar and lends confidence to such measurements at these sizes. The crystal number density for sizes less than about 5 μm was much higher than expected from the number density at larger sizes. This behaviour is similar to that found for systems of higher solubility but raises important questions concerning the reasons for such effects.

Measurement and scale‐up of secondary nucleation kinetics for the potash alum‐water system

AbstractSecondary nucleation kinetics for potash alum in draft‐tube baffled crystallizers have been determined from steady state crystal size distributions measured down to about 5 μm. Differences in nucleation rates were found for crystallizers of similar geometry but different capacity. Models relating the frequency of crystal/propeller collisions to crystallizer geometry and operating parameters were compared. In general, these successfully accounted for the observed differences in nucleation rates.

Some aspects of crystallization theory: Systems that violate McCabe's delta L law

AbstractA new empirical model has been proposed to described crystallization systems where the growth rate increases with increasing crystal size in violation of McCabe's ΔL law. Experimental data for Glauber's salt have been presented to illustrate the fit of the new model to an actual system. Equations for steady state crystal size distributions with the use of the proposed model have been derived. Wider use of population density plots for analysis of crystallization systems along with the coefficient of variation to describe a distribution numerically is recommended.