Empirical Size-Dependent Growth Rate Models

1992 ◽  
Vol 57 (2) ◽  
pp. 309-325 ◽  
Author(s):  
Jerzy Mydlarz
Keyword(s):  
Growth Rate ◽  
Crystal Size ◽  
Size Range ◽  
Growth Models ◽  
Crystal Growth Rate ◽  
Size Distributions ◽  
Large Crystal ◽  
Size Dependent ◽  
Dependent Growth ◽  
Empirical Size

A comparison of well known size-dependent crystal growth rate models has been presented. The models have been verified for crystal size distributions which have been recently presented in the literature. It is shown that for large crystal size range both the Abegg-Stevenson-Larson (ASL) model and the Canning-Randolph (C-R) model can be reduced to the simplest Bransom model. Two another kinetics size-dependent growth rate models have been presented and tested for size distributions which were recently presented by Mydlarz and Jones. Application of the proposed size-dependent growth models gives much better estimation of growth rate than other size-dependent models tested as well as Sikdar and White-Bendig-Larson methods.

Stochastic analysis of the Transient Behavior of an msmpr Crystallizer

1987 ◽  
Vol 1 (4) ◽  
pp. 383-404 ◽  
Author(s):  
S. T. Chou ◽  
J. P. Hsu
Keyword(s):  
Growth Rate ◽  
Steady State ◽  
Size Distribution ◽  
Seed Size ◽  
Crystal Size ◽  
Transient Behavior ◽  
Balance Model ◽  
Mixed Product ◽  
Size Dependent ◽  
Dependent Growth

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.

scholarly journals First size-dependent growth rate measurements of 1 to 5 nm freshly formed atmospheric nuclei

2011 ◽  
Vol 11 (9) ◽  
pp. 25427-25471 ◽  
Author(s):  
C. Kuang ◽  
M. Chen ◽  
J. Zhao ◽  
J. Smith ◽  
P. H. McMurry ◽  
...  
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Survival Probability ◽  
Particle Diameter ◽  
Particle Growth ◽  
Size Distributions ◽  
Electrical Mobility ◽  
Limit Values ◽  
Size Dependent ◽  
Dependent Growth

Abstract. This study presents the first measurements of size-dependent particle diameter growth rates for freshly nucleated particles down to 1 nm geometric diameter. Data analysis methods were developed, de-coupling the size and time-dependence of particle growth rates by fitting the aerosol general dynamic equation to size distributions obtained at an instant in time. Size distributions of freshly nucleated particles were measured during two intensive measurement campaigns in different environments (Atlanta, GA and Boulder, CO) using a recently developed electrical mobility spectrometer with a diethylene glycol-based ultrafine condensation particle counter as the detector. Size and time-dependent growth rates were obtained directly from measured size distributions and were found to increase approximately linearly with size from ~1 to 3 nm geometric diameter, ranging, for example, from 5.6 ± 2.0 to 27 ± 5.3 nm h−1 in Boulder (13:00) and from 5.5 ± 0.82 to 7.6 ± 0.56 nm h−1 in Atlanta (13:00). The resulting growth rate enhancement Γ, defined as the ratio of the observed growth rate to the growth rate due to the condensation of sulfuric acid only, was found to increase approximately linearly with size from ~1 to 3 nm geometric diameter, having lower limit values that approached ~1 at 1.2 nm geometric diameter in Atlanta and ~3 at 0.8 nm geometric diameter in Boulder, and having upper limit values that reached 8.3 at 4.1 nm geometric diameter in Atlanta and 25 at 2.7 nm geometric diameter in Boulder. Survival probability calculations comparing constant and size-dependent growth indicate that neglecting the strong growth rate size dependence from 1 to 3 nm observed in this study could lead to a significant overestimation of CCN survival probability.

Crystal size distributions in continuous crystallizers when growth rate is size dependent

AIChE Journal ◽  
1968 ◽  
Vol 14 (1) ◽  
pp. 118-122 ◽  
Author(s):  
C. F. Abegg ◽  
J. D. Stevens ◽  
M. A. Larson
Keyword(s):  
Growth Rate ◽  
Crystal Size ◽  
Size Distributions ◽  
Size Dependent ◽  
Crystal Size Distributions

scholarly journals Size-dependent growth kinetics of struvite crystals in wastewater with calcium ions

2020 ◽  
Vol 18 (1) ◽  
pp. 196-206
Author(s):  
Nina Hutnik ◽  
Anna Stanclik ◽  
Krzysztof Piotrowski ◽  
Andrzej Matynia
Keyword(s):  
Growth Rate ◽  
Crystal Growth ◽  
Kinetic Parameters ◽  
Calcium Ions ◽  
Crystal Growth Rate ◽  
Experimental Population ◽  
Mixed Product ◽  
Size Dependent ◽  
S 100 ◽  
Dependent Growth

AbstractKinetic parameters describing continuous reaction crystallization of struvite from aqueous solutions containing also calcium ions (from 100 to 2000 mg Ca2+/kg) were estimated. Test results were compared with kinetic data of struvite manufactured from real cattle liquid manure. Kinetic model for ideal MSMPR (Mixed Suspension Mixed Product Removal) crystallizer was used assuming dependence of crystal growth rate G on its size L (size-dependent growth, SDG MSMPR model). Based on nonlinear regression and statistical analysis, one from within five considered G(L) models was selected – Rojkowski exponential (RE) model – rendering the experimental population density distributions the best. It was concluded, that calcium ions influenced all components of struvite manufacturing process disadvantageously. A rise of Ca2+ concentration in a feed from 100 to 2000 mg/kg increased nucleation rate ca. 160-time, whereas growth rate of nuclei up to macroscopic size G0 decreased more than 10-time. Linear (larger) crystal growth rate G¥ was nearly two-times smaller: 1.71·10–8 m/s (100 mg Ca2+/kg) – 9.10·10–9 m/s (2000 mg Ca2+/kg). Resulting in a product with deteriorated quality. Mean size of the crystals decreased nearly two-times (to 18.4 μm), non-homogeneity within product population enlarged and calcium fraction in the product increased. The product, beside struvite MgNH4PO4·6H2O, also contained hydrated amorphous calcium phosphate(V) Ca3(PO4)2·nH2O (ACP). It was observed, that 5-times smaller concentration of phosphate(V) ions in a feed and magnesium ions excess in relation to phosphate(V) and ammonium ions (1.2 : 1 : 1) influenced all kinetic parameters of continuous struvite reaction crystallization advantageously.

Continuous Precipitation of Lead Iodide

1994 ◽  
Vol 59 (7) ◽  
pp. 1503-1510
Author(s):  
Stanislav Žáček ◽  
Jaroslav Nývlt
Keyword(s):  
Growth Rate ◽  
Particle Size ◽  
Crystal Size ◽  
Crystal Growth Rate ◽  
Laboratory Model ◽  
Lead Iodide ◽  
Secondary Nucleation ◽  
Equimolar Ratio ◽  
Continuous Precipitation ◽  
The Mean

Lead iodide was precipitated from aqueous solutions of 0.015 - 0.1 M Pb(NO3)2 and 0.03 - 0.2 M KI in the equimolar ratio using a laboratory model of a stirred continuous crystallizer at 22 °C. After reaching the steady state, the PbI2 crystal size distribution was measured sedimentometrically and the crystallization kinetics was evaluated based on the mean particle size. Both the linear crystal growth rate and the nucleation rate depend on the specific output of the crystallizer. The system crystallization constant either points to a significant effect of secondary nucleation by the mechanism of contact of the crystals with the stirrer blade, or depends on the concentrations of the components added due to the micromixing mechanism.

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