Accuracy and optimal sampling in Monte Carlo solution of population balance equations

Implementation of a Monte Carlo simulation for the solution of population balance equations (PBEs) requires choice of initial sample number (N0), number of replicates (M), and number of bins for probability distribution reconstruction (n). It is found that Squared Hellinger Distance, H2, is a useful measurement of the accuracy of Monte Carlo (MC) simulation, and can be related directly to N0, M, and n. Asymptotic approximations of H2 are deduced and tested for both one-dimensional (1-D) and 2-D PBEs with coalescence. The central processing unit (CPU) cost, C, is found in a power-law relationship, C= aMNb0, with the CPU cost index, b, indicating the weighting of N0 in the total CPU cost. n must be chosen to balance accuracy and resolution. For fixed n, M × N0 determines the accuracy of MC prediction; if b > 1, then the optimal solution strategy uses multiple replications and small sample size. Conversely, if 0 < b < 1, one replicate and a large initial sample size is preferred. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2394–2402, 2015

A priori prediction of aggregation efficiency and rate constant for fluidized bed melt granulation

This paper presents a predictive aggregation rate model for spray fluidized bed melt granulation. The aggregation rate constant was derived from probability analysis of particle–droplet contact combined with time scale analysis of droplet solidification and granule–granule collision rates. The latter was obtained using the principles of kinetic theory of granular flow (KTGF). The predicted aggregation rate constants were validated by comparison with reported experimental data for a range of binder spray rate, binder droplet size and operating granulator temperature. The developed model is particularly useful for predicting particle size distributions and growth using population balance equations (PBEs).

Representing spray zone with cross flow as a well-mixed compartment in a high shear granulator

The spray zone is an important region to control nucleation of granules in a high shear granulator. In this study, a spray zone with cross flow is quantified as a well-mixed compartment in a high shear granulator. Granulation kinetics is quantitatively derived at both particle-scale and spray zone-scale. Two spatial decay rates, DGSDR (droplet-granule spatial decay rate) ζDG and DPSDR (droplet-primary particle spatial decay rate) ζDP, which are functions of volume fraction and diameter of particulate species within the powder bed, are defined to simplify the deduction. It is concluded that in cross flow, explicit analytical results show that the droplet concentration is subject to exponential decay with depth which produces a numerically infinite depth of spray zone in a real penetration process. In a well-mixed spray zone, the depth of the spray zone is 4/(ζDG + ζDP) and π2/3(ζDG + ζDP) in cuboid and cylinder shape, respectively. The first-order droplet-based collision rates of, nucleation rate B0 and rewetting rate RW0 are uncorrelated with the flow pattern and shape of the spray zone. The second-order droplet-based collision rate, nucleated granule-granule collision rate RGG, is correlated with the mixing pattern. Finally, a real formulation case of a high shear granulation process is used to estimate the size of the spray zone. The results show that the spray zone is a thin layer at the powder bed surface. We present, for the first time, the spray zone as a well-mixed compartment. The granulation kinetics of a well-mixed spray zone could be integrated into a Population Balance Model (PBM), particularly to aid development of a distributed model for product quality prediction.

A compartmental CFD-PBM model of high shear wet granulation

The conventional, geometrically lumped description of the physical processes inside a high shear granulator is not reliable for process design and scale-up. In this study, a compartmental Population Balance Model (PBM) with spatial dependence is developed and validated in two lab-scale high shear granulation processes using a 1.9L MiPro granulator and 4L DIOSNA granulator. The compartmental structure is built using a heuristic approach based on computational fluid dynamics (CFD) analysis, which includes the overall flow pattern, velocity and solids concentration. The constant volume Monte Carlo approach is implemented to solve the multi-compartment population balance equations. Different spatial dependent mechanisms are included in the compartmental PBM to describe granule growth. It is concluded that for both cases (low and high liquid content), the adjustment of parameters (e.g. layering, coalescence and breakage rate) can provide a quantitative prediction of the granulation process.

Balance ability of 7 and 10 year old children in the population:results from a large UK birth cohort study

OBJECTIVE: The literature contains many reports of balance function in children, but these are often on atypical samples taken from hospital-based clinics and may not be generalisable to the population as a whole. The purpose of the present study is to describe balance test results from a large UK-based birth cohort study. METHODS: Data from the Avon Longitudinal Study of Parents and Children (ALSPAC) were analysed. A total of 5402 children completed the heel-to-toe walking test at age 7 years. At age 10 years, 6915 children underwent clinical tests of balance including beam-walking, standing heel-to-toe on a beam and standing on one leg. A proportion of the children returned to the clinic for retesting within 3 months allowing test-retest agreement to be measured. RESULTS: Frequency distributions for each of the balance tests are given. Correlations between measures of dynamic balance at ages 7 and 10 years were weak. The static balance of 10 year old children was found to be poorer with eyes closed than with eyes open, and poorer in boys than in girls for all measures. Balance on one leg was poorer than heel-to-toe balance on a beam. A significant learning effect was found when first and second attempts of the tests were compared. Measures of static and dynamic balance appeared independent. Consistent with previous reports in the literature, test-retest reliability was found to be low. CONCLUSIONS: This study provides information about the balance ability of children aged 7 and 10 years and provides clinicians with reference data for balance tests commonly used in the paediatric clinic.

Sampling and measurement methods for a study of childhood refractive error in a UK population

Background There is a paucity of data describing the prevalence of childhood refractive error in the United Kingdom. The Northern Ireland Childhood Errors of Refraction study, along with its sister study the Aston Eye Study, are the first population-based surveys of children using both random cluster sampling and cycloplegic autorefraction to quantify levels of refractive error in the United Kingdom. Methods Children aged 6–7 years and 12–13 years were recruited from a stratified random sample of primary and post-primary schools, representative of the population of Northern Ireland as a whole. Measurements included assessment of visual acuity, oculomotor balance, ocular biometry and cycloplegic binocular open-field autorefraction. Questionnaires were used to identify putative risk factors for refractive error. Results 399 (57%) of 6–7 years and 669 (60%) of 12–13 years participated. School participation rates did not vary statistically significantly with the size of the school, whether the school is urban or rural, or whether it is in a deprived/non-deprived area. The gender balance, ethnicity and type of schooling of participants are reflective of the Northern Ireland population. Conclusions The study design, sample size and methodology will ensure accurate measures of the prevalence of refractive errors in the target population and will facilitate comparisons with other population-based refractive data.

Impaired balance in developmental dyslexia? A meta-analysis of the contending evidence

Background: Developmental dyslexia is typically defined by deficits in phonological skills, but it is also associated with anomalous performance on measures of balance. Although balance assessments are included in several screening batteries for dyslexia, the association between impairments in literacy and deficits in postural stability could be due to the high co-occurrence of dyslexia with other developmental disorders in which impairments of motor behaviour are also prevalent. Methods: We identified 17 published studies that compared balance function between dyslexia and control samples and obtained effect-sizes for each. Contrast and association analyses were used to quantify the influence of hypothesised moderator variables on differences in effects across studies. Results: The mean effect-size of the balance deficit in dyslexia was .64 (95% CI = .44-.78) with heterogeneous findings across the population of studies. Probable co-occurrence of other developmental disorders and variability in intelligence scores in the dyslexia samples were the strongest moderator variables of effect-size. Conclusions: Balance deficits are associated with dyslexia, but these effects are apparently more strongly related to third variables other than to reading ability. Deficits of balance may indicate increased risk of developmental disorder, but are unlikely to be uniquely associated with dyslexia.

Prenatal alcohol exposure and childhood balance ability:findings from a UK birth cohort study

OBJECTIVE: To investigate the association of prenatal alcohol exposure with balance in10-year-old children. DESIGN: Population-based prospective longitudinal study. SETTING: Former Avon region of UK (Southwest England). PARTICIPANTS: 6915 children from the Avon Longitudinal Study of Parents and Children who had a balance assessment at age 10 and had data on maternal alcohol consumption. OUTCOME MEASURES: 3 composite balance scores: dynamic balance (beam-walking), static balance eyes open, static balance eyes closed (heel-to-toe balance on a beam and standing on one leg, eyes open or closed). RESULTS: Most mothers (95.5%) consumed no-to-moderate amounts (3-7 glasses/week) of alcohol during pregnancy. Higher total-alcohol consumption was associated with maternal-social advantage, whereas binge drinking (≥4 units/day) and abstinence were associated with maternal social disadvantage. No evidence was found of an adverse effect of maternal-alcohol consumption on childhood balance. Higher maternal-alcohol use during pregnancy was generally associated with better offspring outcomes, with some specific effects appearing strong (static balance eyes open and moderate total alcohol exposure at 18 weeks, adjusted OR 1.23 (95% CI 1.01 to 1.49); static balance eyes closed and moderate total alcohol exposure at 18 weeks, adjusted OR 1.25 (95% CI 1.06 to 1.48). Similar results were found for both paternal and postnatal maternal alcohol exposure. A Mendelian-randomization approach was used to estimate the association between maternal genotype and offspring balance using the non-synonymous variant rs1229984*A (ADH1B) to proxy for lower maternal alcohol consumption; no strong associations were found between this genotype/proxy and offspring balance. CONCLUSIONS: No evidence was found to indicate that moderate maternal alcohol consumption in this population sample had an adverse effect on offspring balance at age 10. An apparent beneficial effect of higher total maternal alcohol consumption on offspring balance appeared likely to reflect residual confounding.