Four-parameter Function Research Articles

Master Curves for Poroelastic Spherical Indentation With Step Displacement Loading

Abstract Theoretical and numerical analyses are conducted to rigorously construct master curves that can be used for interpretation of displacement-controlled poroelastic spherical indentation test. A fully coupled poroelastic solution is first derived within the framework of Biot’s theory using the McNamee–Gibson displacement function method. The fully saturated porous medium is assumed to consist of slightly compressible solid and fluid phases and the surface is assumed to be impermeable over the contact area and permeable everywhere else. In contrast to the cases in our previous studies with an either fully permeable or impermeable surface, the mixed drainage condition yields two coupled sets of dual integral equations instead of one in the Laplace transform domain. The theoretical solutions show that for this class of poroelastic spherical indentation problems, relaxation of the normalized indentation force is affected by material properties through weak dependence on a single-derived material constant only. Finite element analysis is then performed in order to examine the differences between the theoretical solution, obtained by imposing the normal displacement over the contact area, and the numerical results where frictionless contact between a rigid sphere and the poroelastic medium is explicitly modeled. A four-parameter elementary function, an approximation of the theoretical solution with its validity supported by the numerical analysis, is proposed as the master curve that can be conveniently used to aid the interpretation of the poroelastic spherical indentation test. Application of the master curve for the ramp-hold loading scenario is also discussed.

Modulable 3D-printed plantibody-laden platform enabling microscale affinity extraction and ratiometric front-face fluorescence detection of microcystin-LR in marine waters

A 3D-printed stereolithographic platform for selective biorecognition is designed to enable convective microscale affinity extraction of microcystin-LR (MC-LR) followed by direct solid-phase optosensing exploiting ratiometric front-face fluorescence spectroscopy. For this purpose, a recombinant monoclonal plantibody (recAb) is covalently attached to a 3D-printed structure for sorptive immunoextraction, whereupon the free and unbound primary amino moieties of the recAb are derivatized with a fluorescent probe. The fluorophore-recAb-MC-LR laden device is then accommodated in the cuvette holder of a conventional fluorometer without any instrumental modification for the recording of the solid-phase fluorescence emission. Using Rodbard’s four-parameter sigmoidal function, the 3D-printed bioselective platform features a limit of detection (LOD) of 28 ng L−1 using a sample volume of 500 mL, device-to-device reproducibility down to 12%, and relative recoveries ranging from 91 to 100% in marine waters. Printed prototypes are affordable, just 0.4 € per print and ≤ 10 € per device containing recAb. One of the main assets of the miniaturized immunoextraction device is that it performs comparably well in terms of analytical figures of merit with costly mass spectrometric-based analytical methodologies, such as HPLC–MS/MS. The device is readily applicable to high-matrix samples, such as seawater, as opposed to previous biosensing platforms, just applied to freshwater systems.Graphical abstract

Development of emergent processes and threshold of consciousness with levels of processing.

The transition of experience from unconscious to conscious, the emergent process, is a crucial topic in consciousness studies. Three frameworks exist to explain the process: (1) consciousness arises in an all-or-none manner; (2) consciousness arises gradually; (3) consciousness arises either all at once or gradually, depending on the level of stimulus processing (low- vs. high-level). However, the development of emergent processes of consciousness remains unclear. This study examines the development of emergent processes of consciousness based on the level of stimulus processing framework. Ninety-nine children (5-12 year-olds) and adults participated in two online discrimination tasks. These tasks involved color discrimination as lower-level processing and number magnitude discrimination as higher-level processing, as well as backward masking with stimulus onset asynchronies (SOAs) varying from 16.7 to 266.7 ms. We measured objective discrimination accuracy and used a 4-scale Perceptual Awareness Scale (PAS) to assess subjective awareness. We fit the data to a four-parameter nonlinear function to estimate the center of the slope (threshold) and the range of the slope (gradualness, the measure of emergent process of consciousness) of the model. The results showed the threshold of objective discrimination was significantly higher in 5-6 year-olds than in 7-12 year-olds, but not of subjective awareness. The emergent process of objective discrimination in the number task was more gradual than in the color task. The findings suggest that the thresholds of subjective awareness in 5-6 year-olds and objective discrimination in 7-9 year-olds are similar to those in adults. Moreover, the emergent processes of subjective awareness and objective discrimination in 5-6 year-olds are also similar to those in adults. Our results support the level of processing hypothesis but suggest that its effects may differ across developmental stages.

Constitutive behavior of CFRP-confined normal- and high-strength geopolymer concrete: Experiments and modelling

The constitutive behavior of geopolymer concrete (GPC), which is essential to the design of GPC structures, has not been well explored, especially in multi-axial stress state. This study therefore performs an investigation on the constitutive behavior of normal- and high-strength GPC confined by carbon fiber-reinforced polymer (CFRP) jacket. A total of 36 CFRP-confined GPC specimens are fabricated, which cover six GPC mixtures and three thickness of CFRP jacket. A dataset comprising of 24 specimens is employed to develop the models for ultimate condition and axial stress-strain curve. Two testing datasets, one consisting of the rest 12 specimens and the other composed of six specimens collected from the literature, are exploited to test the models. A linear function of actual confinement ratio is competent for modelling the compressive strength, whereas a nonlinear function is capable of characterizing the ultimate axial strain. The average absolute error of predicted compressive strengths and ultimate axial strains is less than 8.0 % and 15.0 % for both testing datasets. A single four-parameter function originally proposed for OPCC can be extended to represent the axial stress-strain curve of CFRP-confined GPC by formulating expressions for the parameters. The correlation coefficient between predicted and measured curves is more than 0.96.

Directional reflectance of light from landscapes on a long transect in Australia – forest to desert

The reflectance of land and vegetation observed in satellite imagery depends on sun and viewing geometry. This bidirectional reflectance requires correction for monitoring changes in vegetation cover and condition. We used a digital camera mounted in a light aircraft, and fitted with a fisheye lens, to measure directional reflectance of a diverse range of landscapes along a long transect in Australia — between Brisbane and the Simpson desert. All, except one, of the measured directional reflectances were able to be characterised accurately (adjusted r2 > 0.95) by the product of two analytical functions. The first, G(θ1,θ2), which represents volume scattering, is a function of illumination and viewing zenith angles, θ1andθ2, and has one parameter k:

Multistep kinetics in self-induced sol–gel process of amorphous anhydride formation during thermal dehydration of potassium tetraborate tetrahydrate

AbstractThe physico-geometrical mechanism and kinetics of the multistep thermal dehydration of potassium tetraborate tetrahydrate was investigated as a model reaction to produce amorphous anhydrate via the self-induced sol–gel process. The thermal dehydration is composed of three consecutive dehydration steps: (1) a surface reaction in the solid-state accompanied by crack formation; (2) a rapid mass loss process accompanied by liquefaction to form the aggregate of the gel powders; and (3) the dehydration of gel powder aggregate to form a glassy anhydride. The changes in the contribution of the individual dehydration steps to the overall process according to the sample particle size and the heating rate (β) were identified as specific features of the multistep dehydration, which was characterized quantitatively using mathematical deconvolution analysis with log-normal four-parameter functions. The difficulty in determining the apparent activation energy (Ea) of the individual dehydration steps using isoconversional analysis due to the changes in the contribution depending on β values was addressed using modulated temperature thermogravimetry. Using the contributions and the apparent Ea of the individual dehydration steps as initial values, the kinetic description of the multistep thermal dehydration was refined through a kinetic deconvolution analysis using the cumulative kinetic equation. As a result, the individual dehydration steps were kinetically characterized as: (1) a surface reaction described by the first-order rate law with Ea,1 of approximately 68 kJ mol−1; (2) a reaction accompanied by liquefaction controlled by an autocatalytic rate behavior with Ea,2 of approximately 123 kJ mol−1; (3) a diffusion-controlled dehydration of gel powder with Ea,3 of approximately 82 kJ mol−1.

Optimizing Biochar Application Rates to Improve Soil Properties and Crop Growth in Saline–Alkali Soil

There is great demand for the amelioration of saline–alkali soils, which requires efficient and economical amendments. Biochar addition could alleviate the adverse impacts of saline–alkali stress in crops. However, their efficiency and optimal amounts in saline–alkali soil restoration remain contradictory and inconclusive. The objective of this study was to investigate the effects of biochar application on the properties of saline–alkali soil and crop growth, as well as to determine the optimal application rate of biochar. We conducted pot experiments with biochar (B) application rates, including 0 (CK), 1% (B-1%), 2.5% (B-2.5%), 5% (B-5%), and 10% (B-10%), studying the impact of biochar on soil water content (SWC), soil salinity, soil electrical conductivity (EC), soil ion content, soil nutrients, soil enzyme activity, and crop growth. A four-parameter Gaussian function was established for the curves depicting the relationship between soil salinity characteristics and the biochar application rates to determine the most optimal application rate. The results indicated that: (1) Compared to the CK, all biochar treatments improved soil water-holding capacity and reduced soil Na+ content and sodium adsorption ratio (SAR). (2) B-1%, B-2.5%, and B-5% treatments reduced soil content, EC, Cl−, and SO42− content over CK, while the results were reversed for the B-10% treatment. (3) Compared to the CK, all biochar treatments significantly increased soil fertility, enhanced soil enzyme activity (alkaline phosphatase, catalase, and urease activity), and significantly promoted the growth of maize. (4) The results of the Gaussian model suggested that a biochar application rate of 3.16% is the optimal rate for alleviating soil salinity in saline–alkali soils. This research demonstrated the potential of biochar to improve soil properties and promote crop growth and provided useful information on biochar application rates for ameliorating saline–alkali soils.

An approach towards a self-consistent EAM model for bcc metals Lithium and Vanadium

Recently, a four-parameter embedding function that is sufficiently rich to handle both positive and negative Cauchy discrepancy has been produced by modifying the embedding function of Yuan et al. (2003)1, a modification to MEAM92, which failed to produce a reasonable surface energy when applied to bcc Vanadium V. We compute the three low-index surface energies of Li and V using the novel model Generalized Embedding Atom method (GEAM) (Oni-Ojo et al, 2007) parameters (iterated values), and the results fall within the experimental average.

Estimate of lysine nutritional requirements for Japanese quail breeders.

Japanese quail breeders are the basis for genetic improvement and multiplication for commercial layers, however, there have been no known studies on the optimal lysine level for these birds. Thus, study the egg output response to the lysine (Lys) supply using different e-functions and evaluate the that best fit, have allowed the partition the lysine requirements for maintenance, both weight and egg output maximum. The objectives of this study were to identify the responses to various Lys levels, identify the functions related to these responses and determine the ideal Lys intake amount for Japanese quail breeders. A completely randomized design of seven treatments with seven replicated was used. Treatments consisted of diet supplementation by Lys in concentrations of 16.8, 11.8, 8.4, 6.7, 5.0, 3.4, and 1.7 g/kg. Six exponential models were adjusted. The level of Lys was found to affect bird responses (P<0.001). The birds responded to the levels provided, allowing for the creation of a lysine response curve. A monomolecular function with four parameters was balanced against the statistics of adjustment and selection of models. It was possible to estimate the level of lysine required for maintenance as 133 ±2 mg/kg BW0.67, and based an average of 41% efficiency, 22 mg Lys produced 1 g of egg output (EO). The daily intake calculated by the monomolecular factorial model was 284 mg Lys for a bird with 0.170 kg body weight and production of 10 g EO/day. The four-parameter monomolecular function proposed in this study is adequate for interpreting the animal response and calculating lysine intake for breeders.

Use of Danish National Somatic Cell Count Data to Assess the Need for Dry-Off Treatment in Holstein Dairy Cattle.

In Denmark, PCR testing of dairy cattle is commonly used to select animals for the antibacterial treatment of intramammary infection (IMI) during the dry-off period. IMI is associated with a high somatic cell count (SCC), routinely recorded for milk quality control for most commercial dairy herds. This study aimed to compare SCC curves over the lactation among dairy cows with positive vs. negative PCR test results for four major IMI pathogens. Data from 133,877 PCR-tested Holstein cows from 1364 Danish conventional dairy herds were used to fit a nonlinear mixed-effects model using a modified four-parameter Wilmink function. We stratified the data into first, second, third or fourth and later parity and fitted Wilmink curves to all SCC observations between 6 and 305 days in milk. The PCR tests were taken before dry-off at the end of the lactation to investigate which animals qualified for selective dry cow therapy. A PCR Ct-value of 37 and below was used to determine if an animal was PCR positive for any of the following IMI pathogens: Staphylococcus aureus, Streptococcus agalactiae, Str. dysgalactiae and Str. uberis. Our findings showed that mean SCC curve fits were higher for PCR-positive animals in all four parity groups and across lactations. The use of SCC data fitted to the entire lactation for multiple lactations enabled quantification of overall differences in SCC curves between cattle with and without detected IMI, adjusted for parity group and stage of lactation. These findings are relevant to the use of SCC to support treatment decisions.

CERTAIN UNIFIED INTEGRALS INVOLVING A PRODUCT OF THE FOUR-PARAMETER BESSEL FUNCTION AND JACOBI POLYNOMIAL

The present paper is devoted to derive a generalized Oberhettingertype integral formula. The derived form of an integral involving the product of the four-parameter Bessel function and Jacobi polynomial. The outcomes are expressed in terms of the Kamp´e de F´eriet and Srivastava and Daoust functions. Also, four Corollaries of the both Theorems are derived in terms of the Kamp´e de F´eriet and Srivastava and Daoust functions. Some of the significant particular cases are also determined. Furthermore, we drive an interesting relationship between Kamp´e de F´eriet and Srivastava and Daoust functions

Dosimetric response of Gafchromic™ EBT-XD film to therapeutic protons.

EBT-XD model of Gafchromic™ films has a broader optimal dynamic dose range, up to 40 Gy, compared to its predecessor models. This characteristic has made EBT-XD films suitable for high-dose applications such as stereotactic body radiotherapy and stereotactic radiosurgery, as well as ultra-high dose rate FLASH radiotherapy. The purpose of the current study was to characterize the dependence of EBT-XD film response on linear energy transfer (LET) and dose rate of therapeutic protons from a synchrotron. A clinical spot-scanning proton beam was used to study LET dependence at three dose-averaged LET (LETd) values of 1.0 keV/µm, 3.6 keV/µm, and 7.6 keV/µm. A research proton beamline was used to study dose rate dependence at 150 Gy/second in the FLASH mode and 0.3 Gy/second in the non-FLASH mode. Film response data from LETd values of 0.9 keV/µm and 9.0 keV/µm of the proton FLASH beam were also compared. Film response data from a clinical 6 MV photon beam were used as a reference. Both gray value method and optical density (OD) method were used in film calibration. Calibration results using a specific OD calculation method and a generic OD calculation method were compared. The four-parameter NIH Rodbard function and three-parameter rational function were compared in fitting the calibration curves. Experimental results showed that the response of EBT-XD film is proton LET dependent but independent of dose rate. Goodness-of-fit analysis showed that using the NIH Rodbard function is superior for both protons and photons. Using the "specific OD + NIH Rodbard function" method for EBT-XD film calibration is recommended.

Single-time-point dosimetry using model selection and nonlinear mixed-effects modelling: a proof of concept

PurposeThis project aims to develop and evaluate a method for accurately determining time-integrated activities (TIAs) in single-time-point (STP) dosimetry for molecular radiotherapy. It performs a model selection (MS) within the framework of the nonlinear mixed-effects (NLME) model (MS–NLME).MethodsBiokinetic data of [111In]In-DOTATATE activity in kidneys at T1 = (2.9 ± 0.6) h, T2 = (4.6 ± 0.4) h, T3 = (22.8 ± 1.6) h, T4 = (46.7 ± 1.7) h, and T5 = (70.9 ± 1.0) h post injection were obtained from eight patients using planar imaging. Eleven functions were derived from various parameterisations of mono-, bi-, and tri-exponential functions. The functions’ fixed and random effects parameters were fitted simultaneously (in the NLME framework) to the biokinetic data of all patients. The Akaike weights were used to select the fit function most supported by the data. The relative deviations (RD) and the root-mean-square error (RMSE) of the calculated TIAs for the STP dosimetry at T3 = (22.8 ± 1.6) h and T4 = (46.7 ± 1.7) h p.i. were determined for all functions passing the goodness-of-fit test.ResultsThe function f_{4d} left( t right) = A_{1} /left{ {left( {frac{1 - alpha }{{lambda_{1} + lambda_{{{text{phys}}}} }}} right) - left( {frac{alpha }{{lambda_{2} + lambda_{{{text{phys}}}} }}} right) - left( {frac{1 - 2alpha }{{lambda_{bc} + lambda_{{{text{phys}}}} }}} right)} right} cdot e^{{ - lambda_{{{text{phys}}}} t}} cdot left{ {left( {1 - alpha } right) cdot e^{{ - lambda_{1} t}} - alpha cdot e^{{ - lambda_{2} t}} - left( {1 - 2alpha } right) cdot e^{{ - lambda_{bc} t}} } right} with four adjustable parameters and lambda_{bc} = frac{{{text{ln}}left( 2 right)}}{{1;{text{ min}}}} was selected as the function most supported by the data with an Akaike weight of (45 ± 6) %. RD and RMSE values show that the MS–NLME method performs better than functions with three or five adjustable parameters. The RMSEs of TIANLME–PBMS and TIA3-parameters were 7.8% and 10.9% (for STP at T3), and 4.9% and 10.7% (for STP at T4), respectively.ConclusionAn MS–NLME method was developed to determine the best fit function for calculating TIAs in STP dosimetry for a given radiopharmaceutical, organ, and patient population. The proof of concept was demonstrated for biokinetic 111In-DOTATATE data, showing that four-parameter functions perform better than three- and five-parameter functions.

Quantifying COVID-19 recovery process from a human mobility perspective: An intra-city study in Wuhan.

The COVID-19 pandemic has brought huge challenges to sustainable urban and community development. Although some recovery signals and patterns have been uncovered, the intra-city recovery process remains underexploited. This study proposes a comprehensive approach to quantify COVID-19 recovery leveraging fine-grained human mobility records. Taking Wuhan, a typical COVID-19 affected megacity in China, as the study area, we identify accurate recovery phases and select appropriate recovery functions in a data-driven manner. We observe that recovery characteristics regarding duration, amplitude, and velocity exhibit notable differences among urban blocks. We also notice that the recovery process under a one-wave outbreak lasts at least 84days and has an S-shaped form best fitted with four-parameter Logistic functions. More than half of the recovery variance can be well explained and estimated by common variables from auxiliary data, including population, economic level, and built environments. Our study serves as a valuable reference that supports data-driven recovery quantification for COVID-19 and other crises.

Standard GRB Spectral Models “Misused”?

The standard model characterizing the gamma-ray burst (GRB) spectrum invokes a four-parameter empirical function, the so-called the BAND model. An alternative model named cutoff power law (COMP) implements a power law with an exponential cutoff. These functions achieve almost equally good fits on observed spectra, and are adopted in nearly all of the GRB literature. Here, we reanalyze the sample defined in Li et al. (39 bursts including 944 spectra). We classify the spectra by two methods: (1) checking their corner–corner plots of the posteriors to determine well-constrained β (BAND-better) and unconstrained β (COMP-better) categories; and (2) defining the four groups by difference of the deviance information criterion (DIC). We find inconsistent peaks of the parameter distributions between the BAND-better spectra (α = −0.64 ± 0.28 and ) and the COMP-better spectra (α = −0.96 ± 0.33 and ). With the statistically preferred model and vice versa the misused model defined based on DIC statistics, we also find that the fitted parameters obtained by the misused model (COMP) significantly deviate from those obtained by the statistically preferred model (BAND). This means that if a spectrum is statistically preferred, described as the BAND, applying COMP to derive the spectral parameters will prominently deviate from their intrinsic shape, therefore affecting the physical interpretation. Our analysis indicates that the better or statistically preferred model should be duly examined during GRB spectral analysis. In addition, the β distribution exhibits a bimodal structure containing the BAND-better and COMP-better spectra, respectively, implying that BAND and COMP both may have physical origin.

The Polish Provenances of European Larch Overperform the Expected Growth Dynamics Indicated by the Sigmoid Model

This article attempts an unusual interpretation of the observations characteristic of experiments that compare different tree species’ provenances. The focus falls on larch (Larix decidua subsp. polonica). The data came from the experiment established in 1967 at the Siemianice forest experimental station, where 21 Polish larch provenances were compared. The study’s main objective was to compare the basic estimates of growth dynamics, the maximum growth rate and acceleration, and the point in time when these values were achieved. A four-parameter sigmoid growth function was used to model the average stand basal area increments and its first and second derivatives to calculate the indicators of the growth dynamic. The models explained 98% of the observed variations resulting from the 21 inventories. Only one growth parameter showed a statistically significant difference among the compared provenances. The Góra Chełmowa provenance achieved the highest value of the maximum growth acceleration, but it was statistically significantly different only from the three underperformed provenances. However, when the average values for all the experimental plots (n = 86) were compared with those of the deterministic model (the stand volume and yield tables), the maximum growth rate and acceleration values were higher for the former. We discuss the potential factors responsible for this overperformance and point out the potential risks that arise from growth and quality metrics only when deciding on the best-performing provenances. The sigmoid growth model employed in this study might be an excellent alternative for comparing the growth dynamics among different stands or replications in experimental studies. Considering only the early results, the sigmoid growth model proves its limitations, and the conclusions reached should be treated with caution.

A Consolidation Curve Reproduction Based on Sigmoid Model: Evaluation and Statistical Assessment.

In the present study, various shapes of laboratory consolidation curves were numerically reproduced using a four-parametric sigmoid function. Sixteen consolidation curves were selected based on one-dimensional oedometer tests to statistically evaluate the sigmoid model and to determine the appropriate deviation statistics. Comparisons between observed and predicted data were performed using the following statistical metrics: mean error (E), root mean square error (RMSE), mean absolute error (MAE), weighted error (WE), revised Nash–Sutcliffe efficiency index (CE1) and refined index of model performance (dr). The weighted error (WE) was chosen as the optimization target in a first-order iterative optimization algorithm to determine a local minimum of a differentiable function. Comparing the simulated and observed settlements showed close correspondence in the values of CE1 and dr in terms of model performance. Based on statistical assessment, the maximum values of RMSE and MAE for the average degree of consolidation were 0.029 (-) and 0.021 (-), respectively. In turn the settlement data RMSE and MAE were 0.039 mm and 0.025 mm, respectively. These results indicated that the sigmoid expression effectively reproduced the shape of the consolidation curve.

Early and late universe holographic cosmology from a new generalized entropy

We propose a new four-parameter entropy function that generalizes the Tsallis, Rényi, Barrow, Sharma-Mittal, Kaniadakis and Loop Quantum Gravity entropies for suitable limits of the parameters. Consequently, we address the early and late universe cosmology corresponding to the proposed four-parameter entropy function. As a result, it turns out that the entropic cosmology from the generalized entropy function can unify the early inflation to the late dark energy era of the universe. In such a unified scenario, we find that – (1) the inflation era is described by a quasi de-Sitter evolution of the Hubble parameter, which has an exit at around 58 e-folding number, (2) the inflationary observable quantities like the spectral index for primordial scalar perturbation and the tensor-to-scalar ratio are simultaneously compatible with the recent Planck data, and (3) regarding the late time cosmology, the dark energy EoS parameter is found to be consistent with the Planck result for the same values of the entropy parameters that lead to the viable inflation during the early universe. Furthermore, we show that the entropic cosmology from the proposed entropy function is equivalent to holographic cosmology, where the respective holographic cut-offs are determined in terms of either particle horizon and its derivative or future horizon and its derivative.

A comparison of regression models for defining EPA + DHA requirements using the gilthead seabream (Sparus aurata) as a model species

Carnivorous marine fish species such as gilthead seabream (Sparus aurata) require dietary eicosapentaenoic (EPA) and docosahexaenoic (DHA) acids for optimal growth and wellbeing. The rapid growth of global aquaculture, along with increased proportions of dietary oil to increase growth rate of farmed fish, has meant that the supply of marine oils used in aquafeeds has become limited. The shortfall has been satisfied by using vegetable oils that lack EPA and DHA and, therefore, EFA (essential fatty acid) requirements of juvenile marine fish require reassessment. A dietary trial was carried out with gilthead seabream (~25 g) that were fed diets with six EPA + DHA levels ranging from 0.2% - 3.2% diet as fed. For each pellet size, the biometric data (weight gain, daily growth index and feed conversion ratio) were analysed by four different regression strategies, namely split linear, quadratic, the Gompertz function, and the four-parameter logistic function. Over the whole experimental period (two pellet sizes) data suggested the current published requirement (1% of diet) was low and should be increased to at least 1.2%. However, when the first pellet size for fish of 25–80 g was considered, the apparent requirement was at least 1.4% of diet. This demonstrated in a single trial that EFA requirement was a function of fish mass, decreasing as the fish grows. If FCR is considered, the requirement may be as high as 2%. The suitability of different regression models varied, as the data for the first pellet was best fit by curves but, over both pellet sizes, the split linear fit the data best. For asymptotic models (Gompertz and four-parameter logistic functions), a novel way of defining the requirement was presented, the “elbow” calculation as a method to bisect an asymptotic function. Therefore, using the raw data, we illustrate how a range of regression approaches could be explored when determining nutrient requirement estimates as no single model was an ideal fit for all response curves.

The Influential Role of Powder Factor vs. Delay in Full-Scale Blasting: A Perspective Through the Fragment Size-Energy Fan

The fragmentation of 12 full-scale one-row blasts has been measured by sieving a large portion of the muckpiles. The procedure followed, the difficulties encountered and the solutions adopted to construct the fragment size distribution curves are described in detail; 11 curves were finally constructed as production constraints prevented the required measurements on one of the blasts. The blasts covered a powder factor range between 0.42 and 0.88 kg/m3, and were initiated with two significantly different delays, 4 and 23 ms between holes, to assess the influence of both powder factor and delay on fragmentation. The size distributions are well represented by the Swebrec function, which strongly suggests that the dependence of fragmentation with the powder factor can be analyzed by the fragmentation-energy fan. The result is excellent, and the frag-energy fan model in its simplest form (a four-parameter function) is able to predict sizes between percentage passings 92 to 8% with a mean error of 14.4% and a determination coefficient R2 as high as 0.976. The powder factor above grade has been used, in its energy form obtained as the product of the mass powder factor by the explosive energy per unit mass. The incorporation of six more fragment size distributions, also obtained by sieving in a previous blasting project in the same rock mass, but with different layouts, explosives, delay and blast direction, only reduces R2 to 0.968 and increases the mean error to 15.3%. A strength dependence with the size of the blasted block (burden, bench height, etc.) has been tested for inclusion in the fan formulation, with minor improvement compared with the powder factor alone, as the variation in size of the blasts was very limited. Some size descriptors as in-situ block size and fracture intensity have also been tested, though variations were also limited as all blasts were carried out in the same quarry site, not improving the prediction errors when other blast dimensions (e.g., burden) are used. Incorporating the effect of delay in the fragmentation-energy fan model has been attempted with a cooperation function modifying the powder factor, increasing from instantaneous to an optimum delay value, then decreasing as the delay further increases. The effect of such a function is noticeable in terms of improved prediction; the data analyzed, however, do not allow for a definitive statement on an optimum delay value as calculations with different fan characteristics and data result in different optimum values. The effect of the delay on the fragment size varies with the percentile, from about 10–15% for the high percentiles to somewhat more than 30% for the lower percentiles.