Error Function Model Research Articles

Probabilistic Corrosion-free Service Life of the RCC Structures Made with OPC and Ultra-fine Slag-based Binders

The primary factor in the deterioration of reinforced concrete is chloride-induced corrosion. It has a direct impact on the service life of the structure and has been the subject of much research over the past 5 decades. One of the most cost-effective ways to lessen chloride-induced corrosion was identified to be the use of more supplementary cementitious material (SCM). Moreover, when SCMs are ground down to a smaller size, the resulting particles serve as microfillers, which have a significant impact on the improvement of mechanical properties and durability. This paper highlights the influence of grinding slag into an Ultra Fine Slag (UFS) on chloride ion penetration in the mortar specimen. A bulk diffusion test was performed on a mortar replaced with UFS which show a synergic reduction in chloride diffusivity in comparison with control mixes. In addition, an attempt is made on prediction of probabilistic service life estimation by Error function model.

A differential-based method for correcting ultra-short baseline positioning results

The primary factors influencing the positional accuracy of an ultra-short baseline (USBL) system are errors in sound speed, delay, installation deviation, and the bending of sound rays. Premised on the USBL positioning model, an error contribution model including ranging error, delay error, sound speed error and array element installation deviation was constructed. In parallel, an indirect difference correction method with additional coefficients was proposed based on the geometric correlation of the error function model. Specifically, a reference beacon was deployed on the seabed, and differential correction was achieved by directly computing target coordinates, integrating them with the positional data from the reference beacon. The proposed method demonstrated efficacy in mitigating errors associated with gradual variations or pronounced geometric correlations, including discrepancies in sound speed, bending of sound rays, and deviations in the installation of array elements. The correction method was applied to the USBL positioning, and the simulation results show that the proposed method notably enhanced both accuracy and consistency in positioning, particularly in shallow water environments characterised by significant sound speed errors. The sea trial data from the R/V Dayang Yihao in the Western Pacific demonstrate that the proposed method can effectively mitigate the errors induced by sound ray bending in deep-sea environments. The mean absolute error (MAE) and standard deviation (STD) were reduced by 51.3% and 55.3%, respectively, compared with the results of the USBL direct solution.

Optimization method for freeform progressive addition lenses using the coincident degree of weight distributions for power deviation and astigmatism

This study proposed an optimization method for freeform progressive addition lenses (PALs) based on the coincident degree of weight distributions (WD) for power deviation and astigmatism. Compared with the existing methods which were limited in optimizing WDs for power deviation and astigmatism, our proposed approach offers a more refined optimization. In the design phase, the power deviation and astigmatism of these lenses were evaluated using the existing surface shape. Compared with the prescriptions of the patients, the coincident degrees between the obtained distributions and prescriptions of the PAL power and astigmatism were calculated in the multi-view axis condition. Normalization processing of coincident degrees was performed, yielding the corresponding threshold value of WDs and optimizing the allocated coincident degrees. Based on a minimization error function model, two PALs were designed, simulated, machined, and evaluated using a commercial software. The optimized method reduced peripheral astigmatism and improved the optical properties of PALs. The proposed approach optimizes the freeform PALs and enhances their design optimization in optometry.

Fabrication of date palm kernel biochar-sulfur (DPKB-S) for super adsorption of methylene blue dye from water

A novel form of biochar was created by dehydration of Date palm kernel with 85% sulfuric acid. It was examined how the newly produced biochar (DPKB-S) affected the aqueous solution's capacity to extract Methylene Blue (MB) dye. The prepared DPKB-S was categorized by BET, BJH, FT-IR, SEM, EDX, DSC, and TGA analyses. The ideal pH for the MB dye adsorption by DPKB-S is 8. With 0.75 g L−1 of DPKB-S and an initial concentration of 50 ppm MB dye, Date Palm Kernel Biochar-Sulfur (DPKB-S) had the highest removal percentage of 100%. The Langmuir and Freundlich isotherm models were used to investigate the collected data. Freundlich model is the model that best covers MB dye adsorption in DPKB-S at low concentrations (0.75–1.25 g L−1) and the Langmuir model at high concentrations (1.5–1.75 g L−1). The Langmuir model maximum adsorption capacity (Qm) of the DPKB-S was 1512.30 mg g−1. Furthermore, a variety of error function models were applied to investigate the isotherm models derived data, including Marquardt’s percent standard deviation (MPSD), the sum of absolute errors (EABS), the sum of the errors squared (ERRSQ), root mean square errors (RMS), Chi-square error (X2), the average relative error (ARE), average percent errors (APE), and hybrid error function (HYBRID). Kinetic data were calculated by intraparticle diffusion (IPD), pseudo-second-order (PSO), pseudo-first-order (PFO), and film diffusion (FD) models. A PSO rate model with a strong correlation (R2 = 1.00) largely regulated the adsorption rate. The removal mechanism of MB dye by DPKB-S is based on the principle that these positively charged dyes are attracted by electrostatic attraction forces due to the growth in the number of negatively charged regions at basic pH value. According to the results, DPKB-S shows promise as an affordable and competent adsorbent for the adsorption of MB dye. It can be used frequently without experiencing a discernible decrease in adsorption efficiency.

Simple Deconvolution Models for Evaluating the True Microhardness of Thin Nanostructured Coatings Deposited via an Advanced Physical Vapor Deposition Technique

This article discusses the micromechanical properties and true microhardness determination of nanostructured tribological coatings (NTCs) based on a multilayered alternating nitride/carbonitride bilayer substructure for transition metals. The constituent nitride/carbonitride bilayers in the superlattice structure of the NTC were alloyed with refractory metals, denoted as Me = Me1 or Me2= Cr, Hf, Nb, W, and Zr. The resulting NTC coatings were deposited onto 100Cr6 steel substrates using an advanced physical vapor deposition (PVD) technique, referred to here as high-power ion-plasma magnetron sputtering (HiPIPMS). The comprising crystalline nanometer-scale TiAlSiMe1-N/TiMe2-CN nanoparticles strengthened by Me additives significantly increased the NTC microhardness to over 3200 HV. The primary focus of this research was to determine the true microhardness of the NTC film samples. The apparent microhardness (Ha) of the film/substrate system for various NTC samples was measured during microindentation testing using the Vickers method. Nine NTC samples were tested, each generating a corresponding microindentation dataset containing between 430 and 640 imprints, depending on the specific NTC sample. These datasets were analyzed using three distinct empirical approaches: (i) the inverse power-law model (IPL-Model), (ii) the sigmoid-like decay model (SLD-Model), and (iii) the error function model (ERF-Model). The observed solid correlation between the proposed models and experiments suggests that the true microhardness estimates (Hf) obtained through the empirical mathematical modeling approach are reliable.

A Non-Arrhenius Model for Mechanism Consistency Checking in Accelerated Degradation Tests

Degradation models are central to the lifetime prediction of electromagnetic relays. Coefficients of degradation models under accelerated degradation test (ADTs) can be obtained experimentally, and it is customary to map these coefficients back to those describing the actual degradation by the so-called Arrhenius model. However, for some components, such as springs in electromagnetic relays, the Arrhenius model is only appropriate over a certain ADT temperature range, which implies inaccurate mapping outside that range. On this point, an error function model (EFM) is proposed to overcome the shortcomings of the Arrhenius model. EFM is derived from the average vibration energy of the crystal, which is further related to temperature alongside some constants. The empirical part of the paper compares the proposed EFM to the Arrhenius model for the ADT of 28-V–2-A electromagnetic relays. The results show that EFM is superior in describing the temperature characteristics of coefficients in the degradation model. Through mechanism consistency checking, EFM is also shown to be a better option than the Arrhenius model. Moving beyond the case of electromagnetic relays, EFM is thought to have better applicability in the degradation models of capacitors and rubbers.

Isotherm and kinetic investigations of sawdust-based biochar modified by ammonia to remove methylene blue from water

Chemical industry effluent may pose significant environmental risks to both human health and the economy if it is not properly managed. As a result, scientists and decision-makers are paying increasing attention to developing a sustainable, low-cost wastewater treatment technique. This work aims to investigate the adsorption of Methylene Blue (MB) dye present in water using biochar derived from sawdust modified by boiling in an ammonia solution (SDBA). The properties of SDBA were characterized by BET, SEM, XRD, BJH, FT-IR, DTA, EDX and TGA analyses. The presence of –OH and –NH groups in SDBA was confirmed by FTIR, which proved that the NH4OH treatment of biochar successfully added nitrogen groups on its surface. The influence of pH (2 to 12), MB dye initial concentration (20 to 120 mg/L), adsorbent dosage (0.5 to 4.0 g/L) and contact time (0 to 180 min) on the adsorption process has been investigated. The adsorption of MB dye is more favorable at basic pH, with optimum adsorption at pH 8. Using a starting concentration of 20 mg/L of MB dye and a 4.0 g/L SDBA dose, the maximum percent clearance of MB dye was 99.94%. Experimental results were fitted to the Freundlich (FIM), Tempkin (TIM) and Langmuir (LIM) isotherm models (IMs). The FIM fitted the equilibrium data well, with a 643.74 mg/g Qm. Various error function models were used to test the data obtained from IMs. According to Error Function results, experimental data showed that it fits better for LIM and FIM. Kinetic studies indicated that the MB dye adsorption procedure followed pseudo-second-order (PSOM) kinetics based on film diffusion (FDM), pseudo-first-order (PFOM) and intra-particle diffusion models (IPDM). MB dye sorption on the SDBA involved electrostatic interaction, surface participation, hydrogen bond and π–π interactions. The adsorption mechanism of MB dye by SDBA was proposed as physical adsorption via the electrostatic attraction process. SDBA is an effective adsorbent in removing MB dye from water. Six adsorption–desorption cycles of the MB dye were run through the regeneration of SDBA with only a minimal amount of adsorption capacity loss, demonstrating the reusability of manufactured SDBA.

Adaptive NN state error PCH trajectory tracking control for unmanned surface vessel with uncertainties and input saturation

AbstractA novel robust state error port controlled Hamiltonian (PCH) trajectory tracking controller of an unmanned surface vessel (USV) subject to time‐varying disturbances, dynamic uncertainties and control input saturation is presented. The proposed control scheme combines the advantages of the high robustness and energy minimization of the state error PCH approach and the approximation capability of adaptive radial basis function neural networks (RBFNNs). Adaptive RBFNNs are used to the time‐varying disturbances of the environment and unknown dynamics uncertainties of the USV model. The state error PCH control approach is designed such that the system can optimize energy consumption, and the state error PCH technique makes the designed trajectory tracking controller be easy to implement in practice. To handle the effect of the control input saturation, a Gaussian error function model is employed. It has been demonstrated that the proposed approach can maintain the USV's trajectory at the desired trajectory, while the closed‐loop control system can guarantee the uniformly ultimate boundedness. The energy consumption model of the USV is constructed to reveal to the energy consumption. Simulation results demonstrate the effectiveness of the proposed controller.

Fabrication of cellulose acetate/cellulose nitrate/carbon black nanofiber composite for oil spill treatment

There are global challenges in addressing the oil spill treatment. Nanofiber has become a great potential in the oil spill cleaning process because of the environmental friendliness, high efficiency, low cost, and stability of the obtained nanofiber mats. This study presents a novel composite fabricated from cellulose acetate (CA) and cellulose nitrate (CN) nanofibers with the incorporation of carbon black (CA-CN/CB) for efficient oil removal. This nanofiber composite was fabricated in one-step electrospinning of 10% CA and CN solution with different concentrations of carbon black (CB). The morphology and fiber diameter of the CA-CN/CB nanofiber composite were analyzed using scanning electron microscopy (SEM), and they appeared to be smooth, uniform fibers without beads. The average fiber diameter was in nano-meter size and increased with the increasing CB amount in the composite, ranging from 327 to 755 nm. The FTIR results indicated the presence of CA and CN as characteristic peaks of C = O for CA and O-NO2 for CN. The nanofibers mats of the CA-CN, CA-CN/CB0.7, CA-CN/CB1.5, and CA-CN/CB2.2 composites had Brunauer–Emmett–Teller (BET) surface area of 15.29, 38.40, 4.08, and 6.17 m2 g−1, respectively. Under optimal conditions, CA-CN/CB nanofiber mats absorb more than their weight oil in just 30 min. The adsorption result showed that loading 1.5% of CB to CA-CN mats (CA-CN/CB1.5) was more favorable for oil adsorption. The CA-CN/CB1.5 nanofiber showed its reusability for oil adsorption. The Freundlich isotherm model was the most appropriate model among other isotherm models, including Langmuir and Temkin, with a value of correlation coefficient (R2) equal to or closer to unity, and this result was confirmed by the data obtained from studying different error function models. The adsorption kinetics showed that oil adsorption into CA-CN/CB1.5 nanofiber follows a pseudo-second-order kinetics model with R2 close to unity.

Assessing Sb Cross Incorporation in InAs/InAsSb Superlattices

We developed an x-ray diffraction (XRD) method that uses relative satellite peak intensities to assess the width of the alloy interfaces in InAs/InAsSb Type 2 superlattices (T2SLs). Specifically, our method simulates XRD patterns for T2SLs based on a model of alloy cross-incorporation and fits the simulated pattern to experimental data through a small set of model parameters. We model the Sb distribution function with two forms (i) a Gaussian function, or (ii) two error functions. We compared the model with experimental data extracted from the literature. The first example is a T2SL with 50 periods of 7.0 nm thick InAs and 2.3 nm thick InAs1-xSbx with the targeted alloy composition of x = 0.23. The second example is a T2SL with 100 periods of 4.6 nm thick InAs and 1.7 nm thick InAs1-xSbx with the targeted composition of x = 33.3%. The width of the alloy interface is about 2.54 nm for the first example with the Gaussian model, and with the error function model is 2.8 nm. In the second example, we observed a lower width of the alloy interface of about 0.48 nm with the Gaussian model, and 0.5 nm with the error function model.

Comparison of different power distributions for designing freeform progressive addition lenses based on a minimization error function model

We compared different power distributions to design freeform progressive addition lenses (PALs) based on a minimization error function model. We employed straight line, trigonometric, eighth-order polynomial, and directly assigned function curvature laws to process the power distributions over an entire surface and assessed their effects on the PAL design. Four power distribution techniques were constructed to connect far vision and near vision areas. Correspondingly, four PALs were designed, simulated, and machined. The results showed that power distribution from the far vision point to the near vision point, according to the straight line curvature law, caused errors of 0.35 and 0.6 D in the distance and addition power, respectively. The error was reduced by the trigonometric curvature law and further reduced under the eighth-order polynomial curvature law. The assigned curvature law had the lowest error among the four lenses. These results indicate that the outcome of the PAL design is sensitive to different power distributions. The proposed method is expected to help advance the design procedure optimization of PALs in optometry.

Oxygen Diffusion Modeling in Titanium Alloys: New Elements on the Analysis of Microhardness Profiles

This study focuses on the diffusion of oxygen in titanium alloys during high-temperature oxidation. In particular, the model used to obtain thermokinetic coefficients from microhardness profiles was investigated. A literature review shows that microhardness profiles are modeled by a simple error function in the same way as oxygen concentration profiles obtained by microprobe analysis (EPMA). The analysis of literature shows that the hypothesis of a linear relationship between microhardness and oxygen content is not true over the entire oxygen concentration range and that a parabolic relationship is empirically more accurate. A new modeling equation taking into account this parabolic law is proposed as well as a simplified and easier to use form. The relative error of the diffusion coefficients obtained using the simplified equation was then determined. This new model was applied to the experimental microhardness profile of a Ti-6242s sample oxidized at 625 °C. The resulting oxygen diffusion coefficient is in excellent agreement with the one determined from EPMA profile using the classic error function model. Finally, other data from literature were analyzed with the new model to obtain an Arrhenius diagram of oxygen diffusivity in Ti-64 alloy between 550 and 850 °C. This diagram gives thermokinetic coefficients D0=1.1×10−5exp(−191kJ/molRT) that are close to those reported for pure α-Ti in the temperature range 550–850 °C

Empirical development of parsimonious model for international diffusion of residential solar

We develop a new parsimonious model of residential solar diffusion that, with only two regression parameters and one independent variable, reasonably explains empirical observations. Additional solar customers resulting from an increase in Net Present Value (NPV) are modeled as a normal distribution. This leads to adoption as a function of NPV being the integral of the Gaussian, producing the error function, which demonstrates S-curve behavior commonly seen in technology diffusion. Empirical analysis for five regions (three U.S. states: Arizona, California, and Massachusetts; and two countries: Germany and Japan) from 2005 to 2016 shows a consistent relationship between annual adoption per million households and NPV. Non-linear regression indicates good agreement between data and the error function model, the adoption rate peaking at an NPV of $7100/kW with standard deviation of $4110/kW. Consumer purchases of rooftop solar across multiple regions are explained with a single variable, making this model simpler than traditional diffusion approaches. A novel implication of the model is that the subsidy cost to stimulate additional solar adoption increases as the technology becomes cheaper. This is because the same subsidy is paid to all consumers, including those who would have purchased solar without subsidy.

Brilliant Green Dye Sorption onto Snail Shell-rice Husk: Statistical and Error Function Models as Parametric Isotherm Predictors

Brilliant Green Dye Sorption onto Snail Shell-rice Husk: Statistical and Error Function Models as Parametric Isotherm Predictors

Depassivation time estimation in reinforced concrete structures exposed to chloride ingress: A probabilistic approach

Estimation of depassivation time is a key issue in corrosion prevention. A method to get a probabilistic model from a deterministic one is presented and applied to three simple models: square root of time (SRT), error function (EF) and constant flux (CF) models. Probability distributions of the involved random variables are needed as input parameters. Experimental data have been obtained from a concrete structure exposed to the atmospheric marine environment. These data are analysed to obtain the probability distributions of chloride transport parameters: penetration velocity k (SRT model), diffusion coefficient D (EF and CF models), surface chloride concentration CS (EF model), and chloride ingress flux J (CF model). These distributions are used to calculate the depassivation time probability distributions according to the three models and the orientation of the samples respect to the sea. This allows to estimate depassivation time for a given depassivation probability.

The log TSH-free T4 relationship in a community-based cohort is nonlinear and is influenced by age, smoking and thyroid peroxidase antibody status.

The TSH-T4 relationship was thought to be inverse log-linear, but recent cross-sectional studies of selected populations report a complex, nonlinear relationship. The TSH-T4 relationship has not been evaluated in an unselected, community-based cohort, and there are limited data regarding clinical factors which affect it. To analyse the TSH-free T4 relationship in a community-based cohort. In a cross-sectional, retrospective study, we analysed serum TSH and free T4 concentrations from 4427 participants (55% female) in the 1994 Busselton Health Study who were not taking thyroxine. Simple linear, segmented-linear and nonlinear regression models of log10 TSH on free T4 were compared for goodness of fit. All 5 log TSH-free T4 models tested (separate lines, segmented conterminal line, quartic, error function, double-sigmoid curve) fitted significantly better than a simple linear model (each P < 0·01 by Vuong test). Ranking by Akaike information criterion indicated that the segmented conterminal line and double-sigmoid models provided best fit, followed by the error function, quartic and separate lines models. From multiple regression analysis, age tertile, current smoking and TPOAb status each significantly influenced the TSH-free T4 relationship, whereas BMI category and diabetes did not. A sex difference in the TSH-free T4 relationship was apparent only in the lower part of the free T4 reference range. In a community-based setting, the relationship between log TSH and free T4 is complex, nonlinear and influenced by age, smoking and TPOAb status.

Dosimetric comparison between the prostate intensity-modulated radiotherapy (IMRT) and volumetric-modulated arc therapy (VMAT) plans using the planning target volume (PTV) dose–volume factor

AbstractBackgroundWe demonstrated that our proposed planning target volume (PTV) dose–volume factor (PDVF) can be used to evaluate the PTV dose coverage between the intensity-modulated radiotherapy (IMRT) and volumetric-modulated arc therapy (VMAT) plans based on 90 prostate patients.PurposePDVF were determined from the prostate IMRT and VMAT plans to compare their variation of PTV dose coverage. Comparisons of the PDVF with other plan evaluation parameters such as D5%, D95%, D99%, Dmean, conformity index (CI), homogeneity index (HI), gradient index (GI) and prostate tumour control probability (TCP) were carried out.Methods and materialsProstate IMRT and VMAT plans using the 6 MV photon beams were created from 40 and 50 patients, respectively. Dosimetric indices (CI, HI and GI), dose–volume points (D5%, D95%, D99% and Dmean) and prostate TCP were calculated according to the PTV dose–volume histograms (DVHs) of the plans. All PTV DVH curves were fitted using the Gaussian error function (GEF) model. The PDVF were calculated based on the GEF parameters.ResultsFrom the PTV DVHs of the prostate IMRT and VMAT plans, the average D99% of the PTV for IMRT and VMAT were 74·1 and 74·5 Gy, respectively. The average prostate TCP were 0·956 and 0·958 for the IMRT and VMAT plans, respectively. The average PDVF of the IMRT and VMAT plans were 0·970 and 0·983, respectively. Although both the IMRT and VMAT plans showed very similar prostate TCP, the dosimetric and radiobiological results of the VMAT technique were slightly better than IMRT.ConclusionThe calculated PDVF for the prostate IMRT and VMAT plans agreed well with other dosimetric and radiobiological parameters in this study. PDVF was verified as an alternative of evaluation parameter in the quality assurance of prostate treatment planning.

Modeling of the Positioning Error in Ultra-Precision Machine Tools

The positioning error of ultra-precision machine tools is the primary factor directly affecting the machining accuracy. It mainly depends on the machine tools guide. And its variation trend contains the same inertia as that of the machine tools movement. The variation trend of the positioning error is the derivative of a function fitted with the positioning errors of three adjacent testing points. The variation trends function is fitted with the variation trend of many points. The model of positioning error is the integral of the variation trends function. Because the variation trend of positioning error has certain inertia, the model of positioning error which is obtained in this way is more precise and more reliable than error function model which is directly fitted with limited discrete data.

Modeling boundary and edge effect of chloride diffusion for durability design of concrete structures exposed in marine environment

According to the Fick’s second law of diffusion, six analytical solutions of chloride profile in concrete were studied and discussed with regard to different boundary and initial conditions. In those analytical solutions, the most prevailing error-function solution which is based on semi-infinite assumption is the simple one, but may under-estimate the chloride content in concrete and over-rate the life time prediction of concrete structures. The experimental results show that compared with other solutions, the chloride content in concrete predicted by error-function model is the minimum, and the calculation difference produced by different analytical models should not be ignored. The influence of models on chloride content prediction is more than other environment and material coefficients in some time. In order to get a more realistic prediction model, modification to error-function model is suggested based on analysis and calculation examples concerning the boundary and edge effect.

Adsorption of ferrocyanide onto raw and acid-activated clinoptilolite and sepiolite: equilibrium modelling by error minimization

AbstractThe experimental data on adsorption of Fe and CN of a ferrocyanide complex onto raw and acid-activated clinoptilolite/sepiolite on the basis of detention time and particle size was modelled by a linear and a non-linear approach. The linearized best-fit isotherm selection method and non-linear error minimization was applied through Freundlich, Langmuir and Temkin isotherms. ERRSQ, MPSD, HYBRID and ARE error functions were minimized by a developed MATLAB script to determine the isotherm parameters in non-linear optimization. The complex was not adsorbed as whole anions but the Fe and CN were adsorbed separately. 0.65 mg Fe/L. min and 4.84 mg CN/L. min initial adsorption rates were achieved with acid activated clinoptilolite. The Fe adsorption was not as successful as CN. The adsorption of Fe and CN was described by Freundlich and Langmuir isotherms respectively. The differences between the predicted isotherm parameter sets of linear models and minimized error function models indicated that both the best-fit isotherm selection and the isotherm constant determinations can be performed properly by error minimization as well as by conventional linear best fit modelling approach.