The need for a more realistic model follows from the chemical complexity of outdoor contamination. Insulator surfaces collect soluble salts along with non-soluble deposits and are repeatedly wetted by fog, dew and drizzle. In a wet layer, dissolved salts ionize and provide conductive paths for leakage current. NaCl, a highly soluble salt commonly used in artificial pollution tests, behaves differently from CaSO4, whose limited dissolution causes conductivity to approach saturation as its concentration rises. Earlier work showed that flashover voltage can vary substantially with salt composition even at the same equivalent salt deposit density, but dynamic changes in pollution-layer conductivity during heating and evaporation had not been adequately represented.

To characterize that behavior, the team measured the electrical conductivity of analytically pure CaSO4 and NaCl solutions with purities above 99%, as well as mixtures containing different proportions of the two salts. Solutions were heated from 20 to 70 degrees Celsius in five-degree intervals under continuous stirring, with conductivity recorded by an instrument spanning 0.00 microsiemens per centimeter to 200 millisiemens per centimeter. The maximum water-bath temperature was limited to 70 degrees Celsius to limit evaporation-related changes in solution concentration. CaSO4 conductivity followed an exponential saturation curve with concentration, whereas NaCl conductivity increased approximately linearly. Global fits produced coefficients of determination, or R-squared values, of 0.995 for the CaSO4 relationship and 0.997 for the NaCl relationship.

The fitted equations separated the effects of temperature and concentration within the experimental range. For CaSO4, the limiting saturation conductivity was 2.285 millisiemens per centimeter, the dissolution-rate constant was 0.519 liters per gram and the temperature coefficient was 0.0199 per degree Celsius. For NaCl, the conductivity constant was 1.419 and the temperature coefficient was 0.0238 per degree Celsius. Mixed solutions conducted less electricity than a simple sum of their single-component conductivities, requiring a fitted suppression term for interactions among the ions. The mixed-solution interaction constant was 0.01941, with an equivalent temperature coefficient of 0.01891 per degree Celsius and an R-squared value of 0.993. The authors attribute the reduced mixed-solution conductivity to stronger electrostatic interactions involving multivalent calcium and sulfate ions, increased opportunities for oppositely charged ions to form neutral pairs or charged complexes, and the resulting loss of free conductive ions.

The equations were then embedded in a full-process discharge model for an XP-160 porcelain suspension insulator. Rather than using a simplified rectangle, the researchers used a true-form unfolding method that retained the insulator’s actual geometry and converted its creepage path into a planar equivalent. The model divided the surface into differential segments, each with a water-film thickness, temperature and conductivity that could change during the calculation. It coupled leakage-current heating to moisture loss using Dalton’s evaporation law and a balance between Joule heating and evaporative cooling. A dry band formed when a segment’s liquid layer reached zero thickness, concentrating the electric field. Arc inception was triggered when the local field exceeded the adopted surface-air breakdown threshold of 5 kilovolts per centimeter.

After inception, the calculation represented a single arc in series with the remaining pollution-layer resistance, following the Obenaus formulation for an alternating-current discharge. Nonlinear leakage current was solved with the Newton-Raphson method, while the Claverie condition was used to determine whether an arc could reignite. Hampton’s energy criterion and a local arc-propagation criterion assessed whether the discharge could continue. The numerical setup used a spatial increment of 0.1 centimeters, a time increment of 1 millisecond and a voltage increment of 10 volts. For validation, the researchers compared the model with artificial-pollution flashover results from seven-unit insulator strings, converting measured voltages to equivalent single-unit values because flashover voltage was reported to vary approximately linearly with string length for strings of up to seven units.

The validation covered salt deposit densities of 0.06, 0.10 and 0.25 milligrams per square centimeter and mixtures containing 0% to 100% NaCl, with the remainder CaSO4. Across the reported comparisons, calculated flashover voltage decreased as salt deposit density increased, and the calculated values followed a negative power relationship with an R-squared above 0.995. The maximum relative difference between model and test values was 10.0%, while the average was 4.26%. A separate comparison showed why the saturation effect matters: when CaSO4 concentration was modeled as approximately linear below 2 grams per liter, the model predicted lower flashover voltages and increasing divergence from experimental results as salt deposit density rose. Including saturation therefore produced the closer match within the studied range.

Simulations at an ambient temperature of 20 degrees Celsius, a non-soluble-deposit to salt-deposit ratio of 3.5 and an equal salt mixture revealed a sequence of events near the steel pin, where current density and Joule heating were greatest. Conductivity first increased because heating accelerated ion motion and evaporation increased concentration, but it then fell sharply as the aqueous network disappeared. The electric field remained nearly stable during early drying before surging as a dry band approached. It crossed the breakdown threshold, initiating a partial arc. Higher salt deposit density increased initial leakage current, accelerated evaporation and shortened dry-band formation and arc-inception times. After inception, the arc initially grew almost linearly in length and accelerated sharply just before full flashover; higher salt deposit density also reduced the time needed for propagation.

Composition affected the sequence in more complicated ways. At the same salt deposit density, mixtures with more CaSO4 began with lower conductivity but required a higher critical flashover voltage. Near the end of drying, the larger voltage produced a stronger late-stage electric field and more concentrated Joule heating, allowing arc inception earlier in some conditions. Once the arc existed, however, greater CaSO4 content made continuation more demanding because it disrupted conductive continuity in the residual layer, and the arc required longer to propagate. Immediately before complete flashover, the more severe field distortion in CaSO4-rich layers generated a stronger driving force and a higher instantaneous peak propagation velocity. The study therefore presents a modeling framework rather than a field trial: it is based on laboratory solution measurements, a specified porcelain insulator geometry and comparison with artificial-pollution tests, and its applicability to other designs, natural contaminants or operating environments would require further evaluation.

Subject of Research: Dynamic modeling of AC flashover on insulators polluted with calcium sulfate and sodium chloride

Article Title: An AC dynamic discharge model for insulators considering the conductivity concentration saturation characteristics of slightly soluble pollution

Article References: Yang, Z., Zhou, J., Zhang, F., Chen, C., & Zeng, X. (2026). An AC dynamic discharge model for insulators considering the conductivity concentration saturation characteristics of slightly soluble pollution. Results in Engineering, 32, Article 113485. https://doi.org/10.1016/j.rineng.2026.113485

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113485

Keywords: insulator flashover, calcium sulfate, sodium chloride, pollution conductivity, AC discharge, dry-band formation, arc propagation, leakage current, power transmission, dynamic modeling, dynamic, discharge

Researchers Van Thuy Tran, Thi Thanh Truc Nguyen, and Huu Loc Nguyen, affiliated with Ho Chi Minh City University of Technology and Pham Van Dong University in Vietnam, developed a reliability-based framework that combines Monte Carlo simulation with Response Surface Methodology to determine exactly how much adjustment a V-belt drive actually needs. Their central insight is deceptively simple: the geometric parameters that govern a belt drive, including the belt length and the pitch diameters of the driver and driven pulleys, are never exactly what the drawings say they are. Manufacturing tolerances, assembly errors, and material deformation introduce small random deviations, and those deviations propagate through the drive geometry in ways that deterministic formulas cannot capture.

The traditional approach divides the total adjustment of the center distance into two components. The first, often called take-up, is the extra travel needed to stretch the belt to its initial tension and to re-tension it as it elongates over time through creep, wear, and permanent deformation. The second, called slack-off, is the allowance that lets the belt slip over the pulleys during installation despite dimensional deviations. Design manuals typically prescribe fixed values for both, derived from empirical rules rather than probabilistic analysis. The Vietnamese team argued that these fixed values are blind to the statistical reality of the components, and they set out to replace them with intervals defined at a specified confidence level of 99.9 percent.

To do this, the researchers first modeled the three key geometric parameters as random variables following normal distributions. The choice of the Gaussian model rests on the central limit theorem: dimensional variations arise from many independent sources, such as machining errors, measurement uncertainty, and assembly variations, so their combined effect tends toward a normal distribution. Working with standard dimensions for classical Type-B V-belts drawn from ISO 4183, ISO 4184, and JIS K 6323, the team adopted baseline uncertainty levels of plus or minus 0.3 percent for pulley diameters and plus or minus 0.5 percent for belt length. These were treated as modelling assumptions rather than prescribed tolerance limits, and each range was interpreted as roughly plus or minus three standard deviations.

Monte Carlo simulation then became the engine of the analysis. In each of millions of iterations, the method draws a random set of values for belt length and both pulley diameters, computes the resulting center distance from the standard geometric relationship, and records the outcome. Repeated thousands or millions of times, this procedure builds up a full probability distribution of the center distance and of the adjustment range, revealing not just an average value but the extreme quantiles that matter for design. However, running the full geometric model millions of times is computationally expensive, which is where Response Surface Methodology enters the picture.

Using a Box-Behnken experimental design with fifteen runs, the team constructed a second-order regression model that approximates the nonlinear relationship between the input parameters and the adjustment distance. The fitted model proved remarkably accurate, explaining 97.45 percent of the variance in the data, with adjusted and predicted coefficients of determination both at 95.6 percent. Variance inflation factors of exactly 1.0 confirmed that the input variables were free of multicollinearity. The surrogate model then allowed Monte Carlo sampling to run at a fraction of the original computational cost, making a reliability-based design calculation practical rather than prohibitive.

The sensitivity analysis that followed produced a clear hierarchy of influence. Belt length emerged as the dominant factor, followed by the driver pulley diameter, while the driven pulley diameter played a weaker but still non-negligible role. Local partial derivatives at the design center quantified this ranking precisely. More striking were the nonlinear effects: the driver pulley diameter showed a concave response peaking near 160 millimeters, and the driven pulley diameter exhibited an asymmetric, non-monotonic trend shaped by a strong quadratic term and its interaction with the driver pulley. Physically, these curvatures arise because changes in pulley diameters simultaneously alter the effective belt length, the wrap angle, and the contact conditions between belt and pulley, producing effects that no linear approximation can faithfully represent.

The numerical example that anchored the study involved a conveyor transmission system with a rated power of 6.02 kilowatts and an input speed of 986 revolutions per minute. A Monte Carlo simulation with five million realizations showed that the installation slack-off adjustment follows a tight Gaussian distribution centered at 22.01 millimeters with a standard deviation of only 0.037 millimeters. At the 99.9 percent confidence level, the required slack-off range spans just 21.90 to 22.12 millimeters. The take-up adjustment, governed by belt elasticity and initial tension, ranged far more widely, from 41.92 to 96.16 millimeters at the same confidence level. A sensitivity check using truncated normal and uniform distributions changed the results by less than 0.13 percent, confirming that the conclusions do not hinge on the assumed distribution type.

The comparison with conventional practice is where the findings become genuinely provocative. The Bando V-belt Design Manual recommends a mechanical slack-off allowance of 31.75 millimeters for the studied configuration, while the conventional empirical calculation yields 35.58 millimeters. The reliability-based result of roughly 22 millimeters shows that about 30 percent of the traditionally prescribed installation margin is simply unused, representing wasted mechanism travel and unnecessary bulk. Meanwhile, the take-up story runs in the opposite direction: the probabilistic upper bound of 96.16 millimeters substantially exceeds the nominal value of about 61 millimeters, suggesting that traditional methods may underestimate how much tensioning travel is truly needed once variability is accounted for. This asymmetry, in which empirical rules simultaneously overestimate one adjustment component and underestimate the other, is a previously unreported insight and a fundamental critique of deterministic design.

To make the framework usable in practice, the researchers compiled lookup tables of upper design values for both adjustment components across standardized belt lengths from 1600 to 2800 millimeters, driver pulley diameters of 140 to 180 millimeters, and transmission ratios from 1.25 to 6.3. These tables convert millions of simulated realizations into numbers an engineer can read directly, without repeating the simulation. The authors acknowledge that experimental validation remains future work, and that the uncertainty levels were modelling assumptions rather than measured tolerances, but the framework is built to absorb application-specific data as they become available. The broader significance extends beyond belts: the study demonstrates how uncertainty can be transformed from a post-analysis afterthought into a direct design variable, offering a template for reliability-informed decisions across mechanical transmission systems where geometry, tolerance, and chance quietly interact.

Subject of Research: Reliability-based determination of adjustable center distance in V-belt drives using Monte Carlo simulation and response surface methodology

Article Title: Reliability-based determination of adjustable center distance in V-belt drives using Monte Carlo simulation and response surface methodology

Article References: Tran, V. T., Nguyen, T. T. T., & Nguyen, H. L. (2026). Reliability-based determination of adjustable center distance in V-belt drives using Monte Carlo simulation and response surface methodology. Heliyon, 12(15), Article e45554. https://doi.org/10.1016/j.heliyon.2026.e45554

Image Credits: AI Generated

DOI: 10.1016/j.heliyon.2026.e45554

Keywords: V-belt drives, Monte Carlo simulation, Response Surface Methodology, reliability-based design, center distance adjustment, uncertainty quantification, Box-Behnken design, sensitivity analysis, power transmission, mechanical design, probabilistic modeling, belt tension