Range Of Aspect Ratios Research Articles

A Three-Dimensional Modeling Approach for Carbon Nanotubes Filled Polymers Utilizing the Modified Nearest Neighbor Algorithm.

Carbon nanotubes (CNTs) are extensively utilized in the fabrication of high-performance composites due to their exceptional mechanical, electrical, and thermal characteristics. To investigate the mechanical properties of CNTs filled polymers accurately and effectively, a 3D modeling approach that incorporates the microstructural attributes of CNTs was introduced. Initially, a representative volume element model was constructed utilizing the modified nearest neighbor algorithm. During the modeling phase, a corresponding interference judgment method was suggested, taking into account the potential positional relationships among the CNTs. Subsequently, stress-strain curves of the model under various loading conditions were derived through finite element analysis employing the volume averaging technique. To validate the efficacy of the modeling approach, the stress within a CNT/epoxy resin composite with varying volume fractions under different axial strains was computed. The resulting stress-strain curves were in good agreement with experimental data from the existing literature. Hence, the modeling method proposed in this study provides a more precise representation of the random distribution of CNTs in the matrix. Furthermore, it is applicable to a broader range of aspect ratios, thereby enabling the CNT simulation model to more closely align with real-world models.

Modeling of equivalent strain in 2D cross-sections of open die forged components using neural networks

Open die forging is one of the oldest manufacturing methods known to remove defects in the ingot resulting from the casting process. The improved properties of the final component are highly dependent on the strain distribution. Although sinusoidal equations and empirical formulations have been already used to estimate the strain, they have been applied only to the core of the workpiece. In this work, a novel approach is presented to model the equivalent strain distribution in 2D cross-sections, in the direction of the press, of open die forged components using neural networks. The proposed method efficiently combines a parametric sinusoidal function with a neural network to learn the complex relationships between the process parameters and the resulting local strain. The neural network is trained on a dataset of finite element (FE) simulations of rectangular geometries that cover a wide range of aspect ratios, bite ratios, and height reductions. The presented methodology with near real-time prediction capabilities shows good agreement with FE results. Moreover, the parametric function captures the characteristic pattern of the strain distribution and reveals certain physical relationships affecting the deformation of the material. These patterns are later examined by analyzing the parameters identified in the parametric sinusoidal function.

Hydrodynamic force coefficients for spherical triangle shell fragments: Dependence on the aspect ratio and flatness

Euler–Lagrange simulations of particle-laden flow require hydrodynamic models of drag and lift forces for individual particles. Our goal is to develop models that can prescribe these forces for arbitrarily orientated shell objects. Here, we use computational fluid dynamics simulations of steady bottom-boundary layer flow over a series of spherical triangle shell fragments to calculate the hydrodynamic forces. The simulations explicitly resolve the wall boundary layers using grid resolution on the order of y+=1 at the shell fragment surface and use the SST k-omega turbulence closure model. These fragments cover a range of aspect ratio and flatness characteristics. The shell fragments are generated as triangular selections of a spherical shell with azimuthal and longitudinal angles proscribed based on elongation and flatness parameters (varying between 1 to 5, and 0.02 to 0.2 respectively), while characteristic length of the fragment is held constant to define the overall fragment size. Fragment orientations are considered with independently varying pitch, roll, and yaw each ranging from 0 to 180 degrees. The numerical estimates for the forces from all simulations were used to develop robust parameterizations of the drag and lift as a function of aspect ratio and flatness characteristics, as well as orientation of the shell fragments.

Modelling aerodynamic forces and torques of spheroid particles in compressible flows

In the present study, we conduct numerical simulations of compressible flows around spheroid particles, for the purpose of refining empirical formulas for drag force, lift force, and pitching torque acting on them. Through an analysis of approximately a thousand numerical simulation cases spanning a wide range of Mach numbers, Reynolds numbers and particle aspect ratios, we first identify the crucial parameters that are strongly correlated with the forces and torques via Spearman correlation analysis, based on which the empirical formulas for the drag force, lift force and pitching torque coefficients are refined. The novel formulas developed for compressible flows exhibit consistency with their incompressible counterparts at low Mach number limits and, moreover, yield accurate predictions with average relative errors of less than 5%. This underscores their robustness and reliability in predicting aerodynamic loads on spheroidal particles under various flow conditions.

Recycled ionic liquid vs. deep eutectic solvent in cellulose nanocrystals production: Characterization, techno-economic analysis, and life cycle assessment

Three scenarios involving aqueous mixtures of neoteric solvents have been evaluated as solvents and catalysts for the cellulose hydrolysis reaction to obtain CNC in high yields (>70%). The scenarios considered were: scenario 1 (S1) involves a recyclable ionic liquid dilution of [Hmim][(HSO4)(H2SO4)]/H2O (64 wt% IL); scenario 2 (S2) another recyclable dilution of [Hmim][(HSO4)(H2SO4)]//H2O (80 wt% IL) and scenario 3 (S3) a non-recyclable ternary deep eutectic solvent (60 wt% DES: choline chloride: oxalic acid/30 wt% PA/10 wt% water). Experimental analysis, techno-economic, and life cycle analysis (LCA) indicate that S1 emerges as a suitable scenario among the three analyzed. S1 demonstrated the highest competitiveness, with a lower raw material cost per gram of CNC produced (US$0.81/g) and lower environmental contributions concerning climate change and fossil fuel depletion, accounting mainly for its recyclability. Therefore, the recyclable dilution of 64 wt% IL used in S1 appears to be the most viable option for sustainable and environmentally conscious CNC production. Besides, the physicochemical properties of the CNC were revealed: aspect ratio range 7–8, high dispersion stability related to zeta potential > −40 mV, good crystallinity range 50–80%, and thermal stability with Tonset> 300 °C. These results guided towards a scenario with crucial advantages: a diluted IL that retains its acidity after reuse cycles by a facile recovery process, maintaining high CNC production performance and providing low environmental impacts.

Effect of humidity interference on NO2 gas sensing of In2O3 nanoneedles at moderate operating temperature

This study employs both experimental and computational methodologies to examine the gas-sensing performance of needle-like In2O3 nanosensors towards a model NO2 gas in dry (10 % RH) and wet (99 % RH) environments. The single crystalline nanoneedles exhibit distinct surface facets, primarily (110) or (111) crystalline planes, spanning a wide aspect-ratio range of 3.9–7.0. At a relatively low operating temperature of 50 °C and NO2 gas concentration of 10 ppm, the experimental assessment yields sensing-response ratios (i.e., Swet/Sdry) of 0.19 ± 0.18 and 0.23 ± 0.07 for In2O3 nanoneedles featuring (110) and (111) surfaces, respectively. Notably, density-functional theory computations reveal that the adsorption energies of NO2 molecules on both In2O3 (110) and (111) surfaces significantly weaken under moist conditions. A relatively stronger adsorption affinity is observed on the (110) surface than the (111) counterpart. Our calculated electronic band structures reveal conspicuous changes in conductivity upon the adsorption of NO2 and water vapor on the In2O3 surfaces. The observed variations in conductivity, in conjunction with the adsorption-energy calculations, offer valuable insights into the structural mechanisms underlying the experimentally observed sensing-response alterations in humid environments.

Investigation on sheathless inertial focusing within low-aspect ratio spiral microchannel for cascaded microfluidic tumor cell separation

The high-precision and high-purity isolation of circulating tumor cells (CTCs) from whole blood is vital to early cancer detection. Cascaded microfluidic separation is highly efficient because it connects multiple-stage separations in series. Here, we numerically investigated sheathless tumor cell separation with size-dependent cascaded inertial and deterministic lateral displacement (DLD) microfluidic device. The inertial microfluidic is arranged in the first-stage unit for particle focusing and rough sorting, and the cascaded DLD microfluidic is arranged in the second stage for realizing further sorting and purification. A parametric study with flow rate range from 100–600 μl/min and aspect ratio range from 60:100 to 60:300 of the first stage was carried out to optimize channel structure for realizing high-efficiency separation. Then, the pre-separation mechanism within the spiral microchannel was analyzed. The purity of the obtained CTCs and the separation efficiency were further improved using a droplet-type microcolumn DLD microfluidic device as the second unit. The cascade eliminates the need for additional force fields and reduces device complexity while simplifying operation and reducing the chance of sample contamination.

TSAF-Net: a rotated two-stage Cnaphalocrocis medinalis damage detection method based on anchor-free arbitrary-oriented proposal network.

Cnaphalocrocis medinalis (C.medinalis) is an agricultural pest with recurrent outbreaks. The investigation into automated pest and disease detection technology holds significant value for in-field surveys. Current generic detection methods are inadequate due to arbitrary orientations and a wide range of aspect ratios in damage symptoms. To tackle these issues, we put forward a rotated two-stage detection method for in-field C.medinalis surveys. This method relies on an anchor-free rotated region proposal network (AF-R2PN), bypassing the need for hyper-parameter optimization induced by predefined anchor boxes. An in-field C.medinalis dataset is constructed during on-site pest surveys to validate the effectiveness of our method. The experimental results show that our method can accomplish 80% average precision (AP), surpassing the corresponding horizontal detector by 2.3%. The visualization results of our work showcase its exceptional localization capability over generic detection methods, facilitating inspection by plant protectors. Meanwhile, our proposed method outperforms other state-of-the-art rotated detection algorithms. The AF-R2PN module can generate superior arbitrary-oriented proposals even with a decreased number of proposals, balancing inference speed and detection performance among other rotated two-stage methods. The proposed method exhibits superiority in detecting C. medinalis damage under complex field conditions. It provides greater practical applicability during in-field surveys, enhancing their efficiency and coverage. The findings hold significance for pest and disease monitoring, providing important technical support for agricultural production. © 2024 Society of Chemical Industry.

A semi-analytic method for buoyancy-induced thermal stratification in hot water tanks during standby periods

Buoyancy-induced thermal stratification is a spontaneous phenomenon arising from standby periods of hot water tanks. Accurate estimation of temperature levels is crucial for energy-efficient operations of hot water tanks and associated water heating systems. However, temperature estimation by numerical schemes requires sophisticated algorithms and considerable computation resources. This study proposes a novel semi-analytic method to address buoyancy-induced thermal stratification conveniently without iterative calculations. An explicit water temperature function to tank height and time is analytically derived from the modified energy balance equation where the heat convection term is disregarded. The convection effect is compensated for by artificial enhancement of heat conduction, which is indicated by a synthetic parameter, amplification factor. A comprehensive numerical study is carried out to investigate static and transient characteristics of buoyant flows and temperature distribution in a hot water tank during standby periods. The findings imply a strong analogy between buoyancy-induced thermal stratification and a bottom-initiated heat conduction process, which reinforces the physical rationale for the semi-analytic method. The experimental verification of the semi-analytic method is conducted against published experiment data, the proposed method shows excellent estimation accuracy with the rooted mean square error of less than 0.42 °C. Furthermore, suitable case-specific amplification factors are determined with numerical simulations, and a ready-for-use correlation equation of the amplification factor is formulated for a wide range of tank volumes and aspect ratios. This novel semi-analytic method substantially reduces computation time compared with numerical solvers, and the simple form of the water temperature function can help in formulating more detailed hot water tank models in energy system studies.

Research on the anti-collapse performance of steel frame structures with infilled walls in the form of reduced beam sections connections

Steel has more deformation capacity and resistance to compression buckling than concrete materials, which providing steel frames a distinct edge in preventing collapse. This study examines the anti-collapse performance and mechanism of steel frame structures with infilled walls in the form of reduced beam section connections. It also examines the effects of wall thickness, frame aspect ratio, and connection control size on performance and addresses the applicability of current code formulas in the early deformation of the structure. The study discovered that steel frame structures with infilled walls in the form of reduced beam section connections can, during the failure stage, enhance the infilled wall's contribution to the collapse bearing capacity, thereby stimulating the structure's potential performance, in contrast to those with infilled walls in the form of non-reduced beam section connections. Due to the supporting effect of the double strut that forms inside the infilled wall during the failure stage, steel frame structures with infilled walls that have non-reduced beam section connections have higher deformation capacity than corresponding pure frames; however, this effect also decreases the deformation capacity of the structure with reduced beam section. Furthermore, the infilled walls cause the structure to collapse in a same way across a range of aspect ratios, yet their contribution to the structural bearing capacity differs across these same aspect ratios. It is noteworthy that altering the local connection control size can greatly enhance the structure anti-collapse performance, even in unfavorable aspect ratio scenarios. The study also discovered that the bearing capacity of the structure studied in this paper was overestimated by the FEMA356-2000 code and the MSJC code, which were used to predict the initial bearing capacity of infilled frames. The former's overestimation reaches a maximum of 39 %, while the latter's prediction is only accurate within a limited range of aspect ratios. Furthermore, the formula by Qian and Li, which has a maximum error of 9 % in bearing capacity prediction within the standard range of aspect ratios, is better appropriate for the failure mode of the structure described in this study.

Effects of Impact Energy and Aspect Ratio on the Motion of Particle Clouds in Stagnant Water

Abstract A series of laboratory experiments was conducted to investigate the effects of impact energy and other initial controlling parameters on the motion of particle clouds in stagnant water. Experiments were performed for two median sand diameters of D50 = 0.52 mm and 0.74 mm and nozzle diameters of do = 6 mm and 8 mm. Sand masses were converted to an equivalent pipe length with the same diameter as the nozzle, Lo, and a wide range of aspect ratios, Lo/do, between 2 and 93 was tested. The impact energy of sand particles was controlled by the release height of sand particles, and it was quantified by the non-dimensional release height, η, ranging from 1 to 21.5. It was found that particle clouds with higher impact energy had smaller concentration and velocity decay rates. This indicated that by increasing the release height, the momentum transfer between sand particles and the ambient water decreases. The time-series of instantaneous sand velocity were used to determine velocity fluctuations and turbulence intensity of sand particles and a direct correlation was found between sand velocity fluctuations and aspect ratio in particle clouds. The effects of impact energy on the anatomy of the resulted particle clouds were examined in this study. It was found that the cloud width increased dramatically when the impact energy of sand particles with high aspect ratios (i.e., Lo/do > 39) increased. Furthermore, the dispersion of sand particle began earlier as the kinetic energy of sand particles increased at the water surface.

Numerical simulation and modelling of the drag, lift, and pitching torque acting on cylindrical disc particles

This paper aims to improve existing models for drag, lift, and pitching torque coefficients of cylindrical disc particles with a wider range of aspect ratios and particle Reynolds numbers. The full body-fitted direct numerical simulations (DNS) are carried out for three-dimensional flow past cylindrical discs in order to quantify the coefficients at aspect ratios from 0.2 to 0.6, incidence angles from 0°to 90°, and Reynolds numbers from 0.1 to 300. The CFD results are validated against the aerodynamic coefficients of both spherical and non-spherical particles presented in the literature. A full set of new correlations for the coefficients, dependent on aspect ratio, incidence angle, and Reynolds number, is further derived according to the CFD predicted data. The results show that the new correlations have a promising performance for the disc particles considered in this study.

Analysis of NePCM melting flow inside a trapezoidal enclosure with hot cylinders: Effects of hot cylinders configuration and slope angle

Hot cylinders within the cavity can find diverse thermal applications, ranging from electronic devices and heat exchangers to solar systems, nuclear reactors, and the thermal design of passive cooling systems. Consequently, considering the use of phase change materials (PCM) close to hot cylinders emerges as a viable method for both cooling and storing thermal energy. In this work, two hot cylinders are arranged vertically and horizontally in the nano-enhanced phase change material (NePCM) trapezoidal enclosure with varying angles. The novelty of the current study lies in the fact that no prior research has explored the impact of hot cylinders arrangement on the melting process of a NePCM within a trapezoidal cavity, considering various inclination angles. The fixed grid, adaptive mesh refinement, and enthalpy-porosity method are used to model the melting flow. The numerical results indicate that the trapezoidal enclosure with vertically arranged hot cylinders exhibits better thermal performance than the other configuration. It is also found that the full melting time for both arrangements is minimum when the angle of the trapezoidal enclosure is γ=+40°. The reduction in full melting times for the vertical and horizontal arrangements of this angle are 25.3% and 29.6%, respectively, compared to the trapezoidal enclosure with γ=0.0°. Moreover, the NePCM with the graphite nanoplatelets (GNPs) of φ=1.0% has the shortest melting time for 0.0%≤φ≤3.0%. As future research, further exploration into the complexities of non-Newtonian NePCM flow can be pursued. Additionally, there is potential to explore modifying the cylinders' geometry into elliptical forms, encompassing a range of aspect ratios.

Hydrodynamic force and torque models for cylindrical particles in a wide range of aspect ratios

During the pneumatic conveyance of biomass in a coal-fired power station boiler, biomass particles have cylindrical shapes with different aspect ratios. They move through the fluid at any angle and rotate strongly. However, highly accurate and general models of the drag, lift, and torque coefficients (CD, CL, and CT) for biomass particles in a wide range of aspect ratios, especially the CT model and the high aspect ratios, are currently lacking. This paper presents detailed direct numerical simulations of the flow around cylindrical cylinders with varying aspect ratios (6 ≤ AR ≤ 22), Reynolds numbers (100 ≤ Re ≤ 2000), and angles of incidence (0° ≤ θ ≤ 90°). The simulation was conducted using the OpenFOAM solver with the body-fitted mesh method. The flow characteristics and force coefficients of cylindrical particles with different AR were systematically analyzed. New functional correlations between CD, CL, and CT and AR, Re, and θ values were established. The mean squared errors for CD, CL, and CT were 8.8 × 10–2, 2.4 × 10–2, and 4.7 × 10–2, with average relative errors of 5.8%, 3.5%, and 8.17%, respectively. A comparison of the results with other experimental and simulation data in previous literatures showed that the new CD and CL models have considerable higher predictive ability. The generality of the new CD model expanding to low ARs of 1.5 and 3 is verified finally. The new force and torque models are expected to improve the accuracy of Eulerian–Lagrangian simulations of various cylindrical particle-laden flows in the utility of biomass energy.

Sedimentation of spheroids in Newtonian fluids with spatially varying viscosity

This paper examines the rigid body motion of a spheroid sedimenting in a Newtonian fluid with a spatially varying viscosity field. The fluid is at zero Reynolds number, and the viscosity varies linearly in space in an arbitrary direction with respect to the external force. First, we obtain the correction to the spheroid's rigid body motion in the limit of small viscosity gradients, using a perturbation expansion combined with the reciprocal theorem. Next, we determine the general form of the particle's mobility tensor relating its rigid body motion to an external force and torque. The viscosity gradient does not alter the force/translation and torque/rotation relationships, but introduces new force/rotation and torque/translation couplings that are determined for a wide range of particle aspect ratios. Finally, we discuss results for the spheroid's rotation and centre-of-mass trajectory during sedimentation. A steady orientation arises at long time whose value depends on the viscosity gradient direction and particle shape. These results are significantly different than when no viscosity gradient is present, where the particle stays at its initial orientation for all times. We summarize the observations for prolate and oblate spheroids for different viscosity gradient directions and provide plots for the orientation and centre-of-mass trajectory versus time. We also provide guidelines to extend the analysis when the viscosity gradient exhibits a more complicated spatial behaviour.

Topography prediction of high-aspect ratio features milled with overlapping abrasive slurry jet footprints considering fluid confinement effects

Overlapping adjacent passes of abrasive slurry jets can be used to fabricate micro-pockets or blind micro-slots. This study investigates the propagation of surfaces milled using overlapping passes of a high-pressure abrasive slurry jet on a ductile 6061-T6 target over a wide range of aspect ratios. Features such as shallow pockets with low aspect ratios to deep slots with aspect ratios ∼1.5 were fabricated to investigate the correlations between the aspect ratio, degree of pass overlap, and number of repeating passes with the machined surface evolution characteristics. A comprehensive model was presented that combined computational fluid dynamics (CFD) with surface evolution to predict the evolving machined feature topography after each of the overlapping and repeating passes, and to provide insights into experimental observations. The model explicitly considered the CFD-predicted local particle impact angles and velocities as well as effects due to secondary impacts, high sidewall slopes, and stagnation effects. For larger degrees of overlap, measurements revealed a more than linear rate of increase in depth after each repeat of two adjacent overlapped passes, as opposed to a linear depth increase for smaller overlaps and blind pockets. Large overlaps were found to result in asymmetric features. Up to a critical aspect ratio of ∼0.65, the model predicted the surface evolution of the features to within <8.6% of those measured. Beyond that, randomness in the machined feature shape and size made the process challenging to control and the surface evolution difficult to predict. Nevertheless, the model was able to elucidate the reasons for the randomness and other observed phenomena such as the more than linear growth in etch rate, and the occurrence of feature asymmetry. The findings emphasize that flow confinement, increases in the jet's turbulent kinetic energy, and the formation of vortices at the bottom of the machined features are critical factors influencing the machining process, and that in these cases explicit modeling approaches like the one presented in this study must be used.

Bio-Inspired Magnetically Responsive Silicone Cilia: Fabrication Strategy and Interaction with Biological Mucus.

Cilia are biological structures essential to drive the mobility of secretions and maintain the proper function of the respiratory airways. However, this motile self-cleaning process is significantly compromised in the presence of silicone tracheal prosthesis, leading to biofilm growth and impeding effective treatment. To address this challenge and enhance the performance of these devices, we propose the fabrication of magnetic silicone cilia, with the prospect of their integration onto silicone prostheses. The present study presents a fabrication method based on magnetic self-assembly and assesses the interaction behavior of the cilia array with biological mucus. This protocol allows for the customization of cilia dimensions across a wide range of aspect ratios (from 6 to 85) and array densities (from 10 to 80 cilia/mm2) by adjusting the fabrication parameters, offering flexibility for adjustments according to their required characteristics. Furthermore, we evaluated the suitability of different cilia arrays for biomedical applications by analyzing their interaction with bullfrog mucus, simulating the airways environment. Our findings demonstrate that the fabricated cilia are mechanically resistant to the viscous fluid and still exhibit controlled movement under the influence of an external moving magnet. A correlation between cilia dimensions and mucus wettability profile suggests a potential role in facilitating mucus depuration, paving the way for further advancements aimed at enhancing the performance of silicone prostheses in clinical settings.

NEW INSIGHTS INTO THE WEB PANEL COMPONENT

AbstractThe column web panel is a fundamental component of beam‐column joints. This component has been thoroughly studied and codified in the past. In this paper, new insight on its behaviour is obtained using a large amount of high‐quality Finite Element (FE) models, covering a wide range of panel slenderness and aspect ratios. The study shows that the existing models in part 1‐8 of the current and forthcoming Eurocode 3 present a wide scatter in terms of resistance and stiffness, being unconservative in some specific cases.

Numerical Evaluation of the Elastic Moduli of AlN and GaN Nanosheets.

Two-dimensional (2D) nanostructures of aluminum nitride (AlN) and gallium nitride (GaN), called nanosheets, have a graphene-like atomic arrangement and represent novel materials with important upcoming applications in the fields of flexible electronics, optoelectronics, and strain engineering, among others. Knowledge of their mechanical behavior is key to the correct design and enhanced functioning of advanced 2D devices and systems based on aluminum nitride and gallium nitride nanosheets. With this background, the surface Young's and shear moduli of AlN and GaN nanosheets over a wide range of aspect ratios were assessed using the nanoscale continuum model (NCM), also known as the molecular structural mechanics (MSM) approach. The NCM/MSM approach uses elastic beam elements to represent interatomic bonds and allows the elastic moduli of nanosheets to be evaluated in a simple way. The surface Young's and shear moduli calculated in the current study contribute to building a reference for the evaluation of the elastic moduli of AlN and GaN nanosheets using the theoretical method. The results show that an analytical methodology can be used to assess the Young's and shear moduli of aluminum nitride and gallium nitride nanosheets without the need for numerical simulation. An exploratory study was performed to adjust the input parameters of the numerical simulation, which led to good agreement with the results of elastic moduli available in the literature. The limitations of this method are also discussed.

Theoretical estimation of sound absorption characteristics in a collection of rectangular holes: Comparison with experimental results

This study quantifies the sound absorption properties of small rectangular or square pores that are less than a few millimeters per side, based solely on their geometric dimensions. Analytical solutions for the sound absorption coefficient of narrow tubes with rectangular holes have not been obtained. Therefore, alternative methods of predicting the sound absorption coefficient, such as using the circular tube approximation or the two-plane approximation, are possible. In this study, the sound absorption coefficient was estimated by several theoretical models that are considered useful for estimating the sound absorption coefficient in rectangular tubes. The results were then compared with experimental values. For square holes, the accuracy of the sound absorption coefficient increased in the order of the model considering two-sided circumference only, the circular tube model, and Allard's method based on Stinson's. In rectangular holes, Allard's method based on Stinson's showed good agreement with the experimental data across a wide range of aspect ratios. For rectangular holes, the accuracy of Stinson's two-plane approximation was superior to that of the circular tube approximation when the aspect ratio was 4. Furthermore, when the aspect ratio was 8, the accuracy of Stinson's two-plane approximation became comparable to that of Allard's method based on Stinson's.