Parabolic Surface Research Articles

Study on heat transfer aspects of solar aircraft wings for the case of Reiner-Philippoff hybrid nanofluid past a parabolic trough: Keller box method

Solar energy is the major source of heat energy from the sun having immense utilization in technologies like photovoltaic cells, solar energy plates, solar street lights, solar water pumping, and so on. How to maximize the heat energy which is later on used as electrical energy for many purposes in solar aircraft is the hot topic of research in this decade. The current paper is about the study of entropy generation analysis and hybrid nano-solid particles impact on parabolic trough surface collector (PTSC) located inside the solar aircraft wings. homo/heterogeneous reactions have been taken. Non-Newtonian Reiner-Philippoff model along with porous medium and Darcy-Forchheimer effects have been taken into account for the present study. The heat transfer performance of the solar aircraft wings has been enhanced with the inclusion of effects like hybrid nanoparticles, thermal radiations, variable thermal conductivity, and viscous dissipation. By utilizing the proper conversions the demonstrated partial-differential equations are renovated into ordinary-differential formulas and tackled numerically with the utilization of well established numerical scheme termed as Keller-Box method. The impact of dimensionless sundry parameters on velocity, shear stress, temperature, homo/heterogeneous as well as surface drag and heat transfer rates are sketched through tables and figures. A positive variation in thermal radiation, thermal conductivity, and viscous dissipation effects enhances the heat transfer rate inside the solar aircraft wings.

A Validated Study of a Modified Shallow Water Model for Strong Cyclonic Motions and Their Structures in a Rotating Tank

A joint theoretical and numerical study was carried out to investigate the fluid dynamical aspect of the motion of a vortex generated in a rotating tank with a sloping bottom. This study aims at understanding the evolution of strong cyclonic motions on a β-plane in the Northern Hemisphere. The strong cyclonic vortices were characterized by four nondimensional parameters which were derived through a scale analysis of the depth variations of fluid. By simplifying the model flow field and the prototype flow field, respectively, through the conservation of potential vorticity, two sets of dynamic similarity conditions are derived. This study proposed a sophisticated modified shallow water model (MSWM) to investigate the flow features of such strong vortices. A detailed numerical calculation adopted by multidimensional positive definite advection transport algorithm (MPDATA) was carried out to validate those effects considered in the MSWM model, including sloping bottom, parabolic free surface deformation, and viscous dissipation. Close agreements were found between the experimental and numerical results, including the streamlines patterns and the vortex trajectory. Comprehensive simulations for strong cyclonic vortices over different sloping bottoms were investigated to understand the impact of planetary β effect on vortex. The results calculated by MSWM demonstrate a variety of flow features of interactions between the primary vortex and induced secondary Rossby wave wakes that were essential and prominent in environmental geophysical flows.

Solar Radiation on a Parabolic Concave Surface

Curved structures are used in buildings and may be integrated with photovoltaic modules. Self-shading occurs on non-flat (curved) surface collectors resulting in a non-uniform distribution of the direct beam and the diffuse incident solar radiation along the curvature the surface. The present study uses analytical expressions for calculating and analyzing the incident solar radiation on a general parabolic concave surface. Concave surfaces facing north, south and east/west are considered, and numerical values for the annual incident irradiations (in kWh) are demonstrated for two locations: 32° N (Tel Aviv, Israel) and 52.2° N (Lindenberg, Germany). The numerical results show that the difference in the incident global irradiation for the different surface orientations is not very wide. At 32° N, the irradiation difference between the south and north-oriented surface is about 15 percent, and between the south and east surface orientation it is about 9.6 percent. For latitude 52.2° N, the global irradiation difference between the south and north-oriented surface is about 16 percent, and between the south and east orientation it is about 3 percent.

Estimation of sediment thickness by solving Poisson's equation with bedrock outcrops as boundary conditions

AbstractSediment thickness and bedrock topography are vital for the terrestrial hydrosphere. In this study, we estimated sediment thickness by using information from digital elevation models, geological maps, and public databases. We discuss two different approaches: First, the horizontal distances to the nearest bedrock outcrop were used as a secondary function in kriging and cokriging. Second, we applied Poisson's equation to estimate the local trend of the sediment thickness where bedrock outcrops were used as boundary conditions. Differences between point observations and the parabolic surface from Poisson's equation were minimized by inverse modelling. Ordinary kriging was applied to the residuals. These two approaches were evaluated with data from the Øvre Eiker, Norway. Estimates derived from Poisson's equation gave the smallest mean absolute error, and larger soil depths were reproduced better if the local trend was included in the estimation procedure. An independent cross-validation was undertaken. The results showed the best accuracy and precision for kriging on the residuals from Poisson's equation. Solutions of Poisson's equation are sensitive to the boundary conditions, which in this case were locations of the bedrock outcrops. Bedrock outcrops are available for direct observations; hence, the quality of the estimates can be improved by updating input from high-resolution mapping.

The high-temperature oxidation resistance properties of magnesium alloys alloyed with Gd and Ca

This work studied the high-temperature oxidation resistance properties of Mg alloys with Gd and Ca. The oxidation property of Mg-3.5Gd-xCa (x = 0, 0.5, 1.0 and 3.0 wt%) alloys followed parabolic oxidation kinetics at 500 °C. A stable oxide film was formed on the alloy surface and limited further oxidation. The oxide film was composed of Gd2O3, MgO and MgCO3 for Mg-3.5Gd and was mainly CaO and CaCO3 with increasing Ca content for Mg-3.5Gd-xCa alloys. The high oxidation affinity of Ca caused it to be oxidized preferentially and the fast diffusion of Ca in Mg alloys provided a source for the continuous oxidation of Ca at the surface. The parabolic rate constant, grains size and surface roughness of the oxide film increased with the increasing Ca content in Mg-3.5Gd-xCa alloy. The understanding of multi-element alloying on the oxidation behavior of Mg alloy is helpful for the design of multi-element oxidation-resistant Mg alloys.

An algorithm on dimensional range of machining axisymmetric aspheric surface for machine tool

Based on coordinate transformation method, the mathematical model for machining axisymmetric aspheric surface is set up. Parabolic surface is studied. The functional relationships between workpiece parameter, tool parameter, cutting state parameter and machine tool performance parameter are analyzed. The variation laws of key technical indexes such as the position of tool locating in vertical direction, the maximum vector height and aperture of parabolic surface, swing range of tool, the minimum arbor distance of tool locating tool carrier in lateral are deduced. An algorithm on dimensional range of machining axisymmetric aspheric surface for machine tool is presented. It can be estimated swiftly that whether the machine tool could do it before machining axisymmetric aspheric surface by means of inputting structure parameters of axisymmetric aspheric surface and motion ranges of machine tool. The simulation experiment results show the validity of the algorithm, which facilitates the machining of axisymmetric aspheric surface.

On differential invariants of parabolic surfaces

We study equivalence classes of local graphed analytic surfaces $\{ u = F(x,y)\}$ in $\mathbb R^3$ under the action of the special affine group ${\sf SA}_3(\mathbb R)$, assuming that their Hessian matrices $\bigl( \begin{smallmatrix} F_{xx} & F_{xy

Numerical Simulation of Static Seal Contact Mechanics Including Hydrostatic Load at the Contacting Interface

A finite element model of a static seal assembled in its housing has been built and is utilized to study how the seal deforms under varying loading conditions. The total contact load on the sealing surface is balanced by the sealed fluid pressure and the friction between the seal and the housing sidewall perpendicular to the sealing surface. The effect of the sealed fluid pressure between the sealing surfaces was investigated and the simulation showed that the surface profile is distorted due to the hydrostatic pressure. We study the distorted contact profile with varying sealed fluid pressure and propose five parameters to describe the corresponding contact pressure profile. One of these parameters, overshoot pressure, a measure of the difference between maximum contact pressure and the sealed fluid pressure, is an indicator of sealing performance. The simulations performed show different behaviors of the overshoot pressure with sealed fluid pressure for cosinusoidal and parabolic surfaces with the same peak to valley (PV) value.

Parabolic revolution surfaces of finite type in simply isotropic 3-spaces

In this paper, we classify parabolic revolution surfaces in the three-dimensional simply isotropic space [Formula: see text] under the condition [Formula: see text] where [Formula: see text] is the Laplace operator with respect to first and second fundamental form and [Formula: see text], [Formula: see text] are some real numbers. Also, as an application, we give some explicit examples for these surfaces.

Focusing of Light Pulses by Different Types of Wavy Parabolic Mirrors

The propagation of an ultra-short light pulse is studied in the framework of scalar diffraction theory. Light pulses are focused by different types of wavy parabolic surfaces. The temporal-spatial behavior of the two-dimensional wave field is computed in the vicinity of the focal plane. It is shown that the slightly perturbation from the perfect parabolic shape leads a space-time dispersion of the pulse in the neighborhood of the focus.

Laser Scanner-Based 3D Digitization for the Reflective Shape Measurement of a Parabolic Trough Collector

In concentrated solar power technology, the precise shape of the reflective surfaces is crucial for efficiency. Considering the geometry and size of a parabolic trough collector, measuring the actual shape is not trivial and some techniques can only be adopted during the assembly operations, evaluating only the manufacturing and alignment processes. The method proposed and tested in this work exploits a laser scanner-based three-dimensional digitization technique that can be used without any marker or other tools, and is attached to the structure. This technique is particularly suitable for assessing the behavior and the optical efficiency of the collectors under load and for validating a finite element model of the structure. The method defines the shape of the parabolic surface by collecting a 3D point cloud of the parabolic surface using a laser scanner. The measured form can then be compared with the ideal shape obtained from a finite element analysis of the structure subject to the gravity field. The comparison can also be performed when the collector is loaded by known forces or torques, with the finite element model reproducing the actual loading scenario. The object of the case study of this work was a 12 m wide full-scale prototype trough collector manufactured at the Politecnico di Milano. The uncertainty of the 3D measurements, acquiring twelve images in different positions, was verified to be less than 3.6 mm.

Image and Ray Tracing Analysis of a Parabolic Dish Collector to Achieve High Flux on a Solar Volumetric Reactor

The need to achieve a uniform distribution of concentrated solar flux in the photovoltaic, thermal or any other receivers is a common problem; therefore, the optical characterization of the concentration system is necessary to determinate the physical characteristics of the receptors. In this work, a parabolic dish concentrator of 1.65x1.65 m2, developed by research from the University of Arizona, is optically characterized under normal operating conditions, also known as environmental conditions that refer to non-controlled conditions as solar radiation, environmental temperature and wind velocity that could affect slightly, by thermal and mechanical efforts, the distribution profiles of the concentrated solar radiation. The set used for the evaluation consisted of the parabolic mirror and Chilled Lambertian Flat Surface installed in the focal point on the optical axis of the mirror. The evaluation was divided into two parts: a theoretical part that consist on using ray tracing simulation and an experimental part that corresponds to image analysis. The used methodology in this work has been stablish in many researches, so this is a reliable method. The global optical error was 2.3 mrad under normal operating conditions.

Axilenses: Refractive micro-optical elements with arbitrary exponential profiles

We report on the experimental demonstration of refractive micro-optical elements with arbitrary exponential surface profiles. Refractive optical elements such as lenses and axicons have parabolic (power-exponent of two) or conical (power-exponent of one) surface profiles, respectively. Here, we analyze micro-optical elements with non-parabolic surface profiles characterized by both integer and fractional power-exponents between 0.5 and 3.25. The experimentally measured beam shaping properties of the components were compared with numerical simulations and shown to be in excellent agreement. The generalized family of optical elements with non-parabolic power-exponent elements allows variation of the optical performance within the same device form-factor and enhancement of various optical properties, such as the extension of the depth-of-focus or resolution improvement.

Construction of freeform mirrors for an off-axis telecentric scanning system through multiple surfaces expansion and mixing

In this paper, a direct design method for an off-axis two-mirror telecentric scanning system with a linear field of view (FOV) is proposed. A single freeform mirror structure is firstly considered, in which the aberration free geometry of the off-axis parabolic (OAP) surface is leveraged to provide the focusing function and build surface contour of the sub-region on the mirror for each FOV. Multiple OAP surfaces for construction of the freeform mirror are located at an OAP base to satisfy the telecentric condition. The imaging distortion of this single freeform mirror structure is analyzed and found unavoidable due to the unsymmetrical geometry of the OAP base. A freeform reflective corrector is supplemented, and it is constructed from multiple plane surfaces located at a curved base to fulfill the f-theta scanning geometry. Thus, a two-mirror structure composed of one freeform primary mirror and one freeform reflective corrector is established. Each plane-OAP surfaces pair corresponds to a specific FOV. These multiple OAP surfaces and multiple plane surfaces are then expanded and mixed respectively, to construct the freeform primary mirror and freeform reflective corrector. An f-theta two-mirror freeform scanning system with ±10.4° linear FOV is designed using the proposed construction method. The design result is diffraction-limited, and a scanning error less than 5 μm and telecentricity angle less than 0.2° are achieved.

Low-density, W-free Co–Nb–V–Al-based superalloys with γ/γ’ microstructure

Previously-reported metastable γ′-L12 Co3(Nb0.65V0.35) precipitates in the Co–6Nb–6V (at. %) ternary alloy are stabilized by additions of Al, Ti, Ni and Cr to the alloy. We identify two such multinary γ/γ′ alloys - with compositions of Co–10Ni–6Ti–5Al-xCr-3Nb–3V-0.04B at.% (with x = 0 and 4% Cr) - with γ′-precipitates remaining stable for up to 1000 h at 850 °C, with no additional phases present. Decreasing the Ti concentration from 6 to 2%, two more γ/γ′ superalloys - Co–10Ni–5Al-xCr-3Nb–3V–2Ti-0.04B (with x = 4 and 8% Cr) – are created with stable γ′-precipitates (measured for 168 h at 850 °C) with morphologies more cuboidal than for the first two alloys with 6%Ti. These submicron cuboidal γ′ precipitates are arranged into crystallographically-oriented sheets with small (<50 nm) γ′-spacing within sheets and larger (~100 nm) γ′-spacing between sheets. The alloy with the highest 8% Cr concentration shows, after aging at 850 °C for 24 h, γ′-nanoprecipitates with (Co0.85Ni0.15)3(Ti0.13Al0.25 Nb0.24Cr0.21V0.16B0.01) composition (assuming full segregation onto the two sublattices), with Al and Ti replacing at a similar rate both Nb and V in the ternary Co3(Nb0.65V0.35). This high-Cr alloy exhibits the best oxidation resistance, as seen by a reduction in the parabolic growth rate constant and surface oxide thickness. Also, the present W- and Ta-free, Cr-containing superalloys show good creep resistance at 850 °C, which is comparable to other recent Cr-containing Co-base γ/γ′ superalloys with higher densities: (i) W-containing Co–9W–9Al–8Cr (at.%), and (ii) Ta-containing Co–10Ni–5Al–4Cr–3Ta–3V–2Ti-0.04B (at.%). This is the first report of a family of Co–Nb–V–Al-based γ/γ′-superalloys with low density (<8.0 g/cm3).

The Projection Technique for Two Open Problems of Unconstrained Optimization Problems

There are two problems for nonconvex functions under the weak Wolfe–Powell line search in unconstrained optimization problems. The first one is the global convergence of the Polak–Ribiere–Polyak conjugate gradient algorithm and the second is the global convergence of the Broyden–Fletcher–Goldfarb–Shanno quasi-Newton method. Many scholars have proven that the two problems do not converge, even under an exact line search. Two circle counterexamples were proposed to generate the nonconvergence of the Polak–Ribiere–Polyak algorithm for the nonconvex functions under the exact line search, which inspired us to define a new technique to jump out of the circle point and obtain the global convergence. Thus, a new Polak–Ribiere–Polyak algorithm is designed by the following steps. (i) Given the current point and a parabolic surface is designed; (ii) An assistant point is defined based on the current point; (iii) The assistant point is projected onto the surface to generate the next point; (iv) The presented algorithm has the global convergence for nonconvex functions with the weak Wolfe–Powell line search. A similar technique is used for the quasi-Newton method to get a new quasi-Newton algorithm and to establish its global convergence. Numerical results show that the given algorithms are more competitive than other similar algorithms. Meanwhile, the well-known hydrologic engineering application problem, called parameter estimation problem of nonlinear Muskingum model, is also done by the proposed algorithms.

Compensation Measurement Study for Aspherical Primary Mirror of Space Camera with CGH

With the development of modern optical technology, demands on the performance of optical systems are increasing. Aspherical surfaces are widely used in space optical system considering its excellent ability in correcting aberrations, reducing the optical system weight and simplifying the optical construction. However, the interferometry of this kind of aspherical surface is difficult. To overcome the difficulty in testing aspherical primary mirror of space camera system, we provide a compensation method to accomplish the null testing of the aspherical primary mirror with CGH (Computer generated hologram). Our proposed method is introduced in detail in the theory part. In addition, we provide a detailed design for an actual parabolic surface. The results show that our proposed method can accomplish the testing of aspherical primary mirror in large aperture with satisfactory accuracy.

Diamond grinding of carbide punches by grinding wheels with a multilayer composite electrolytic coating

The finish diamond grinding of a parabolic surface of a hard alloy punch with a grinding wheel with a multilayer composite electrolytic coating is considered. The cutting forces acting on a single grain are calculated. The cutting conditions that provide high tool life and the required quality of the machined surface are determined. Keywords diamond grinding, grinding wheel, multilayer composite electrolytic coating, quality, machined surface, cutting forces. katapu@mail.ru

A Finite Element Method for a Fourth Order Surface Equation With Application to the Onset of Cell Blebbing

A variational problem for a fourth order parabolic surface partial differential equation is discussed. It contains nonlinear lower order terms, on which we only make abstract assumptions, and which need to be defined for specified problems. We derive a semi-discrete scheme based on the surface finite element method, show a-priori error estimates, and use the analytical results to prove well-posedness. Furthermore, we present a computational framework where specific problems can be conveniently implemented and, later on, altered with relative ease. It uses a domain specific language implemented in Python. The high level program control can also be done within the Python scripting environment. The computationally expensive step of evolving the solution over time is carried out by binding to an efficient C++ software back-end. The study is motivated by cell blebbing, which can be instrumental for cell migration. Starting with a force balance for the cell membrane, we derive a continuum model for some mechanical and geometrical aspects of the onset of blebbing in a form that fits into the abstract framework. It is flexible in that it allows for amending force contributions related to membrane tension or the presence of linker molecules between membrane and cell cortex. Cell membrane geometries given in terms of a parametrisation or obtained from image data can be accounted for by the software. The use of a domain specific language to describe the model makes is straightforward to add additional effects such as reaction-diffusion equations modelling some biochemistry on the cell membrane. Some numerical simulations illustrate the approach.

Flexible Dual-Band Dual-Beam Radiation of Reflector Antennas by Embedding Resonant Phase Alignment Elements for Power Refocusing

This article presents a dual-band composite reflector antenna by embedding an array of metallic phase alignment elements (PAEs) on a selected area of reflector. As a result, this phase-configurable reflector may radiate two relatively arbitrary beams at two widely separated frequency bands, respectively. In this concept, a conventional parabolic surface is first created for ordinary field radiation by a focal feed in the low-frequency band. Afterward, a feed of high-frequency band is offset from the focal point to result in defocused radiation due to the phase distortion of reflection from the parabolic reflector. Metallic PAEs are subsequently implemented on the reflector to configure their phases of reflections for distortion compensation, and thus to refocus the radiation power along a desired direction. Here, the resonance of PAEs also specifies a valid high-frequency range. In this article, the methodology is first presented with radiation discrepancy evaluated to validate the design concept, which is further validated by full-wave simulations. Numerical examples are presented to demonstrate the feasibility.