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Concept# Parabola

Summary

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.
The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved. The distance betw

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Due to the variations in the local solidification conditions in typical industrial casting processes, dendrites grow under transient rather than steady-state conditions. In this study, the phase-field method was used to study the evolution of secondary dendrite arms of Fe-0.3 wt.% C alloy during transient directional solidification imposed by decreasing the pulling velocity. We find that the dendrite under transient growth conditions is different from the steady-state dendrite, with smaller selection parameter e and the dendrite envelope inside the parabola scaled by the tip radius. The secondary arms undergo a ripening process in which other secondary arms remelt by shrinking from their tips, rather than by detachment from the primary stalk. The surviving arms are finer than those found under steady-state growth conditions, and the size of the surviving arms decreases with decreasing growth velocity.

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Assyr Abdulle, Doghonay Arjmand, Edoardo Paganoni

This paper aims at an accurate and efficient computation of effective quantities, e.g. the homogenized coefficients for approximating the solutions to partial differential equations with oscillatory coefficients. Typical multiscale methods are based on a micro-macro-coupling, where the macromodel describes the coarse scale behavior, and the micromodel is solved only locally to upscale the effective quantities, which are missing in the macromodel. The fact that the microproblems are solved over small domains within the entire macroscopic domain, implies imposing artificial boundary conditions on the boundary of the microscopic domains. A naive treatment of these artificial boundary conditions leads to a first-order error in epsilon/delta, where epsilon < delta represents the characteristic length of the small scale oscillations and delta(d) is the size of microdomain. This error dominates all other errors originating from the discretization of the macro and the microproblems, and its reduction is a main issue in today's engineering multiscale computations. The objective of this work is to analyze a parabolic approach, first announced in A. Abdulle, D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019, for computing the homogenized coefficients with arbitrarily high convergence rates in epsilon/delta. The analysis covers the setting of periodic microstructure, and numerical simulations are provided to verify the theoretical findings for more general settings, e.g. non-periodic microstructures.

Confronted to resources depletion and the global warming, the humanity has to reduce its fossil fuels dependence. The renewable energy is an interesting option to reach this aim. The most powerful one is the solar energy since in few hours, the Earth receives more energy than the humanity consumes in one year. In order to convert this energy to electricity, the concentrated solar power plants (CSP) are more advantageous than photovoltaics (PV) because of the possibility to store the energy. Indeed, they concentrate with numerous mirrors the sunlight to heat up a ﬂuid which can be stored in a big insulated tank. Then, the heat energy is transferred to a steam which drives a turbine and a generator to produce electricity. Nowadays, this type of power plants has a real potential. The aim of this work is to test and conﬁrm that the CSP engender less environmental impacts than other types of power plants over their entire life cycle including production, operation and end of life. Furthermore, four different technologies of CSP (tower, parabolic trough, Fresnel and dish) are compared in order to ﬁnd the most efﬁcient one for the environment and identify their strong and weak points. Finally, multi-objectives optimizations, which consider economic and thermodynamic aspects, are performed to ﬁnd the optimal conﬁguration for each power plant. In this work, the environmental impacts are calculated by carrying out a life cycle analysis (LCA) with the Impact 2002+ method. It takes into account the damages on resources, climate change, human health and ecosystem quality. According to this study, the results indicate that CSP produce lightly less impacts than PV but much less impacts than fossil fuels energy. The comparisons between the different CSP reveal that the dish technology engenders less damages than other due to its high efﬁciency and less facilities. However, its investment costs are too important to be competitive. From the other hand, Fresnel and parabolic technologies produce too much environmental impacts for lower costs. The weak point of Fresnel is its very low efﬁciency and the disadvantage of parabolic is the use of a speciﬁc synthetic oil which is very harmful for the environment, without speaking of the possible spills. Moreover, the hybridation with natural gas of both technologies reduces their costs but increases considerably the impacts on climate change and resources categories. Finally, the tower technology is the most advantageous CSP regarding the impacts, the electricity production and costs. It does not need hybridation to have low costs and have an important storage capacity to produce electricity during nights and cloudy days. However, the investment is still too important to be more attractive than fossil fuel power plants. Therefore, there are two possibilities: to implement a CO2 tax of around 60$/MWh or to improve CSP technology. For the last option, the steel of the mirrors structure, which is the most harmful material according to LCA results, can be reduced and thus, the impacts and the costs will be lower. It can also be possible to use recycled steel or another material. Another hint is to build CSP in a very sunny regions with a DNI of at least 1750 kWh/m2 /yr with short transmission lines. Finally, it is important to increase as much as possible the efﬁciency of CSP to produce more energy and decrease the environmental impacts too.

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