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Gravity Concentration
Published in S. Komar Kawatra, Advanced Coal Preparation and Beyond, 2020
Spirals have several positive features, such as having low capital and operating cost, being easy to operate and maintain, and taking up relatively little floor space for a given capacity. For example, a 76 cm (30 in) diameter spiral has a capacity of 2–5 metric tons/h, but only occupies approximately 0.84 m2 (9 ft2) of floor space. They are quite well-suited for removing pyrite from coal, as shown in Table 5.6, provided that the pyrite liberates at a sufficiently coarse size. The particle size range over which spirals are effective is similar to that of tables, with a top size of about 1 cm and a lower limit of approximately 50 µm. Spirals are typically used in conjunction with either desliming cyclones or fine screens, to remove the fines that the spiral cannot handle. The desliming operation can be either before or after the spirals.
Design
Published in Wanda Grimsgaard, Design and Strategy, 2023
The golden spiral is constructed based on the golden ratio principle. We find it in many places in nature, such as in a conch shell, in a wave, in the human ear. The golden spiral has an outer shape with the same proportions as the golden rectangle. The same proportions are repeated in all the rectangles from which the spiral is built, all the way into the core. The golden spiral is proportional both vertically and horizontally, and can be built from the outside in or from the inside out infinitely. Or put in another way: It can be divided into smaller golden rectangles or more golden rectangles can be added. (4.8.5 The golden ratio, 4.9.9 Grid system)
Spiral Dynamics Algorithms
Published in Nazmul Siddique, Hojjat Adeli, Nature-Inspired Computing, 2017
Logarithmic spiral based on Equation 7.23 is shown in Figure 7.10. Logarithmic spiral can be distinguished from Archimedean spiral by the fact that the distance between the arms of a logarithmic spiral increases in a geometric sequence while in Archimedean spiral the distance is constant.
The Conceptual Design of Bridges: Form Finding and Aesthetics
Published in Structural Engineering International, 2021
This ratio has certain algebraic and geometric properties and is a transcendental number similar to “π” (the ratio of the perimeter to the diameter of a circle). The proportions are infinitely divisible, where each subdivision retains its original proportion and is harmonically related not only to whole but to all subdivisions. For example, when a series of Golden Rectangles are assembled on each other and their outside corners are connected by a smooth curve as depicted in Fig. 4, the culmination is a Golden Spiral. Luca Pacoli, friend of Leonardo da Vinci, called this the “divine proportion”, and Kepler called the ratio 1.618 “one of the two jewels of geometry” expressing it as a positive root of x2 = x+1. Thus, the Golden Section satisfies the relationships ϕ2 = ϕ + 1 = 2.618, ϕ3 = ϕ2 + ϕ = 4.236 and so on. Thus it has a mathematical nature of equipartition (symmetry), succession (order) and continuous proportion (regularity), which can give rhythm to any art or physical form.
Analytical confinement model for square section confined with circular ties
Published in Australian Journal of Structural Engineering, 2020
Chen, Feng, and Yin (2012) and Sun et al. (2017) argued that the compressive behaviour of reinforced concrete columns confined by multispiral hoops, which showed the strength and ductility of the column can be greatly enhanced due to better confinement. Spirals, because of their shape, are in hoop tensile zone and provide a continuous confining pressure around the circumference. It is generally accepted that spiral reinforcement results in increased strength and ductility of confined concrete and this has been widely validated by many researchers who have done extensive testing to prove this theory (Sakai and Sheikh 1989; Priestley 1984; Mander, Priestley, and Park 1988; Sheikh 1982; Park, Priestley, and Gill 1982). It is also accepted that concrete confined by rectilinear ties leads to increased ductility but there is a division of opinion on both the magnitude of this ductility and on the contribution from the ties to the enhancement of the strength of confined concrete (Sheikh 1982; Asha and Sundararajan 2017).
Polar eigenvectors: a better way?
Published in International Journal of Mathematical Education in Science and Technology, 2022
There are two classic types of spirals: Archimedean and logarithmic. In essence, Archimedean spirals are of the form and logarithmic spirals are of the form (see Dennis Lawrence, 1972). The spirals resulting from systems of differential equations don’t fall neatly into either of these two types, but they are closest the logarithmic type. They are found when (but ) in Equation (4), which is rewritten here for convenience: and .