Analytical view factor solutions of a spherical cap from an infinitesimal surface

Sasaki, Kaname; Sznajder, Maciej

doi:10.1016/j.ijheatmasstransfer.2020.120477

Cited by 12 publications

(10 citation statements)

References 26 publications

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“…From there, the corresponding factors with the surrounding fragment of the sphere are obtained [9] and are considered innovative and important contributions of the author [13,29,30] (Figure 8). Other important form factors that are obtained from Equation ( 36) are those of F 13 and F 31 , (41) with which we confirm that some of the expressions found are as simple as they are elegant.…”

Section: Discussionsupporting

confidence: 80%

“…The sections may not even cross at the center of the sphere since the third postulate remains unchanged and the solution is determined for any fragment of the sphere and two arbitrary limiting elements that completely enclose an inner volume in the void (Figures 16 and 17). Possible inter-reflections between the three involved surfaces are addressed in Appendix A [41,42]. As a concluding remark, we need to stress that due to the author's original formulation, the limiting surfaces to the spherical segment do not need to be equal [38][39][40].…”

Section: Discussionmentioning

confidence: 99%

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Innovative Tool to Determine Radiative Heat Transfer Inside Spherical Segments

Lainez

2023

Applied Sciences

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The classic equations used to find the form factor inside fragments of spheres are often unassailable. The main difficulties that they present lie in iterative integrations effected over curved surfaces. The typical simulation software for this kind of issue is not capable of tackling the drawbacks that appear in the process, among them we could cite the impossibility of discretizing curved shapes with equal matching tiles, whether triangles or rectangles, especially when we arrive at the contour elements. The current type of cylindrical tiles employed for the calculation of spheres, due to incoherence in curvature, presents a significant array of gaps that render the whole procedure inadequate and inconsistent. To countermeasure this drawback, the recent finding of some innovative principles by the present author has provided a sure and exact path towards the solution of the problem in the frequent case of a volume enclosed within a spherical fragment and two limiting sections of the said sphere placed at arbitrary positions. The coherent application of such postulates by virtue of form factor algebra leads to an encompassing expression which solely requires the input of the surface areas of the involved shapes and, thus, avoids the lengthy resort to integration. A relevant number of cases in radiative heat transfer simulation, that cannot be solved by any other method, become feasible and accurate. Since the new tool can be implemented as an algorithm for simulation software, pivotal advances emerge in the complex domain of radiation which are applicable for the lighting industry, building simulations, and aerospace technologies, among others.

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Section: Discussionsupporting

confidence: 80%

Section: Discussionmentioning

confidence: 99%

Innovative Tool to Determine Radiative Heat Transfer Inside Spherical Segments

Lainez

2023

Applied Sciences

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“…cos 2 α tan 2 β = z 2 cos 2 β (A20) Thus, we arrive at the miraculous equation of the intersection between the former cone and a sphere of unit radius with its center in the vertex of the cone (Equation (A20)); this curve that we have found is called Tomomi by Cabeza-Lainez [10] and yields the following parametric equations for its coordinates (Figure A7),…”

Section: Appendix C the Recurrent Problem Of Circular Emittersmentioning

confidence: 92%

“…A much wider repertoire of possible radiative exchanges would be achieved in this fashion, with relative simplicity and accuracy and without alternative. The repercussions of such findings would be wide-ranging in fields as diverse as aerospace technologies [10], Light Emitting Diodes (LED), radiotherapy medicine, risk prevention, acoustics and architecture [11]. On an industrial basis, we consider that the vast field of artificial lighting and design of luminaries will immediately benefit from our methods.…”

Section: Conclusion and Future Aimsmentioning

confidence: 99%

New Computational Geometry Methods Applied to Solve Complex Problems of Radiative Transfer

Andújar

Lainez

2020

Mathematics

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Diverse problems of radiative transfer remain as yet unsolved due to the difficulties of the calculations involved, especially if the intervening shapes are geometrically complex. The main goal of our investigation in this domain is to convert the equations that were previously derived into a graphical interface based on the projected solid-angle principle. Such a procedure is now feasible by virtue of several widely diffused programs for Algorithms Aided Design (AAD). Accuracy and reliability of the process is controlled in the basic examples by means of subroutines from the analytical software DianaX, developed at an earlier stage by the authors, though mainly oriented to closed cuboidal or curved volumes. With this innovative approach, the often cumbersome calculation procedure of lighting, thermal or even acoustic energy exchange can be simplified and made available for the neophyte, with the undeniable advantage of reduced computer time.

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“…Being non planar, the fraction of energy that the radiating conoid exchanges with itself is, F33 = 1-0.6680-0.2032= 0.1288 (15) Not merely radiative heat transfer in the figure under study has been solved by this procedure, but also light transmission when it originates at conoidal skylights like those constructed by Ilja Doganoff [11] in 1957 in Bulgaria.…”

Section: Conflicts Of Interestmentioning

confidence: 99%

An Integral Expression for Pi Obtained by Virtue of Ramanujan’s Approximation in Conoidal Areas: Applications of the Resultant Figure in Engineering

Lainez¹

2023

Preprint

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unlike the volume, the expression for the surface area of a regular conoid has not yet been obtained by means of direct integration or a differential geometry procedure. As this non-developable form is relatively used in engineering, the difficulty to determine its surface, represents a serious shortcoming for several problems which arise in radiative transfer, lighting and construction, to cite just a few. In order to solve the problem, I conceived the surface as a set of linearly dwindling ellipses which remain parallel to a circular directrix, a typical problem appears when searching the length of such ellipses. I employed a new procedure which, in principle, consists in dividing the surface into infinitesimal elliptic strips to which we have successively applied Ramanujan’s second approximation for the longitude of the ellipse. In this manner, we can obtain the perimeter of any transversal curve pertaining to the said form as a function of the radius of the directrix and the position of the ellipse’s center on the X-axis. Integrating the so-found perimeters of the differential strips for the whole span of the conoid, an unexpected solution emerges through the newly found number psi (ψ) which seems like a refined approximation to the third decimal of Pi but derived from a definite integral equation. As the strips are in truth slanted in the symmetry axis, their width is not uniform and we need to perform some adjustments in order to complete the problem with sufficient precision, but this is discussed separately in the annexes. Relevant implications for mathematical symmetry applied to multifarious architectural and engineering forms can be derived from this finding

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Analytical view factor solutions of a spherical cap from an infinitesimal surface

Cited by 12 publications

References 26 publications

Innovative Tool to Determine Radiative Heat Transfer Inside Spherical Segments

Innovative Tool to Determine Radiative Heat Transfer Inside Spherical Segments

New Computational Geometry Methods Applied to Solve Complex Problems of Radiative Transfer

An Integral Expression for Pi Obtained by Virtue of Ramanujan’s Approximation in Conoidal Areas: Applications of the Resultant Figure in Engineering

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