Innovative tool to determine radiative heat transfer inside spherical segments
Abstract
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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Citation: Cabeza-Lainez, J. Innovative Tool to Determine Radiative Heat Transfer Inside Spherical Segments. Appl. Sci. 2023, 13, 8251. https://doi.org/10.3390/ app13148251 Academic Editors: Lioua Kolsi, Walid Hassen and Patrice Estellé Received: 13 June 2023 Revised: 6 July 2023 Accepted: 15 July 2023 Published: 16 July 2023 Copyright: © 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). applied sciences Article Innovative Tool to Determine Radiative Heat Transfer Inside Spherical Segments Joseph Cabeza-Lainez Department of Architectural Composition, University of Seville, Av. Reina Mercedes 2, 41012 Sevilla, Spain; [email protected] Featured Application: Radiative form factor calculations without integrals. Abstract: 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. Keywords: radiative heat transfer; form factor calculation tools; spherical geometry; design of LED sources 1. Introduction View or form factors are a relevant entity in all the phenomena that involve radiation [ 1 ]. Since this particular process of heat transfer, unlike the common belief, does not imply any kind of continuity or contact between the affected bodies, pure form and position become pivotal for the transmission of energy [ 2 ]. Diffuse radiation is inextricably correlated to surface area [ 3 ]. In basic elementary shapes (cuboid), no particular problem appears with such a concept [ 3 ], but when it is required to extend the well-known differential equations to concrete and finite forms arbitrarily deployed, we have to deal with two sets of different and often complex double integrals [ 4 , 5 ]. It is known that to obtain an exact fourfold primitive, even of the simplest elements, is a mathematical feat and often a serendipity, to say the least [6]. Even recent studies have failed to overcome this severe hindrance for the completion of important problems in radiative transfer in the void [ 3 ]. Experts have frequently resorted to numerical or statistical approaches such as Monte Carlo methods [ 7 ], albeit they are not conclusive in most cases, not only because of the many errors that arise in the process but also due to the impossibility of validating their accuracy as they are in fact the only method available to try to surmount this shortcoming. In the case that we are addressing, Monte Appl. Sci. 2023,13, 8251. https://doi.org/10.3390/app13148251 https://www.mdpi.com/journal/applsci
Appl. Sci. 2023,13, 8251 2 of 18 Carlo methods are not available due to the presence of double curvatures which invalidate any form of regular or coherent tiling of the surfaces. The present author was able to perform the said quadruple integration in an exact manner solely for three kinds of simple forms: rectangles (whether perpendicular or parallel), parallel coaxial circles, and the same for a sphere and a circle [ 8 , 9 ]. It seems puzzling that if the rectangles simply form an angle different to π /2 or zero between them, or if the circles are slightly offset or placed in a perpendicular fashion, we cannot even reach an exact solution to the problem of finding the form factor between these rather common elements. If the forms are incomplete and fragmented, then the problem increases significantly because we need to remark that, until the author discovered it, there was no exact solution available even for the form factor due to a triangle [ 10 ]. Only the simpler expression of the configuration factor (a double integral entity which regulates the exchange on a point to point basis, but not including the whole surface) had been identified and solved for particular points [11]. Furthermore, another degree of complexity arises when the exchanges take place in a three-dimensional environment or in other words, the surfaces involved are not planar but curved or warped [12]. It is clearly improbable for the time being to find the form factor of a set of arbitrarily curved surfaces by integration alone [ 3 ]. However, other numerical methods based on discretization of the said areas in differential units are also not viable since such kinds of surfaces cannot be divided into any form of regular, concentric, or symmetrically patterned tiles. Once again we arrive at the previous impasse. With enough fortune and considerable effort, the author has discovered ways to advance in the finding of exact solutions by means of innovative principles of their own creation [ 13 ], suitable for algebraic treatment and eventually offering exact expressions that altogether avoid integration and are currently being programmed in algorithms. In the following sections we will explain and develop one such property, perhaps the most relevant of all. 2. Materials and Methods—General Departing Equation The canonical equation that regulates a general exchange of radiant energy for arbitrary and non-planar surfaces, as depicted in Figure 1, can be written in the following form (Equation (1)): dφ12 =(E1−E2)cos θ1cos θ2 dA1dA2 πr2 12 (1) Appl.Sci.2023,13,xFORPEERREVIEW2of18 onlymethodavailabletotrytosurmountthisshortcoming.Inthecasethatweareaddressing,MonteCarlomethodsarenotavailableduetothepresenceofdoublecurvatures whichinvalidateanyformofregularorcoherenttilingofthesurfaces. Thepresentauthorwasabletoperformthesaidquadrupleintegrationinanexact mannersolelyforthreekindsofsimpleforms:rectangles(whetherperpendicularorparallel),parallelcoaxialcircles,andthesameforasphereandacircle[8,9].Itseemspuzzling thatiftherectanglessimplyformanangledifferenttopi/2orzerobetweenthem,orifthe circlesareslightlyoffsetorplacedinaperpendicularfashion,wecannotevenreachan exactsolutiontotheproblemoffindingtheformfactorbetweentheserathercommon elements. Iftheformsareincompleteandfragmented,thentheproblemincreasessignificantly becauseweneedtoremarkthat,untiltheauthordiscoveredit,therewasnoexactsolution availableevenfortheformfactorduetoatriangle[10].Onlythesimplerexpressionofthe configurationfactor(adoubleintegralentitywhichregulatestheexchangeonapointto pointbasis,butnotincludingthewholesurface)hadbeenidentifiedandsolvedforparticularpoints[11]. Furthermore,anotherdegreeofcomplexityariseswhentheexchangestakeplacein athree-dimensionalenvironmentorinotherwords,thesurfacesinvolvedarenotplanar butcurvedorwarped[12]. Itisclearlyimprobableforthetimebeingtofindtheformfactorofasetofarbitrarily curvedsurfacesbyintegrationalone[3].However,othernumericalmethodsbasedon discretizationofthesaidareasindifferentialunitsarealsonotviablesincesuchkindsof surfacescannotbedividedintoanyformofregular,concentric,orsymmetricallypatternedtiles.Onceagainwearriveatthepreviousimpasse.Withenoughfortuneandconsiderableeffort,theauthorhasdiscoveredwaystoadvanceinthefindingofexactsolutionsbymeansofinnovativeprinciplesoftheirowncreation[13],suitableforalgebraic treatmentandeventuallyofferingexactexpressionsthataltogetheravoidintegrationand arecurrentlybeingprogrammedinalgorithms.Inthefollowingsectionswewillexplain anddeveloponesuchproperty,perhapsthemostrelevantofall. 2.MaterialsandMethods—GeneralDepartingEquation Thecanonicalequationthatregulatesageneralexchangeofradiantenergyforarbitraryandnon-planarsurfaces,asdepictedinFigure1,canbewritteninthefollowingform (Equation(1)): dϕ EEcosθcosθdAdA πr (1) Figure1.Radiationinterchangesforasetofarbitrarycurvedsurfaces,calledA1andA2. Figure 1. Radiation interchanges for a set of arbitrary curved surfaces, called A1and A2. Equation (1) derives from the reciprocity theorem first stated by Lambert [ 14 – 16 ] and it gives the rate of diffuse radiant energy, in W/m 2 , that is transmitted into the void from each of the surfaces, namely, E 1 and E 2 . The angles θ1 and θ2 are formed between the normal and the respective surfaces and the segment that links any arbitrary couple of points termed rij (see nomenclature).
Appl. Sci. 2023,13, 8251 3 of 18 Thorough solving of Equation (1) implies four rounds of integration [ 4 ]. The present author has been able to finish them with much difficulty for the following shapes: perpendicular rectangles with a common edge, parallel rectangles, parallel coaxial disks, and a parallel coaxial circle under a sphere [ 8 ]. The last two are possible since the polar coordinates coincide for the involved coaxial elements. Other authors have reported a similar solution but they have not fully demonstrated their integral procedures [ 16 – 18 ], and some of them seem inconclusive or incomplete although they are frequently quoted [19–21]. For other different positions of the surface sources, an exact solution has not been achieved yet [ 5 ]. Neither for triangles or irregular fragments of rectangles and circles or even for the familiar form of the sphere. Intermediate steps to proceed only with the first and the second round of integration have been proposed [ 22 ]. They usually end at a single point or elementary area that belongs to the receiving surface. Some of them are recorded under the misleading manner of nomograms at catalogues of radiation view factors [ 23 , 24 ]. Such results are somewhat helpful but they are by no means general and can be confusing in many situations. Subsequently, if we try to apply the former calculations to the interior of a sphere, we can identify two important postulates for this domain. To begin with, suppose for instance that the fragment under study is a simple hemisphere (Figure 2), in this situation the half-sphere encloses a space made of two surfaces, that is, a cap and a circular base [ 25 ]. Their respective areas amount to A 1 = 2 π R 2 (cap) and A 2 = π R 2 (disk). Under such circumstances, it is not feasible to extend double integration to the disk and the hemisphere at the same time, or at least not in the way that we would apply for parallel rectangles running only through coordinates x and y, because Cartesian coordinates are interchangeable and polar coordinates are not, since they need to originate from the same central point and that makes them unmanageable. However, by the principle of conservation of energy, we know that all radiant interchanges inside a closed volume must add to 100% or the unity [ 6 ]. Thus, the orderly array of form factor algebra for the two surfaces implies, F11 + F12 = 1; (2) F21 + F22 = 1. (3) Appl. Sci. 2023, 13, x FOR PEER REVIEW 4 of 18 Figure 2. Form factor calculation applied to a hemisphere, Surf. 1 (eq: x2 + y2 + z2 = R2, with origin at 0,0,0) with a circular base, Surf. 2 (eq: x2 + y2 = R2) at the same origin. An unexpected fact has happened in front of our eyes. The appearance of the former result implies that the amount of energy diffusely radiated from the hemisphere to itself or F11 has a definite value, other than null, and the highest probability of particles reaching the emitting source is equal to the area of the same fragment of the sphere divided by the total extension of the theoretical complete sphere, 4πR2 (Figure 3). Unlike planar surfaces, their curved counterparts present the rare property of “seeing” themselves under radiative fields as a result of their own curvature. Such a relevant geometric fact has been largely ignored in the heat transfer literature. Nonetheless, the author has demonstrated that the said relationship holds true for all kind of fragments of sphere regardless of their size or relative position within the surface [26]. Becoming an invariable and significant relationship, previously unknown, such a finding has been named Cabeza-Lainez’s first principle. From Figure 3, it can be expressed algebraically as a simple ratio which gives the maximum probability of energy attainment in this configuration (Equation (6)) [27]: F =A A=A 4π R (6) Figure 3. Half-sphere with outlined section of sphere A3 in red mapping, which is employed to account for the finding of F33, or the form factor of the section over itself (color codes selected for clearer visualization). The second form factor given in Equation (5), namely, F =𝐴/𝐴=1/2, amounts to the ratio between the area of the disk and that of the enclosing surface (the cap). This represent the maximum theoretical probability for the quanta of energy from the covering source to reach the base circle. In this case, the previous invariance property applies to Figure 2. Form factor calculation applied to a hemisphere, Surf. 1 (eq: x 2 + y 2 + z 2 = R 2 , with origin at 0,0,0) with a circular base, Surf. 2 (eq: x2+ y2= R2) at the same origin. However, the disk is planar, and in Equation (3) we know that F 22 equates to zero. Thus, F21 must be one in order to fulfil the equation.
Appl. Sci. 2023,13, 8251 4 of 18 By the reciprocity theorem mentioned in Equation (1), and the said algebra, we also obtain A2F21 = A1F12; (4) F12 = A2/A1F21; (5) with this, F12 needs to be 1/2 and so F11 = 1/2. An unexpected fact has happened in front of our eyes. The appearance of the former result implies that the amount of energy diffusely radiated from the hemisphere to itself or F 11 has a definite value, other than null, and the highest probability of particles reaching the emitting source is equal to the area of the same fragment of the sphere divided by the total extension of the theoretical complete sphere, 4 π R 2 (Figure 3). Unlike planar surfaces, their curved counterparts present the rare property of “seeing” themselves under radiative fields as a result of their own curvature. Such a relevant geometric fact has been largely ignored in the heat transfer literature. Nonetheless, the author has demonstrated that the said relationship holds true for all kind of fragments of sphere regardless of their size or relative position within the surface [26]. Appl.Sci.2023,13,xFORPEERREVIEW4of18 Figure2.Formfactorcalculationappliedtoahemisphere,Surf.1(Eq.:x2+y2+z2=R2,withoriginat 0,0,0)withacircularbase,Surf.2(Eq.:x2+y2=R2)atthesameorigin. Anunexpectedfacthashappenedinfrontofoureyes.Theappearanceoftheformer resultimpliesthattheamountofenergydiffuselyradiatedfromthehemispheretoitself orF11hasadefinitevalue,otherthannull,andthehighestprobabilityofparticlesreaching theemittingsourceisequaltotheareaofthesamefragmentofthespheredividedbythe totalextensionofthetheoreticalcompletesphere,4πR2(Figure3).Unlikeplanarsurfaces, theircurvedcounterpartspresenttherarepropertyof“seeing”themselvesunderradiativefieldsasaresultoftheirowncurvature.Sucharelevantgeometricfacthasbeen largelyignoredintheheattransferliterature.Nonetheless,theauthorhasdemonstrated thatthesaidrelationshipholdstrueforallkindoffragmentsofsphereregardlessoftheir sizeorrelativepositionwithinthesurface[26]. Becominganinvariableandsignificantrelationship,previouslyunknown,sucha findinghasbeennamedCabeza-Lainez’sfirstprinciple. FromFigure3,itcanbeexpressedalgebraicallyasasimpleratiowhichgivesthe maximumprobabilityofenergyattainmentinthisconfiguration(Equation(6))[27]: F A AA 4πR(6) Figure3.Half-spherewithoutlinedsectionofsphereA3inredmapping,whichisemployedtoaccountforthefindingofF33,ortheformfactorofthesectionoveritself(colorcodesselectedfor clearervisualization). ThesecondformfactorgiveninEquation(5),namely,F A/A1/2,amounts totheratiobetweentheareaofthediskandthatoftheenclosingsurface(thecap).This representthemaximumtheoreticalprobabilityforthequantaofenergyfromthecovering sourcetoreachthebasecircle.Inthiscase,thepreviousinvariancepropertyappliesto Figure 3. Half-sphere with outlined section of sphere A 3 in red mapping, which is employed to account for the finding of F 33 , or the form factor of the section over itself (color codes selected for clearer visualization). Becoming an invariable and significant relationship, previously unknown, such a finding has been named Cabeza-Lainez’s first principle. From Figure 3, it can be expressed algebraically as a simple ratio which gives the maximum probability of energy attainment in this configuration (Equation (6)) [27]: F33 =A3 As =A3 4πR2(6) The second form factor given in Equation (5), namely, F12 =A2/A1= 1 / 2, amounts to the ratio between the area of the disk and that of the enclosing surface (the cap). This represent the maximum theoretical probability for the quanta of energy from the covering source to reach the base circle. In this case, the previous invariance property applies to any arbitrary surface that surrounds the circle entirely and not just to the spherical cap (Figure 4). Accordingly, it is called the second postulate of radiant energy of Cabeza-Lainez.
Appl. Sci. 2023,13, 8251 5 of 18 Appl. Sci. 2023, 13, x FOR PEER REVIEW 5 of 18 any arbitrary surface that surrounds the circle entirely and not just to the spherical cap (Figure 4). Accordingly, it is called the second postulate of radiant energy of Cabeza- Lainez. Figure 4. A circular disk of radius a (surf. 2) topped by a paraboloid of height h (surf. 1), used to illustrate the invariance of Cabeza-Lainez’s second postulate. 3. Results Let us now apply the previous findings to the problems of exchange in the interior of a sphere. To this aim, consider a volume formed by three surfaces (Figure 5), a fragment of a sphere (surface 3), and two limiting planar elements (areas 1 and 2). The form factor algebra demonstrated in references [1,5,12,17] gives that, F11 + F12 + F13 + … F1N = 1 (7) Figure 5. The three surfaces appearing in radiative exchanges for the inside of a sphere. Two planar sections (not necessarily circular nor equal) and a spherical fragment as an enclosure. As defined in the first principle (Equation (6)), F33 stands for the relation between the areas of the spherical section (however irregular in shape) and the total area of the sphere, to which surface 3 pertains. F33 is, in this way, constant for any derived surface as far as it belongs to the enveloping sphere [22]. Figure 4. A circular disk of radius a (surf. 2) topped by a paraboloid of height h (surf. 1), used to illustrate the invariance of Cabeza-Lainez’s second postulate. 3. Results Let us now apply the previous findings to the problems of exchange in the interior of a sphere. To this aim, consider a volume formed by three surfaces (Figure 5), a fragment of a sphere (surface 3), and two limiting planar elements (areas 1 and 2). Appl.Sci.2023,13,xFORPEERREVIEW5of18 anyarbitrarysurfacethatsurroundsthecircleentirelyandnotjusttothesphericalcap (Figure4).Accordingly,itiscalledthesecondpostulateofradiantenergyofCabeza- Lainez. Figure4.Acirculardiskofradiusa(surf.2)toppedbyaparaboloidofheighth(surf.1),usedto illustratetheinvarianceofCabeza-Lainez’ssecondpostulate. 3.Results Letusnowapplythepreviousfindingstotheproblemsofexchangeintheinteriorof asphere.Tothisaim,consideravolumeformedbythreesurfaces(Figure5),afragment ofasphere(surface3),andtwolimitingplanarelements(areas1and2). Theformfactoralgebrademonstratedinreferences[1,5,12,17]givesthat, F11+F12+F13+…F1N=1(7) Figure5.Thethreesurfacesappearinginradiativeexchangesfortheinsideofasphere.Twoplanar sections(notnecessarilycircularnorequal)andasphericalfragmentasanenclosure. Asdefinedinthefirstprinciple(Equation(6)),F33standsfortherelationbetweenthe areasofthesphericalsection(howeverirregularinshape)andthetotalareaofthesphere, towhichsurface3pertains.F33is,inthisway,constantforanyderivedsurfaceasfarasit belongstotheenvelopingsphere[22]. Commented[M6]:TheformattingofA1,A2andA3 isnotconsistent,pleaserevise. Commented[U7R6]:Yesthereisanewfigure now Figure 5. The three surfaces appearing in radiative exchanges for the inside of a sphere. Two planar sections (not necessarily circular nor equal) and a spherical fragment as an enclosure. The form factor algebra demonstrated in references [1,5,12,17] gives that, F11 + F12 + F13 + . . . F1N = 1 (7) As defined in the first principle (Equation (6)), F 33 stands for the relation between the areas of the spherical section (however irregular in shape) and the total area of the sphere, to which surface 3 pertains. F 33 is, in this way, constant for any derived surface as far as it belongs to the enveloping sphere [22]. In this case, if we use the previous algebra (Equation (7)) to solve the remaining factors, we will obtain the following system of three equations: F12 + F13 = 1; (8) F21 + F23 = 1; (9)
Appl. Sci. 2023,13, 8251 6 of 18 F31 + F32+ F33 = 1. (10) Remembering that, as the planar surfaces 1 and 2 do not “see” each other in the radiation field, there is neither F11 nor F22 in this set of equations as they amount to zero. Consequently, F12 = 1 −F13; (11) F13 = 1 −F12; (12) F23 = 1 −F21. (13) However, the reciprocity theorem [3] implies that: A3F32 = A2F23 (14) Thus, successively, F32 =A2 A3 (1−F21); (15) F31 =A1 A3 (1−F12). (16) From Equation (10), we know that F31 = 1 −F32 −F33. (17) Substituting (15) and (16) into (17), A1 A3 (1−F12)=1−A2 A3 (1−F21)−F33. (18) Additionally, by changing F21 to its reciprocal F12 in Equation (18): A1 A3 (1−F12)=1−A2 A3 +A1F12 A3−F33. (19) In turn, multiplying both terms by A3, we would obtain A1−A1F12 =A3−A2+A1F12 −A3F33. (20) Additionally, simplifying, 2A1F12 =A1+A2−A3+A3F33. (21) Thus, working out F12, Equation (21) yields, F12 =1 21+A2−A3 A1 +A3 A1 F33. (22) Formerly, this equation would have remained unresolved, but thanks to the patient research embodied in Cabeza-Lainez’s first principle we are in the position of substituting F 33 , equal to A 3 /A s , from Equation (6) and attaining, for the first time, a physical solution to the problem and a major finding that encompasses many other unaffordable problems. Then, Equation (22) is completely solved with perfect ease. The exchange of energy for any couple of surfaces that enclose an arbitrary spherical fragment is found as, F12 =1 21+A2−A3 A1 +A3A3 A1As. (23)
Appl. Sci. 2023,13, 8251 7 of 18 Derived from Equation (23), Cabeza-Lainez’s third postulate gives, F12 =1 2 1+A2−A3 A1 +A2 3 A1As!. (24) Only the areas of the three involved surfaces plus that of the complete sphere appear in this expression. The problem of finding the radiative exchanges inside the volume composed of three curved figures is completely solved in an elegant manner without resorting to integration. The finding of the areas for 1 and 2 is normally simple, but determining the lateral area of the sectioned sphere may require managing a special kind of trigonometry and the curved angles that appear in it. The theorem of Girard [ 28 ] is often useful for this task. In the first example proposed, the planar figures possess a common tangent point, but this is not necessary in all cases as the postulate is perfectly general. In fact, the sections of the sphere can be detached, connected, or intersected as in the case of two semicircles. Surfaces 1 and 2 can be different in size and shape, as determined by the enveloping sphere [29]. The rest of the factors required in the calculation, F 13 , F 21 , F 23 , F 31 , and F 32 can be found without difficulty from Equations (11)–(16). For example, F13 =1−1 2 A1+A2−A3 A1 +A2 3 A1As!=1 2− A2−A3 2A1 +A2 3 2A1As!; (25) F31 =A1 2A3− A2−A3 2A3 +A2 3 2A3As!=A1 2A3−A2−A3 2A3 +A3 2As; (26) F21 =A1 2A2 A1+A2−A3 A1 +A2 3 A1As!=A1+A2−A3 2A2 +A2 3 2A2As; (27) F23 =1−A1+A2−A3 2A2−A2 3 2A2As =1 2− A1−A3 2A2 +A2 3 2A2As!. (28) 4. Discussion To continue, we will discuss in this section, the applicability of the results derived from the new form factor equations for several usual configurations. In the first place, we will study the expressions deduced in Equations (24)–(28) for a new case of two tangent circles which form an arbitrary angle from 0 to π (Figure 6). If the surfaces A1and A2are equal, the postulate reduces to F12 =2A1+A3(F33 −1) 2A1 =1 2 2A1−A3 A1 +A2 3 AsA1!=1+A3 2A1A3 As−1(29) F13 =A3 2A11−A3 As(30) Should the angle between the said circles stand at π /2 (perpendicular), by common geometry rules we can substitute the respective values of the magnitudes in Equation (25) to obtain F12.
Appl. Sci. 2023,13, 8251 8 of 18 Appl. Sci. 2023, 13, x FOR PEER REVIEW 8 of 18 Figure 6. Form factor calculation between two circles with a tangent point forming an arbitrary angle α. Should the angle between the said circles stand at π/2 (perpendicular), by common geometry rules we can substitute the respective values of the magnitudes in Equation (25) to obtain F12. The area of the two equal circles A1 = A2 is πR2/2, the former Equation (29) can then be cleared to F=1+A(F−1) πR =1+ A AπR−A πR (31) A3 is obtained as the total area of the sphere, 4πR2, minus two spherical segments whose lateral area equates to π(a2 + h2), where a represents half of the base of the segment and h is the perpendicular distance from the base to the top of the segment. Therefore, by simple trigonometry, h is R(1−√2/2) and a equates to R√2/2. a=R 2 ; h=R3 2 − √ 2=R 2 3 − 2 √ 2 (32) a+h=R 2 4 − 2 √ 2 A=4πR−2πR 2 1+3− 2 √ 2=4πR−2πR2− √ 2=πR2 √ 2 (33) Squaring this area in accordance, we obtain, A =8πR (34) Substituting into Equation (29), F =1+ 8πR 4πRπR−πR2 √ 2 πR (35) Thus, the sought factor F12 is obtained as 𝐹 =1+2−2 √ 2=3−2 √ 2=0.1715 (36) For a general case of any subtended angle α=2θ, the area of the circles as compared to the sphere yields A=A=πRcos θ (37) To compare with the above perpendicular case, remember that cosθ was there equal to cos(π/4) = √2/2, and A1 = A2 = πR2/2. Figure 6. Form factor calculation between two circles with a tangent point forming an arbitrary angle α. The area of the two equal circles A 1 = A 2 is π R 2 /2, the former Equation (29) can then be cleared to F12 =1+A3(F33 −1) πR2=1+A2 3 AsπR2−A3 πR2(31) A 3 is obtained as the total area of the sphere, 4 π R 2 , minus two spherical segments whose lateral area equates to π (a 2 + h 2 ), where a represents half of the base of the segment and h is the perpendicular distance from the base to the top of the segment. Therefore, by simple trigonometry, h is R(1−√2/2) and a equates to R√2/2. a2=R2 2; h2=R23 2−√2=R2 23−2√2(32) a2+h2=R2 24−2√2 A3=4πR2−2πR2 21+3−2√2=4πR2−2πR22−√2=πR22√2 (33) Squaring this area in accordance, we obtain, A2 3=8π2R4(34) Substituting into Equation (29), F12 =1+8π2R4 4πR2πR2− πR22√2 πR2(35) Thus, the sought factor F12 is obtained as F12 =1+2−2√2=3−2√2=0.1715 (36) For a general case of any subtended angle α = 2 θ , the area of the circles as compared to the sphere yields A1=A2=πR2cos2θ(37) To compare with the above perpendicular case, remember that cos θ was there equal to cos(π/4) = √2/2, and A1= A2=πR2/2. In this more encompassing case, the spherical caps resulting from the limit sections, present the parameters a = R/2 cosθand h = R(1 −sinθ).
Appl. Sci. 2023,13, 8251 9 of 18 Therefore, if we subtract the area of those caps from the total area of the sphere, A 3 is equal to 4 π R 2 sin θ , F 33 gives sin θ , Equation (28) is adapted accordingly and the entirely new form factor yields, after considerable manipulation, F12 =1+2sin θ(sin θ−1) cos2θ=F21 =1+2 sin2θ−2sin θ 1−sin2θ(38) Or writing it in terms of the double angle α, F12 =1+2 sin α 2sin α 2−1/ cos2α 2(39) This represents a totally new postulate to determine the form factor between tangent circles that form any angle from 0 to π. It is an undeniable finding. The value of the curve for α = π /2 is 0.17, as found previously in Equation (43). Such a curve has been named “Herodias” by the author. It is represented in Figure 7. Appl. Sci. 2023, 13, x FOR PEER REVIEW 9 of 18 In this more encompassing case, the spherical caps resulting from the limit sections, present the parameters a = R/2 cosθ and h = R(1 − sinθ). Therefore, if we subtract the area of those caps from the total area of the sphere, A3 is equal to 4πR2 sinθ, F33 gives sinθ, Equation (28) is adapted accordingly and the entirely new form factor yields, after considerable manipulation, F =1+2sin θ (sin θ −1) cos θ =F =1+2sin θ −2sin θ 1−sin θ (38) Or writing it in terms of the double angle α, F =1+2sinα 2sinα 2−1/cosα 2 (39) This represents a totally new postulate to determine the form factor between tangent circles that form any angle from 0 to π. It is an undeniable finding. The value of the curve for α = π/2 is 0.17, as found previously in Equation (43). Such a curve has been named “Herodias” by the author. It is represented in Figure 7. Figure 7. Form factor calculation between two circles with a tangent point forming an arbitrary angle. Other important form factors that are obtained from Equation (36) are those of F13 and F31, F =2sin θ (1−sin θ ) cos θ =F (40) F=1−sin θ 2=F (41) with which we confirm that some of the expressions found are as simple as they are elegant. A second discussion case, inspired by the first, deals with two semicircles which share a common edge. In this case the symmetric planar sections start at the center of the containing sphere, while in case 1 they departed from a pole or extreme of the said sphere. The author could obtain by deduction the form factor due to half-disks of identical radius R with a shared straight edge, as a function of the subtended angle α between the semicircles. 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). Figure 7. Form factor calculation between two circles with a tangent point forming an arbitrary angle. Other important form factors that are obtained from Equation (36) are those of F 13 and F31, F13 =2sin θ(1−sin θ) cos2θ=F23 (40) F31 =1−sin θ 2=F32 (41) with which we confirm that some of the expressions found are as simple as they are elegant. A second discussion case, inspired by the first, deals with two semicircles which share a common edge. In this case the symmetric planar sections start at the center of the containing sphere, while in case 1 they departed from a pole or extreme of the said sphere. The author could obtain by deduction the form factor due to half-disks of identical radius R with a shared straight edge, as a function of the subtended angle α between the semicircles. 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).
Appl. Sci. 2023,13, 8251 16 of 18 investigations. Raul Limon has been instrumental in diffusing these findings. I would like to thank Francisco J. Guevara García and Carlos Rubio Bellido for their kindness in allowing me to present the work at an early phase. Scenography of the pool at Strauss’ opera Salome has been a powerful source of inspiration. Conflicts of Interest: The authors declare no conflict of interest. Nomenclature E 1 and E 2 are the amounts of energy (in W/m 2 ) emitted in Lambertian mode by elements 1 and 2. θ1 and θ2 represent the angles (in radians) formed by the normals to the unit area elements of areas dA 1 and dA 2 . r 12 is the arbitrary vector (in metres) that connects the surfaces dA1and dA2. Appendix A In the particular situation of analyzing a curved space made by three surfaces, by virtue of the third postulate presented we may need to supplement the radiative balance by considering the possible internal reflections between the said surfaces. Such an issue has been addressed by the author with the ensuing deductions [13,31]: The final balance of energy is based on the expression: Etot =Edir +Eref (A1) In it, E dir stands for the ratio of directly emitted energy and E ref represents the amount of reflected energy. Adding both magnitudes, we find the total balance of radiated energy Etot. When more than two surfaces appear in the process of reflection, it is customary to arrange the set of necessary equations into an array. Two types of square matrices are instrumental, they are named F r and F d [ 3 , 31 ], the number of rows and columns coincides with the surfaces involved in the calculation: Fd= 0 F12ρ2F13ρ3 F21ρ10 F23ρ3 F31ρ1F32ρ2F33ρ3 (A2) Fr= 1−F12ρ2−F13ρ3 −F21ρ11−F23ρ3 −F31ρ1−F32ρ21 (A3) Factors F ij have been previously found, they give the energy emission exchanged by the surfaces under consideration. In this case, ρi stands for the ratio of reflection associated with a given surface i [3,13]. Notice, in the array defined as F d (Equation (A2)), the last coefficient of the diagonal is non-zero since the F ii factors or self-factors for the coincidental sub-indexes have definite values for curved surfaces such as the sphere, unlike cuboids or planar faces [2]. Once we are able to obtain the value of the elements in the matrices (2) and (3), we can find different formulas (Equations (A4)–(A6)) to link the direct radiation with that given by reflection [1]. FrEref =FdEdir (A4) Frd =F−1 rFd(A5) Eref =FrdEdir (A6) In this manner, the issue of radiative energy interchanges in the void for curved spaces is eventually settled.
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