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Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 272–276

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Notes

New configuration factor between a circle, a sphere anda differential area at random positions

Jose M. Cabeza-Lainez, Jesus A. Pulido-Arcas n, Manuel-V. CastillaHigher Technical School of Architecture, Higher Polytechnic School, University of Seville, Spain

a r t i c l e i n f o

Article history:Received 23 April 2013Received in revised form18 June 2013Accepted 29 June 2013Available online 10 July 2013

Keywords:Configuration factorsCircles and semicirclesSphereRadiative transferComputer simulationEngineering applications

73/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.jqsrt.2013.06.027

esponding author. Tel.: +34 676 307 409.ail addresses: jesus.a.pulido@gmail.com,us.es (J.A. Pulido-Arcas).

a b s t r a c t

In this Note, a configuration factor between a circle and the three coordinate directions ispresented. This new factor brings considerable independence and versatility to radiativetransfer analysis as it allows determining the radiative energy for points lying on a planeat an arbitrary position. By virtue of the radiation vector theory, the calculation isperformed whether the angles that the plane forms with the coordinate directions areknown. If the said plane cuts the disk in two halves a new factor is also presented for theradiant interchange between a half disk and a differential element. Besides, by extensionof the case of the circle, the view factor between a sphere and a differential element in arandom position is deducted.

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1. Introduction

Configuration factors that involve circular emitters anddifferential elements, irrespective of their relative positionsin space, have posed a remarkable issue for both designersand scientists. Exact expressions derived from double inte-gration were not accessible, and likewise it happened tothose referring to a differential element in a reference systemthat does not coincide with the one of the emitting disk.

These configuration factors recall an interesting ques-tion about the radiative performance of circular emitters,fragments of them, like semicircles, and related volumessuch as the sphere, as well as its interaction with surfacesrandomly placed with respect to the source. Any figurethat radiates in a three-dimensional fashion can be decom-posed into circular or even spherical finite elements, andsuch finding will allow for unexpected possibilities ofreliable and fast radiative transfer simulations. The authorshave so far experimented with this possibility.

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Radiative properties for the said emitters have beenunderstood partially, mainly because available solutionsfor the configuration factor between an emitting disk anda differential element were limited to rather particularpositions, where the element was aligned in some plane oraxis with respect to the source.

We can quote in this sense, available expressions byNaragi and Chung [1,2], which give the configurationfactor between an emitting disk and a differential elementrotated an arbitrary angle Θ but only when the elementlies on the x–z plane and is aligned with respect to thecentre of the emitting circle.

In a similar fashion, Hollands [3] proposes the config-uration factor between an emitting circle and a differentialarea, rotated an arbitrary angle to a disk bisected by the y–z plane, only when the plane that contains the elementdoes not intersect the disk.

For any portion of the circle some approximations weremade by Chung and Sumitra [9], Sparrow [10] and Minn-ing [11]. However, the particular case of the semicircle wasnot properly approached.

Eventually, regarding the three dimensional extensionof the circular emitter, that is, the sphere, Naraghi [12]proposed an expression for plane elements to spheres, but

J.M. Cabeza-Lainez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 272–276 273

they were limited to a single plane, parallel or perpendi-cular to the emitter.

Complete references about the existing factors, withdetailed explanation and graphical references for all theproposed expressions can be found in [4].

In this note, an approach to these unanswered ques-tions is made by virtue of both the configuration factor andthe radiance vector theory. This notion apparently origi-nated with Mehmke in 1898, and since then Fock [13],Yamauchi [14–16], Moon [17,18] and others [6] haveemployed the concept of radiance vector, with severaladvantages, those ones referring to the calculation of viewfactors for inclined surfaces being the most remarkable.Thanks to them, we are provided with simpler expressionsfor our configuration factors, using elemental vector theory.

Fig. 2. Calculations parameters for F1d�z.

2. Configuration factor between a circle and a differentialelement, placed in a parallel and perpendicular position

According to previous researches by the authors [5–7],the configuration factor between an emitting circle and adifferential element located in a perpendicular or parallelplane for the three directions of space has been deductedvia analytical exact expressions, obtaining comparableresults to former authors [8] when available. Thus, theirthree values for coordinate axes are, according to Figs. 1–3:

Fd1�y ¼12

1� x2 þ y2 þ z2�r2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr2 þ y2 þ z2Þ2�4r2ðx2 þ z2Þ

q0B@

1CA ð1Þ

Fig. 1. Calculations parameters for F1d�y.

Fig. 3. Calculations parameters for F1d�x.

In the y direction.

Fd1�z ¼yz

2ðx2 þ z2Þr2 þ x2 þ y2 þ z2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðr2 þ x2 þ y2 þ z2Þ2�4r2ðx2 þ z2Þq �1

264

375

ð2Þ

Fig. 5. Calculation parameters for a differential element placed at a randomposition.

J.M. Cabeza-Lainez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 272–276274

In the z direction.

Fd1�x ¼yx

2ðx2 þ z2Þr2 þ x2 þ y2 þ z2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðr2 þ x2 þ y2 þ z2Þ2�4r2ðx2 þ z2Þq �1

264

375ð3Þ

In the x directionThese two last factors, previously unavailable in the

literature, have been derived via analytical exact expressionsby the authors [5] and are presented in this simple Cartesianform for the first time. Also, they are valid for any givenlocation of the differential element, which has not to beplaced in any particular position with respect to the source.

When a magnitude shows very different values for themain directions of space it is not scalar and thus, it is oftenconvenient to treat it as a vector [6,17]. Taking the expressions[1–3] as its coordinate components, we are able to constructthe radiation vector at any point, according to Fig. 4:

F←¼ ðFd1�x; Fd1�y; Fd1�ZÞ ð4Þ

In the same fashion, the modulus of the vector can beobtained using the expression:

jFj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðFd1�xÞ2 þ ðFd1�yÞ2 þ ðFd1�zÞ2

qð5Þ

Such modulus is precisely the configuration factor at adifferential area whose normal follows the same directionof the vector [6,18].

After these operations, all the elements that are neces-sary to define the radiative field produced by an emittingdisk are presented: the value of the three components ofthe vector, that is, the configuration factors for the threecanonical directions of space; the vector itself, with a givendirection in the said coordinate system; and finally the

Fig. 4. The three components of the radiation vector.

modulus, which equates the configuration factor for adifferential element whose normal coincides with thedirection of the vector. All these expressions depend onlyon geometric parameters x, y, z and r.

3. Configuration factor between a circle and a differentialelement, placed in a random position

Once we have defined the vector for a parallel andperpendicular position with respect to the circular emitter,it is logical to derive the question to a differential elementplaced in a random position; this situation is described inFig. 5:

In this case, it is necessary to find the mathematicalrelation between the previous vector F

!and the normal to

the chosen element, which is given, according to thefigure, by the angles α, β and γ. The operation is simple,as the modulus of F

!can be projected multiplying by the

cosine (dot product) onto the normal to the unit surface,obtaining the desired factor F′ at any plane.

j F!j¼ cos αFd1�x þ cos βFd1�y þ cos γFd1�z ð6Þ

In the same way as in point 2, based on vector theory,the configuration factor for a differential element in arandom position with respect to the emitting disk hasbeen defined.

4. Configuration factor between a semicircle and adifferential element

A sequel to the former appears when the calculationplane cuts the circle in any section; in this case, it is difficult

Fig. 6. Calculation parameters for the semicircle.

Fig. 7. Calculations parameters for the sphere and a differential elementat a random position.

J.M. Cabeza-Lainez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 272–276 275

to find an exact expression to address the factor. For theparticular situation of a semicircle we propose the following.

According to Fig. 6, the integral equation which solvesthe problem can be outlined in terms of r and θː

F1d�x ¼Z r

0

Z π

0

yr2 sin θ

ðr2 þ z2 þ y2�2zr cos θÞ2dθ dr ð7Þ

With the integration limits (0, π), that is, half of thecircle, and (0,r), i.e. from the centre to the extreme of theradius. If we first integrate with respect to θ, in thenumerator we would find the derivative of cos θ, �sin θ.

Eq. (7) can be expressed as

F1d�x ¼Z r

0

Z 1

�1

�yr2

ðr2 þ z2 þ y2�2zrtÞ2dt dr ð8Þ

In turn, integrating with respect to r, the latter equationresponds to the form of arc of tangent and logarithm. Thus,the final result can be expressed as

F1d�x ¼12π

arctanr þ zy

þ arctanr�zy

� �

þ y4πz

lnr2 þ y2 þ z2�2rzr2 þ y2 þ z2 þ 2rz

ð9Þ

We can check that this equates the area subtended by acircular sector that encompasses the diameter of theemitting disk and the corresponding sector of a hyperboladefined by the intersection of the unit sphere andthe cone.

If z¼0 the expression is undetermined and we have topass to the limit with l'Hôpital's rule, obtaining the familiarresult [6].

F1d�x ¼1π

arctanry� ryr2 þ y2

� �ð10Þ

Eq. (9) is entirely new and has never been mentioned inthe literature; Eq. (10) describes the situation in which thereceiving point is aligned with the centre of the half-disk.

Thus, the view factor F1d�x represents a considerableadvance, as we are able to know the radiant distributiondue to a semicircle in a plane that constitutes its base.

In the parallel direction the factor F1d�y is obviously 1/2of the one found in [1].

5. Extension to three dimensional emitters.Configuration factor between a sphere and a differentialelement placed at random positions

After analyzing the previous form factors for the circle,a new question can be deducted. Let us consider asemitting source a sphere of radius r, and a differentialelement, placed randomly in space at a distance (x, y, z),referenced to the three coordinate directions as shown inFig. 7.

The differential element, as in former cases, is definedby its normal, and we need to find the radiation vectorimpinging on it F

!r . Obtaining the modulus (configuration

factor) is a direct operation, since we already know theangles formed by the unit element:

jF←

rj ¼ Fd1�x cos αþ Fd1�y cos β þ Fd1�z cos γ ð11Þ

J.M. Cabeza-Lainez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 272–276276

And expanding each of them,

Fd1�x ¼r2xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðx2 þ y2 þ z2Þ3q

Fd1�y ¼r2yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðx2 þ y2 þ z2Þ3q

Fd1�z ¼r2zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðx2 þ y2 þ z2Þ3q ð12Þ

6. Conclusions

New configuration factors for 2D and 3D geometriesrelying on the basic shape of the circle have been found.The energy exchange between the source and a freedifferential element in a random position can be assessedusing accessible mathematical expressions. With the aid ofCAD software and simulation programs, also devised bythe authors, this procedure is readily facilitated.

We would like to emphasize that every three-dimensional shape can be decomposed into circular oreven spherical finite elements, and in this fashion we canobtain with greater accuracy the configuration factor dueto any irregular or occluded form.

Starting from these premises, the particular case of thesemicircle has also been solved; a new expression for theconfiguration factor between this source and its base hasbeen obtained. This new finding shows many applicationsin engineering and architectural design issues, as theproposed factor responds to the widely known form oftunnels and vaults, composed by a planar base and twoemitting extremes of semicircular shape.

The three dimensional extension of the case of thecircle, that is, the energy exchange between an emittingsphere and a differential element in a random position,leads to a more accurate and fast simulation for sphericalsources; this presents several applications in lightingengineering, where the simplification of the point sourceis not always valid, especially when addressing LEDluminaires of any shape or dimension.

The authors believe that these findings will produce auseful contribution to the field of radiative heat transfer,providing numerous applications in the engineering andarchitectural domains and, in general, in every scientificrealm where determination of energy exchange betweensurfaces has become an issue.

Acknowledgements

The authors want to dedicate their findings to Dr. DominaEberle Spencer a constant source of inspiration in the questfor light. Dr. Jose M. Cabeza-Lainez is indebted to ProfessorFelix Ruiz de la Puerta, his wife Carmen and their proud sonand daughter. Dr. Manuel-V. Castilla would like to thank hisfamily for their constant support. Dr. Jesus A. Pulido-Arcaswould like to recognise the labour of Dr. Jose M. Cabeza-Lainez, who has greatly helped him to take the path ofscience.

References

[1] Naraghi MHN. Radiation view factors from differential plane sourcesto disks—a general formulation. J Thermophys Heat Transfer 1988;2(3):271–4.

[2] Naraghi MHN, Chung BTF. Radiation configuration between disksand a class of axisymmetric bodies. J Heat Transfer 1982;104(3):426–31.

[3] Hollands KGT. On the superposition rule for configuration factors.J Heat Transfer 1995;117(1):241–5.

[4] Howell John R. A catalogue of radiation heat transfer configurationfactors. 3rd edition. On-line version available at ⟨http://www.engr.uky.edu/rtl/Catalog/⟩.

[5] Cabeza-Lainez JM, Pulido-Arcas JA. New configuration factors forcurved surfaces. J Quant Spectrosc Radiat Transfer 2013;117:71–80.

[6] Cabeza Lainez JM. Fundamentos de Transferencia Radiante Lumi-nosa. (Including software for simulation). Spain: Netbiblo; 2010.

[7] Cabeza Lainez JM. Solar radiation in buildings. In: Babatunde ElishaB, editor. Transfer and simulation procedures. InTech; 2012.Isbn:978-953-51-0384-4. [Chapter 16 of Solar Radiation]. On-lineversion available at ⟨www.intechopen.com⟩.

[8] Hamilton DC, Morgan WR. Radiant-interchange configuration factors.NASA TN 2836; 1952.

[9] Chung BTF, Sumitra PS. Radiation shape factors from plane pointsources. J. Heat Transfer 1972;94(3):328–30.

[10] Sparrow EM. A new and simpler formulation for radiative anglefactors. J Heat Transfer 1962;85(2):81–8.

[11] Minning CP. Calculation of shape factors between parallel ringsectors sharing a common centerline. AIAA J 1976;14(6):813–5.

[12] Naraghi MHN. Radiation view factors from spherical segments toplanar surfaces. J Thermophys Heat Transfer 1988;2(4):373–5.

[13] Fock V. Zur Berechnung der Beleuchtungsstärke. St. Petersburg:Optisches Institut; 1924.

[14] Yamauchi J. Theory of field of illumination. Tokyo: Researches of theElectro-technical Laboratory; 1932. ([p. 339]).

[15] Yamauchi J. Further study of geometrical calculation of illuminationdue to light from luminous sources of simple forms. Tokyo:Researches of the Electrotechnical Laboratory; 1927. ([p. 194]).

[16] Yamauchi J. Geometrical calculation of illumination due to light fromluminous sources of simple forms. Tokyo: Researches of the Electro-technical Laboratory; 1924. ([p. 148]).

[17] Moon PH, Spencer DE. The photic field. Cambridge, Massachussets:The MIT Press; 1981.

[18] Moon PH. The scientific basis of illuminating engineering. New York:Dover Publications; 1962.