Efficient Evaluation of Diffuse View Factors For

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Efcient Evaluat

o Diuse ViewFactors or

Radiation


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ABSTRACT

•

This paper presents a numerical method of evaluating view factoplanar surfaces.

• The method, which is based on Gaussian quadrature to perform integration is etended to surfaces with curved boundaries.

•As an application, the shape factor between two elliptic surfaceevaluated.

• The following discussion will consider onl! the case of radiant between di"usel! re#ecting and absorbing surfaces with no absorintervening air.


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• Thermal radiation is t!picall! as importconvection or conduction in the overall heat babuildings.

• )henever we tal* about net radiation energ! ebetween surfaces, +t starts with the calculationfactors.

• The problem with view factors is not that t

inherentl! dicult to compute, but that the ca


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•Radiation heat transfer betweensurfaces depends on theorientation of the surfaces relativeto each other as well as theirradiation properties and

temperatures, as illustrated in -ig.


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3 The fraction of the radiationleaving surface i that stri*es surface 4 directl!.


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The view factor based on theassumption that the surfaces aredi"use emitters and di"use re#ectorsis called the di"use view factor.

• And the view factor based on theassumption that the surfaces aredi"use emitters but specularre#ectors is called the specular viewfactor.


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• Consider a situation involving ' surfaces. Since each supotentiall! interact with ever! other surface, there interactions, or view factors.

• /ven simpli8cations, such as the reciprocit! relation and tha #at surface cannot view itself, reduce the number of vionl! to '9':; or =occlude>< the views between surface pairs, it is to chec* ':7 surfaces as possible obstructing surfaces for factor. This gives '9':;


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• Since anal!tical solution is not possible for man!geometries of practical interest, accurate determof view factors b! numerical means has been a t

research.


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•

Three methods which are considered in this papevaluate the integrals in the equation developedare

• Trapeoidal,

• Simpson and

• Gaussian uadrature


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•

simplest of all the quadrature formulas• needs two points on each contour element.


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• The function to beF integrated is approimated bquadratic function through three points within thof integration


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•

Gaussian quadrature is the most accurate methgiven number of points on the elements.

• An n point quadrature formula approimafunction b! 97n: ;


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•

A two point formula 9GA&SS7< approimafunction with a pol!nomial of degree three and point formula 9GA&SS?< with degree 8ve.

• To get the same accurac!, the S+5 needs four

points, respectivel!. +n general, for a desired aS+5 needs four times more computation time coto the Gaussian method.


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The Gaussian quadrature needs the integration limits to bebetween : ; and ; and hence demands a transformation ofglobal coordinates to local ones.

Referring to -ig. ;, the global coordinates of the points on the twocontours are transformed using the following F


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The present method, is capable of computing viefactors for plane surfaces which are located arbistraight or curved and share common edges prawith an! desired accurac!. The method iscomputationall! ver! ecient.

• &se of higher order quadrature is recommendwhen the surfaces share a common edge or if thsurfaces are touching an!where.


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• &se of higher order transformation is recommenwhen the surfaces have curved contours.

• +t is found that the computational e"ort will be l

when the number of elements is increased b! qutransformation than when the order of transformincreased.


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Efficient Evaluation of Diffuse View Factors For