Comparison of calculation methods and models in software for computer graphics and radiative heat exchange

Transcription

1 Comparson of calculaton methods and models n software for computer graphcs and radatve heat exchange Insttute of Electrcal and Electroncs Engneerng Poznan Unversty of Technology ul. Potrowo 3A, Poznań Poland konrad.domke@put.poznan.pl Abstract: The paper descrbes the physcal backgrounds for modellng n vsualsaton usng computer graphcs software and n programs of radatve heat exchange. Physcal phenomena n both cases (generaton, propagaton, reflecton and absorpton of lght and radatve heat flux) are qute smlar, provded wth basc equatons, typcal smplfcaton assumptons and the most frequently appled methods of resolvng such ssues. Varous models appled n radatve heat transfer and computer graphcs software are compared. The smlarty and dfferences of algorthms for both tasks are determned and replaceablty condtons are enumerated. Formulas are provded to convert thermal boundary values (temperature and power densty) nto lumnous or radant quanttes (lumnance) and vce versa. A possble scope of shared computatons s also outlned. Keywords: computer graphcs, radatve heat transfer, renderng, modellng, radosty, ray tracng, boundary condtons Introducton Computer graphcs s a sub-feld of computer scence and s concerned wth dgtally synthessng and manpulatng vsual content usng computatonal technques. As an academc dscplne, t focuses on the mathematcal and computatonal foundatons of mage generaton and processng rather than purely aesthetc ssues. The begnnngs of computer graphcs go back to the 950s, however as computers and specalst graphcs equpment were farly expensve and ndvdual unts had very lmted processng capacty at the tme, computer graphcs used to be a doman explored only at robust research centres, n maor companes and government nsttutons. The breakthrough came n the 980s wth the advent of relatvely cheap and fast personal computers wth suffcent processng power. At present, computer graphcs s commonplace and natural both n professonal and home applcatons, smlarly to spreadsheets and word processors. The most common applcatons of computer graphcs: vsualsaton of measurement data, vsualsaton of computer smulatons, medcal dagnostcs, smulators and tranng machne, computer-aded desgn (CAD), desktop publshng (DTP), specal effects n flms, computer games, archtectural presentatons. Computer graphcs s often employed to obtan photorealstc (photographc) mages of real or magnary obects or people. It s worthwhle to note that the achevement of a subectve sensaton of realty s only possble f generaton algorthms of such mages duly take nto account physcal laws governng the emsson, propagaton, reflecton and absorpton of lght,.e. electromagnetc wave wth the wavelength λ ( ) µm. Passng on to dscuss radatve heat exchange, t must be noted that besde convecton and conducton, radatve heat transfer s one of three basc ways of heat exchange. It s a common phenomenon, based on the exchange of thermal energy (power) va electromagnetc radaton (chefly nfrared radaton wth the wavelength λ ( ) µm) between nontransparent surfaces or translucent volumes of a temperature hgher than 0K. In non-vacuum systems, for a surface wth a temperature wthn the range of 0 50 o C, radaton accounts for ca. 5-0% of total heat transfer and rses substantally at hgher temperatures. We can see t on Fg.. It s also the only way of heat exchange n vacuum systems. In many electrcal or electronc devces, determnaton of thermal felds assocated wth normal ISSN: Issue 4, Volume 3, October 008

2 work and operaton s the basc means of verfcaton whether the desgn s sutable, and serves as the bass for determnng maxmum allowed electrcal loads. Heat exchange n electrcal or electronc devces s usually of a mxed nature, made up of conducton, convecton and radaton. Snce numercal methods of calculatng thermal conducton are relatvely well developed, t s all the more mportant to work out numercal computaton methods for modellng radatve heat transfer. % convecton radaton temperature of element n [ o C] Fg. Share of convecton and radaton n general heat transfer wth typcal condtons Computer graphcs tasks, descrpton and basc technques The basc task of computer graphcs s to generate mages on D screen 3D photorealstc mages of real or magnary obects or people usng dgtal technques. Two maor mage generaton methods can be dstngushed, whereby computer graphcs s broadly dvded nto raster graphcs and vector graphcs. In computer raster graphcs, an mage s created as a btmap, where a data structure represents a rectangular grd of pxels vewable va a montor, on paper or usng other dsplay devces. Vector graphcs (also called obect-orented graphcs) uses geometrcal prmtves such as ponts, lnes, curves, and polygons, whch are all based on mathematcal equatons to represent mages n computer graphcs. At present, vector graphcs s essentally lmted to the programme layer and user nterface. It s appled because of ts leadng feature of mantanng unchanged mage qualty durng transformatons (e.g. rotaton or scalng). However as present-day dsplays are of the raster type, applcatons based on vector graphcs are forced to represent perfect geometrc fgures n a fnte resoluton. Generally, the process of generatng an mage from a model, usng dedcated computer programmes, s called renderng [, 6, ] The model s a descrpton of vrtual 3D obects n a strctly defned language or data structure. It would contan geometry, vewpont, texture, lghtng, and shadng nformaton. The mage s a dgtal mage or a raster graphcs mage. The orgn basc renderng equaton, as proposed by Kaya [6], s the followng: ( x,x ) = g( x,x ) e( x,x ) ( x,x,x ) I( x,x ) dx I ρ () V where: I ( x,x ) sum of ntensty of the radaton emtted and reflected n pont x n the drecton of pont x, g ( x,x ) factor dependant from the system s geometry, e ( x,x ) radatve emsson from pont x n the drecton of pont x, ρ ( x,x,x ) specular reflectance of radaton I ( x,x ) n pont x, ncdent from pont x, and reflected n the drecton of pont x. The equaton () can be transfer n form, whch descrbes ths phenomenon n terms of radaton: L( x, ω) = L ( x, ω) o L( x, ω )& ρ( x, ω, ω) cosθ ω Ω () where: L(, ω) - total radance n pont x (see defnton n eq. x (5)) and drecton ω, whle ρ& & represents bdrectonal reflectve dstrbuton functon (BRDF). Integraton extends over all the surfaces of the scene Ω. Radance n pont x conssts of own emsson L o resultng from the temperature n pont x ) and reflected radance L ref radance ncdent on x from any drecton ω and reflected n drecton ω. Determnng the nteracton of the ray and the surface (reflecton) n pont x as the ntegral operator ζ collectng radaton from the entre surface Ω: ω Ω L ( x, ω )&& ρ( x, ω, ω) cosθ = ζ ( L( x, ω )) = L ( x, ω) (3) the formula gven n () can be expressed as L ( x, ω) = L ( x, ω) L ( x, ω) = o ref L ( x, ω ) ζ ( L( x, ω )) (4) = o ref ISSN: Issue 4, Volume 3, October 008

3 In (4) the unknown radance L s both nsde and outsde of the operator ζ, whch renders the problem very complex. Typcally, the problem s solved usng a smple formal trck: the ntegral part should be substtuted n the unknown functon. Own emsson can be regarded as zero reflecton. Ths transforms the ntegral equaton nto the followng nfnte seres: L( x, ω) = n= 0 n ζ L( x, Ω ω Ω ) ζ ( L) ζ ( L) ζ ( L) K (5) L o The elements n (5) represent own emsson, a sngle reflecton of own emsson, followed by a secondary reflecton and so on. Smple renderng algorthms (ray castng) do not even try to solve ths equaton. The local llumnaton method tres to solve only the frst two elements of the seres, ray tracng elmnates the ntegral from every element, makng the seres much smpler. A number of renderng algorthms have been put forth, and software used for renderng may employ a varety of dfferent technques to obtan the fnal mage. Most advanced software combnes two or more of avalable technques of renderng to obtan goodenough results at a reasonable cost. Standard renderng technques, currently consdered the most effectve, nclude: ray castng local llumnaton ray tracng (backward ray tracng) radosty or many others technques, as photon mappng, Metropols-Hastngs, Monte-Carlo methods. Ray castng and ray tracng are based on the same fundamental prncple: determnaton of the vsblty of surfaces by tracng magnary rays of lght from the vewer's eye (or the camera) to the obect n the scene. The algorthm employed n ray castng nvolves the followng steps:. Ray proecton from the observaton pont towards the proecton plane, usually equvalent to the dsplay.. Determnaton of the frst pont of ntersecton wth an obect n the scene. 3. Determnaton of lumnosty of the pont, based on the prncple that the smaller the angle η between a vector normal to the obect surface and the secton (camera, pont), the more lumnous the pont. 3 Fg. Ray castng prncple Ths s a smplfed method that does not take nto consderaton lght sources and the law of reflecton. Furthermore, t does not model shadows, concave or convex walls and pont lghtng. Smplfed mages are only obtaned for specfc obects (e.g. flat perpendcular walls) and a specfc type of lghtng. Local llumnaton s commonly used n smple 3D graphcs programs. It s assumed that the look of scene surfaces can be determned by calculatng drect rays and sngle reflecton n pont x of lght sources only (see equaton (5) and (6)). The rest of the nfnte seres s expressed n the so-called constant ambent lght. L ( x, ω) = L ( x, ω) M = L o sources ( x, ω )& ρ( x, ω, ω)cosθ L (6) Summaton (whch, n ths formula, replaces ntegraton used n ()) s performed over M sources, not over space Ω. In ths equaton the renderng equaton () s reduced to the sum of ust a few easly computable terms. The local llumnaton method s used because n the case of artfcal envronments domnant lght sources determne the look of obects, whle ntersurface lght reflectons are only of secondary mportance. Ths method s physcally ncorrect and t mght be unweldy, however n the producton envronment the method has proven ts usablty n a wde range of cases. Ray tracng s an extenson of the ray castng and local llumnaton methods. A smple castng algorthm s completed by secondary rays generated n ponts of ntersectons whch are used as startng ponts for new rays n order to determne reflectons, refractons, and shadows. In the conventonal ray-tracng method, meshng or other pre-processng of the scene are not needed. The colour of an approprate pxel s computed accordng to the ray-obect nteractons [5]. Ray tracng s a technque for generatng an mage by tracng the path of lght through pxels n an mage plane. The technque s capable of producng a very η amb ISSN: Issue 4, Volume 3, October 008

4 hgh degree of photorealsm; usually hgher than that of typcal scanlne renderng methods, but at a greater computatonal cost. Ths makes ray tracng best suted for applcatons where the mage can be rendered slowly ahead of tme, such as n stll mages and flm and televson specal effects, and s less suted for real-tme applcatons lke computer games where speed s crtcal. The algorthm for ray tracng ncludes the followng elements:. The prmary ray s sent out from the pont n whch the observer s located; the ray ntersects the proecton plane.. The closest pont of ntersecton wth obects wthn the scene s sought out. Items - are the same as those nvolved n ray castng, but addtonally: 3. Takng nto account each lght source defned n a gven scene, total lumnosty s determned at the pont of ntersecton, as a sum of drect (prmary) radaton and reflected (secondary) radaton n complance wth the adopted model of lght reflecton (Lambert s, Phong s) on other surfaces. 4. If the pont of ntersecton belongs to a lghtreflectng or transparent obect, the pont sends out secondary rays (reflected or refracted ray) and the algorthm s repeated recursvely startng n step. The number of reflectons adopted for programmes s usually restrcted. 5 4 Fg. 3 Ray tracng prncple: a forward tracng (lght), b backward tracng (vsblty ray) a b Ths can be mathematcally expressed as []: L ( x, ω) = L ( x, ω) M = L sources o ( x, ω )& ρ( x, ω, ω)cosθ Nrl Nrr x, ω )&& ρ( x, ω, ω) x Ω x Ω L ( L( x, ω )& τ ( x, ω, ω) L (7) where N rl number of reflected lght rays and N rr number of refracted rays. In order to ncrease accuracy, the ray tracng algorthm s appled to all pxels of an mage, wth several rays traced through each of the pxels. A very sgnfcant advantage of the method s the fact that t makes t easy to parallelse programmes: each prmary ray can be processed ndependently of others; smlarly, secondary rays are ndependent of one another. Ray tracng however, s computatonally costly, as the number of computatons s proportonal to mage resoluton, the degree of complexty of the scene: the number of lght sources, the number and shape of obects and ther reflecton characterstcs. A natural course of ray from the source to the observer s presented n Fg. 3a. It would be hghly neffcent to process a system wth such ray courses, as t s only some of generated rays whch, drectly or ndrectly or followng multple reflectons, reach the observer. It must be stressed, though, that only such rays are useful because they produce desrable vsual sensatons. It s not possble (wthout sutable computatons) to select only useful rays out of the total number of rays. To reduce the amount of superfluous rays that tend to be processed, the prncple of backward ray tracng was ntroduced, whereby hypothetcal rays (backward rays) are modelled runnng from the observer s eye and httng the lght source, ether drectly or va reflectons. In ths case, the number of analysed, though superfluous, rays s markedly lower, as llustrated n Fg. 3b. Ray tracng s not a perfect technque for generatng photorealstc mages, ether. Above all, t fals to take due account of dffused lght (the number of reflectons s lmted) and analysng ndvdual rays t cannot correctly model dffracton, nterference and other wave phenomena resultng from the smultaneous mutual nteracton of a number of waves. Radosty as a method of renderng usng a global llumnaton algorthm. Global llumnaton n 3D graphcs s a lghtng model n whch each obect wthn a scene s llumnated by lght emtted drectly from the lght source, but also lght repeatedly reflected from other obects wthn the scene. The number of reflectons s essentally unlmted, whle amb ISSN: Issue 4, Volume 3, October 008

5 the ray s suppressed when ts assgned energy drops below a defned level. Radosty s an applcaton of the fnte element method to solvng the renderng equaton for scenes wth purely dffuse surfaces. Unlke Monte Carlo algorthms (such as path tracng) whch handle all types of lght paths, typcal radosty methods only account for paths whch leave a lght source and are reflected dffusely for a number of tmes before httng the eye. More correctly, radosty M s the energy leavng the patch surface per dscrete tme nterval and s the combnaton of emtted and reflected energy [5,]: M ds = E ds ρ M ϕ ds (8) where: M s the radosty of patch, E s emtted energy, ρ s the reflectvty of the patch, ϕ s the constant-value of vew factor for the radaton leavng path S and httng patch S. For dffuse radaton, the followng holds: and when the ntegral s replaced and unform radosty s assumed over the patch, creatng the smpler: N ρ M = E M ϕ (0) Ths equaton can be appled to each patch. The equaton s monochromatc n form, so colour radosty renderng requres calculaton for each of the requred colours. A more developed verson of the method s progressve radosty. Ths method solves the system teratvely n such a way that after each teraton we obtan ntermedate radosty values for the patch. These ntermedate values correspond to bounce levels. That s, after one teraton we know how the scene looks after one lght bounce, after two passes, two bounces, and so forth. Progressve radosty s useful for gettng an nteractve prevew of the scene. Also, the user can stop the teratons once the mage looks good enough, rather than wat for the computaton to numercally converge. A comparson of dfferent computer graphcs models s presented n Table. ϕ S = ϕ S (9) Table. Methods used s computer graphcs. Methods employed n computer graphcs Ray castng Local llumnaton Ray tracng Radosty (global llumnaton) Rays are cast and traced n groups based on some geometrc constrants. Only prmary lght sources are tested. Prncple Each ray s traced separately, number of reflecton s lmted, every pont on the dsplay s traced by at least one ray. Physcal correctness Each ray s traced separately, so that every pont on the dsplay s traced by at least one ray. None Very lmted Partal Nearly complete Closer perpendcular surfaces are lghter. Lmted by smple geometrc shapes. Resultng mage s not very realstc. Only parallel pont lghtng. Selected lght sources, almost any shape. Moderate for selected lght types. Assumpton Only one (or, less commonly, several) dffuse reflectons, no refracton. Scene Almost any shape can be rendered. Qualty Resultng mage s realstc. Complete model of reflectons and sources, meshng s requred, mnor detals may dsappear. Any shape, though wthout very fne detals. Very hgh, dependng on meshng accuracy. ISSN: Issue 4, Volume 3, October 008

6 3 Radatve heat exchange tasks, physcal descrpton, basc equaton, resoluton methods The basc task n accountng for the phenomenon of radatve heat transfer s determnng the dstrbuton of temperature and radatve power penetratng the set area Ω. The area, referred to as the radatve heat exchange system, s lmted by the boundary surfaces S. Physcal foundatons of thermal radaton and radatve heat transfer can be summed up as follows: each physcal body of a temperature hgher than 0K generates electromagnetc radaton nto the surroundng half-space. For the black body, the phenomenon s defned wth Planck s law [7]: m 5 bb, = c λt c λ λ () e whch expresses the relatonshp between densty of black body monochromatc radosty m bb,λ (monochromatc surface densty of power) and ts temperature T for any wavelength λ. Some of the radaton wthn the sub-range of λ ( )µm conveys substantal amounts of energy and, due to ts locaton n the spectrum, s referred to as nfrared radaton. It s the cause of radatve heat exchange. The formula n () defnes the radaton of a black body, whle real bodes are descrbed wth the followng relaton m = () λ ε λm bb, λ where ε λ represents monochromatc emssvty of radatng surface. On the other hand, total quanttes, whch do not take nto consderaton the spectral nature of radaton, are defned wth the formula: M = ε m dλ (3) λ λ bb, λ where M stands for radosty of the radatng surface, measured n W/m. Radaton analyses should typcally ncorporate the drecton of propagaton. Ths s why the noton of densty of monochromatc radance l m s ntroduced, defned as densty of monochromatc radosty m measured at a set sold angle : dmλ l λ = (4) By the same token, for total quanttes whch do not take nto account the relatonshp wth the wavelength λ, radance L can be defned as: dm L = ldλ = (5) λ whch can also be expressed wth the equaton: d Φ L = ds p d Φ = ds cosη (6) where L represents radance measured n [W/(m sr)] (see also renderng equaton ()). Total radaton spreadng from any pont x of surface S n a defned drecton ω conssts of own emsson and radaton propagatng from the entre half-space towards any gven pont and reflected n the analysed drecton (compare to ()), as llustrated n Fg. 4. Fg.4 Radance L at pont x and drecton ω Consequently, for any pont wthn the surface one can express the basc equaton governng radatve heat exchange as [,7,]: L( x, ω, = ε( x, ω, L ( x, Ω reflected radaton: & ρ ( x, ω, ω, L( x, ω, cosη own radaton: L ( x, ω, = ε( x, L ( x, & ρ ( x, ω, ω, L ( x, ω, cosη (7) Ω In (7), ρ& & stands for BRDF, the bb ndexes refer to radaton emtted by a black body, to ncdent radaton or pont x. Integraton s performed over the entre half-space Ω. The equaton (7) s the same as governng computer graphcs equaton () or (). o bb total radaton: L (, ω, x bb ISSN: Issue 4, Volume 3, October 008

7 The equaton n (7) results drectly from the prncple of conservaton of energy. Ths s a Fredholm ntegral equaton type. The notaton means that for any pont x on the boundary surface S, whose locaton s defned wth pont x, radance L n drecton defned by vector ω, conssts of total radance of own emsson (the frst component of the formula n (7)) and total of reflectons n the drecton ω of radaton ncdent from the entre half-space Ω (the next component, ntegral of the formula (7)). The equaton gven n (7) descrbes radaton from a sngle pont, but practcal applcatons nearly always nvolve a whole radatve exchange system. Boundary surfaces of the system are usually defned wth the socalled boundary condtons. The condtons are defned ether as Drchlet condtons or Neumann condtons. The former case requres a functon of the dstrbuton of temperature t n the analysed boundary surface S [,5] t ( x ) = t( x0, z0) = f ( x ) x S (8) The latter case, on the other hand, requres a specfcaton of surface power densty p proportonal to the dervatve of functon t [,7] penetratng from the outsde nto the system va the surface boundary: t p ( x ) = p( x0, y0, z0 ) = λ = g( x ) (9) r x S In ths case, as shown n (9), only the dervatve of functon t s set, whch means that temperature s defned wth accuracy to a constant. Consequently, n order to determne absolute temperature dstrbuton wthn the system, t s necessary to specfy a Drchlet condton for at least one boundary surface. We can see t on Fg. 5. S frst element of the sum n () equals p. In both cases, the equaton () comes down to the form of Fredholm s ntegral equaton of the second knd, usually presented n the followng form: φ ( x) = f(x) K( x,x ) φ( x )ds (0) S where: K ( x,x ) kernal of the ntegral equaton, f( x) known functon, φ ( x ) quested functon (radance L or heat flux densty p). So, we obtan a system of N Fredholm ntegral equatons (7) together wth boundary condtons (8) and (9), whch s practcally unsolvable algebracally [8]. There are two related technques (methods) that can be used to solve the ntegral equaton (7): fnte element methods, stochastc modellng and smulaton, zonal, dscrete ordnate, Monte-Carlo methods, and many others. Computatons usng the fnte element method (net or zonal method) are based on a pror dscretsaton and lnearsaton of basc equatons and smplfcatons of these equatons (7)-(9), whch makes t possble to obtan a system of lnear equatons. Stochastc methods, n turn, are developed wth the basc equaton (7) and boundary condtons (8)-(9). Ths approach s adopted e.g. n the conventonal Monte-Carlo method and ray tracng. The bass for dscretsaton of surfaces S partcpatng n heat transfer s the assumpton that the boundary condtons, as well as emssvty and reflectvty propertes for the entre surface S or a part of t, S. Mathematcally speakng, ths comes down to assumng that t ( S x ) = t( x, y, z ) = t( x ) = const x ( x ) = p( x0, y0, z0 S p ) = p( x ) = const () x Fg. 5. Graphc nterpretaton of boundary condtons: a) Drchlet s, b) Neumann s Drchlet s condton makes t possble to calculate the value of radance L o n the equaton (), for Neumann s condton t can be assumed that the and & ρ ( x, ω, ω, = & ρ(s, ω, ω, const () = for each r 0 belongng to the analysed fragment S. As a result, the ntegral equaton (7) applcable to selected S can be expressed as the followng total: ISSN: Issue 4, Volume 3, October 008

8 L ( s, ω, = ε (s, ω, (s, N = L bb & ρ (s, ω, ω, L (s, ω, cosη (3) p where N stands for the number of S fragments n the entre system. After ntroducng the noton of confguraton coeffcent ϕ specfed as [5,7]: = ds ds S cos η cos η ϕ (4) πd S S and defnng mutual geometrcal relatons of S and S surface fragments, as well as assumng monochromatcty of radaton or greyness of surfaces ( ε ( s, ω, = ε ( s ) = const ) nvolved n radatve heat exchange, the equaton system (3) s presented n the form of a system of lnear equatons [5,7]: δ ε N N 4 ( ϕ ) pout, = ( δ ϕ ) σt = ε ε = (5) for =,,...,N The equatons gven n (5), together wth boundary condtons n (8) and (9), create a unquely determned system of lnear equatons whch can be solved ether usng classcal methods (for low N) or teratve methods (for hgh N). The other group of methods s based on stochastc modellng of radatve heat transfer. Knowng the surface arrangement of a heat exchange system, and emssvty and reflectvty characterstcs of surfaces, t s possble to model the propagaton of radaton by tracng the hstory of each ray from ts emsson, through reflectons on boundary surfaces, untl ts fnal absorpton, as shown n Fg. 6. P a ste of reflecton: dvson nto reflected rays and absorbed power ste of emsson of own radaton Fg. 6 Radaton traced n the classc Monte-Carlo method In ths manner, t s possble to track the flow of thermal power between system surfaces and defne requred temperature dstrbutons. In a classc form, ths s acheved by the Monte-Carlo method whch, theoretcally, makes t possble to render any heat exchange system. Unversalty (whereby any system can be modelled) s an unquestonable and mportant advantage of such solutons. Unfortunately, assumng that t s necessary to trace each emtted and each reflected ray, the number grows exponentally (see Fg. 6) and even contemporary computers seem too Wthn ths scope, research s focused on developng methods that would allow a reducton of the number of rays to be analysed, whle mantanng the requred accuracy of stmulatons. One of such methods s a technque based on nvestgatng the course of rays. Smlarly to computer graphcs, t conssts of tracng rays from the moment of emsson (from the source) untl dsappearance (e.g. absorpton). In terms of prncple, the method s dentcal as the one appled n computer graphcs. Key dfferences are an effect of a dfferent descrpton of the lght system and the thermal system. The range of acceptable smplfcatons s also dfferent. 4. Smlartes and dfferences between models Smlartes and dfferences of models descrbng the phenomena analysed here (llumnaton, exchange of thermal energy) from the vewpont of computer graphcs and radatve heat exchange are lsted n Table below. Table. Computer graphcs model and radatve heat exchange model Model of computer graphcs Model of radatve heat exchange Propagaton of electromagnetc radaton (vsble radaton) λ ( ) µm Photorealstc mage of the scene vewed by the observer Dstrbuton of llumnaton on observed surfaces Phenomenon analysed Ultmate goal of modellng Propagaton of electromagnetc radaton (nfrared radaton) λ ( ) µm Requred ntermedate stage Dstrbuton of temperatures (or power exchanges) on the surfaces Dstrbuton of absorbed powers on analysed surfaces ISSN: Issue 4, Volume 3, October 008

9 Table cont. Generaton and propagaton of radaton Dffuse reflecton, specular reflecton Essental prmary phenomena Generaton and propagaton of radaton Dffuse reflecton, absorpton Secondary phenomena taken nto account Drect reflecton, dffracton, multchromatcs Absorpton, nterference, polarsaton Dsregarded phenomena Specular or drect reflecton Polarsaton, nterference, dffracton Other factors underlyng dfferences n approach to computer graphcs and radatve heat transfer nclude dfferng smplfcaton assumptons adopted (often wthout lstng) n developng software for both task types. A relevant lst s provded n Table 3 below. Table 3. Basc smplfcatons assocated wth analysed models Model of computer graphcs Model of radatve heat exchange Ether source of radaton (lght) or reflectve surface No power balance for surfaces and the entre system Absorpton dsregarded Illumnaton boundary condtons only for lght sources Model of computer graphcs Source surfaces are ndependent of one another Does not carry energy No change of energy status of the surface Essental role played by colour (wavelength) Boundary surfaces Radaton Radaton source and radaton-reflectng surface at the same tme Power balance for each surface and the entre system Absorpton calculated Thermal boundary condtons for each boundary surface Model of radatve heat exchange Each surface affects other surfaces Carres energy Change of energy status of the surface Wavelength often nsgnfcant Phenomena whch are mportant for reflecton Surface texture, colour, gloss, halo Absorpton of energy 5 Smlartes of algorthms used n computer graphcs and radatve heat exchange Although goals pursued by computer graphcs applcatons (vsualsaton) and programmes developed for radatve heat exchange modellng (determnaton of heat transfer) are dfferent, there s a sgnfcant convergence between selected stages of both types of applcatons, manly the phase of generaton, propagaton and reflectons of radaton on boundary surfaces. Methods employed to trace the course (hstory) of ndvdual rays also tend to be dentcal, as shown n Fg. 7. Data on lght sources or boundary condtons Smulaton of radaton emsson Smulaton of multple reflectons Completon of heat transfer task: calculated transfers of power and temperature Data on scene geometry and surface emssvty Data on surface reflectvty Procedures shared by computer graphcs and radatve heat exchange Absorpton analyss Calculaton of power absorbed Illumnaton analyss Completon of llumnaton task: lumnances determned observer data Vsualsaton End of vsualsaton task: known mage of the scene Fg. 7 Functonal dagram of the modellng process: of radatve heat exchange and vsualsaton. [3,4] 6 Replaceablty condtons Analysng the basc equatons of computer graphcs and radatve heat exchange n (() and (6)), t s evdent that they are nearly dentcal. Ths results from the fact that they both descrbe the same phenomenon,.e. generaton and reflecton of the electromagnetc ISSN: Issue 4, Volume 3, October 008

10 wave n a set drecton. Computer graphcs procedures and models are a subset of radatve heat exchange procedures and models, as n solvng tasks related to the lghtng technology one can adopt much more farreachng smplfcatons than n radatve exchange. However, t s computer graphcs that has been growng rapdly n recent years, wth new mproved computatonal procedures and smulaton models beng developed on an ongong bass. It seems an nterestng endeavour to explore the possblty of usng exstng tools n the form of computer graphcs software to compute radatve heat transfer. Not every computer graphcs applcaton can be adusted for modellng radatve heat exchange. There are a number of precondtons whch can be dvded nto two man groups: necessary and desrable. The former group of factors determne the sutablty of a gven computer graphcs programme for adaptatons allowng smulatons of radatve heat exchange n general. The latter category ncludes factors that markedly facltate the task and ncrease the potental accuracy of any computatons that may be performed n the future. Absolutely necessary precondtons nclude the followng [,3,4]: relance on the basc equaton () or () n llumnaton modellng, modellng of multple reflectons, possblty to enter ntermedate results n the form of fles (not btmaps) contanng data on the dstrbuton of radance (or an equvalent quantty) on set surfaces. Desrable precondtons nclude: possblty to defne reflectve characterstcs of surfaces n the form of BRDF, compatblty of the computer graphcs programme wth CAD-type applcatons, possblty of any arrangement of emsson characterstcs of radaton sources, relance of the graphcs programme on the modellng of radant, not lumnous quanttes. Study [] shows that boundary surface S (wth defned Drchlet condton (8),.e. temperature T=t73) can be replaced wth an energetcally equvalent surface of the source of radaton wth radance L and emssvty ε defned wth the formula []: Usng the energy conservaton law, t s possble to prove that for the surface, for whch Drchlet s condton has been specfed (8) the equvalent surface of the emssvty ε has the radance L, as descrbed by the formula (6). 4 σ ε ( s ) T ( x0, z0 ) L( s, x0, z0 ) = (6) π cosϕ n0 s where: cosϕ =, no the normal to the surface n n0 s pont r o =(x o, y o, z o ) and drecton s. Smlarly, surfaces wth defned Neumann condton (9) an equvalent surface wth emssvty ε can be assgned radance L []: p( x0, z0) s Esum ( x0, z0 ) ε ( s ) L( s, x0, z0 ) = π (5) where E sum represents total rradaton of the area analysed, resultng from the remanng surfaces of the system. As a consequence, usng the formulas gven n (6) and (7), the radatve heat transfer task can be transformed nto a lghtng technology task, as llustrated n Fg. 8. Radatve exchange task; set temperatures and powers T, p Converson nto radant quanttes L, ε 7 Converson of thermal quanttes nto radant quanttes Radatve heat exchange s descrbed by means of thermal quanttes (temperature, power densty). Computer graphcs applcatons process lumnous quanttes (lumnance) or radant quanttes (radance). An adustment of computer graphcs programmes for the modellng and smulaton of radatve heat transfer requres thermal quanttes to be converted nto radant quanttes. Ths apples n partcular to boundary condtons descrbng radatve heat transfer systems. Computer graphcs applcaton: nvestgaton of ray hstory L, Ε Power absorpton procedures T, P a, Completon of the radatve exchange task Fg. 8. Stages n modellng radatve heat exchange usng computer graphcs programmes ISSN: Issue 4, Volume 3, October 008

11 8 Sample thermal computatons usng computer graphcs programmes There are a number of computer graphcs packages that can be appled for scene vsualsaton purposes. One of such programmes s Radance. The system s ray trackng procedures, complemented wth other specfc functonaltes, n effect produce a tool (an applcaton) whch makes t possble to model radatve heat transfer systems. A number of computatons have been carred out, focused n partcular on thorough tests of smple systems for whch precse analytcal solutons are avalable. Detaled results are dscussed n a range of publcatons, ncludng [,4]. All cases analysed reveal a sgnfcant smlarty of results of smulatons (though ncludng a certan element of randomness) and precse computatonal results. The problems presented n ths paper, partcularly computer graphcs algorthms, are only sketchly descrbed. A more exhaustve revew of these ssues can be found n broad lterature on both computer graphcs [6] and radatve heat exchange [7,]. 9 Conclusons Recent development of computer graphcs technologes markedly surpasses the level of research and the range of avalable applcatons of radatve heat exchange programmes. Nearly all computer graphcs programmes can be used n radatve heat transfer computatons. Programmes based on the radosty method can be used wthout maor modfcatons, wth only mnor elements added. On the other hand, ray tracng-based methods requre more extensve adaptatons. In all modfcatons, startng thermal data must be transformed nto lumnous quanttes, whle output lumnous data must be converted nto thermal data. Some computer graphcs programmes (e.g. Radance) meet the condtons allowng ther applcaton not only n lghtng smulatons, but also n resoluton of the most complex tasks nvolved n radatve heat exchange. The programmes have procedures for emsson modellng and the procedure of multple reflectons of vsble radaton. Complance wth condtons ndcated n the paper above s a precondton for employng such programmes to perform the basc functon of a modellng programme,.e. nvestgate the propagaton of rays n a preset system. An addton of specfc functons for converson of thermal quanttes nto lumnous quanttes (and vce versa), and a functon for addng up power carred by each ray, helps to acheve a good tool for modellng radatve heat transfer. Computatons performed to verfy the proposed soluton have confrmed a hgh degree of smlarty between results of smulatons of radatve heat exchange (effected usng the programmes) and analytcal computatons [,3,4]. References: [] Celakovsk S., Preda M., Kaladzsk S., Davcev D, Preteux F.: Renderng Strateges for Dsplayng MPEG-4 3D Graphcs Obects. WSEAS Transactons on Informaton Scence & Applcatons, Issue 0, Volume 3, October 006 [] Domke K., Modelowane Symulaca Badane Radacyne Wymany Cepła w Środowsku Radance, ser. Rozprawy nr 378, Wyd. Pol. Pozn., Poznań, 004, (n polsh). [3] Domke K.: Use of graphcs software n radatve heat transfer smulaton. n: Advanced Computonal Methods n Heat Transfer IX, ed. Sunden B., Brebba C.A. WITpress, Southampton, Boston, pp. -30, 006. [4] Domke K. & Hauser J., Applcaton of RADIANCE procedures for radatve heat transfer modelng: n: Computer Ad Desgn of Electroheat Devces ed. Herng M., Sadak Cz. & Wcślk M., Wyd. Pol Śląske, Glwce, pp. 3-49, 00. [5] Fala P., Kadlecova E.: Progressve Methods n Numercal Modelng by Fnte Elements. n: Proceedngs of the 7 th WSEAS Int. Conf. On Automatc Control, Modelng and Smulaton. Praque, 005 [6] Kaya J., The Renderng Equatons, Computer Graphcs, 0 (4), 986. [7] Modest M. F., Radatve Heat Transfer, II ed., Academc Press Amsterdam, Boston, London, N. York and Sydney, 003. [8] Rahbar S.: Solvng Fredholm Integral Equaton Usng Legendre Wavelet Functons. WSEAS Transactons on Matematcs, Issue 3, Volume 3, July 004 [9] Sllon F. X. & Puech C., Radosty and Global Illumnaton, Morgan Kaufmann Publshers Inc., San Francsco and Calforna, 994. [0] Segel R. & Howell J. R., Thermal Radaton Heat Transfer, Mc-Graw Hll Book Co., N. York, 97. [] Vass G., Dffuse and Specular Interreflecton wth Classcal, Determnstc Ray Tracng. Dept. of Control Engneerng and Informaton Technology Techncal Unversty of Budapest. ISSN: Issue 4, Volume 3, October 008