Temperature Distribution and Heat Radiation of Patterned Surfaces at Short Wave Lengths
TTemperature Distribution and Heat Radiation of Patterned Surfacesat Short Wave Lengths
Thorsten Emig
1, 2 Massachusetts Institute of Technology, MultiScale Materials Science for Energy and Environment,Joint MIT-CNRS Laboratory (UMI 3466), Cambridge, Massachusetts 02139, USA Laboratoire de Physique Th´eorique et Mod`eles Statistiques, CNRS UMR 8626,Bˆat. 100, Universit´e Paris-Saclay, 91405 Orsay cedex, France (Dated: August 28, 2018)We analyze the equilibrium spatial distribution of surface temperatures of patterned surfaces.The surface is exposed to a constant external heat flux and has a fixed internal temperature thatis coupled to the outside heat fluxes by finite heat conductivity across surface. It is assumedthat the temperatures are sufficiently high so that the thermal wavelength (a few microns at roomtemperature) is short compared to all geometric length scales of the surface patterns. Hence theradiosity method can be employed. A recursive multiple scattering method is developed that enablesrapid convergence to equilibrium temperatures. While the temperature distributions show distinctdependence on the detailed surface shapes (cuboids and cylinder are studied), we demonstrate robustuniversal relations between the mean and the standard deviation of the temperature distributionsand quantities that characterize overall geometric features of the surface shape.
PACS numbers:
I. INTRODUCTION
Planck’s law describes the intensity of radiation of ablack body with temperature T at a given wavelength[1]. Integration over all wavelengths yields the Stefan-Boltzmann law [2] for the total power P emitted by theblack body P = σAT (1)where A is the surface area of the body, and σ = π k B / (60 (cid:126) c ). For real materials Eq. (1) is modified bymultiplying σ with the emissivity of the material. How-ever, recently various modifications of the radiation lawdue to size and shape of the body have been exploredand new general approaches based on scattering theoryhave been developed [3]. In general, the (effective) emis-sivity of an object depends on its size and shape due toself-scattering of the emitted radiation. Recent scatter-ing approaches, however, assume that the bodies’ surfacehas a spatially constant temperature. In general, this isnot strictly justified due to self-absorption of heat emit-ted by a body with a non-planar surface.Information about the temperature distribution onpatterened objects and the resulting transport of energyby heat radiation [4] is important to many science andengineering applications: radiating micro-structured sur-faces, transfer in combustion chambers and heat exchang-ers, climate phenomena like the spatial variation of landsurface temperatures [5], solar energy utilization and thedesign of sustainable buildings. Modeling of heat radi-ation and radiative heat transfer in large-scale, complexgeometries consisting of many shapes, objects and mate-rials presents enormous challenges due to the long-rangewave nature of electromagnetic radiation. Most precisesolution requires numerical solution of the electromag-netic wave equation to obtain the scattering of electro- magnetic waves at all surfaces. However, for large com-plex geometries, the computing time and lack of preci-sion of this methods increases [6]. Hence, it is desirableto identify universal scaling laws that can predict howshape and geometry influences spatial variation of tem-peratures and heat radiation. This work attempts topropose a step in this direction by considering surfaceswith various geometric patterns.We assume that the thermal wavelength λ T = (cid:126) c/ ( k B T ) is short compared to all geometric length scalesof the surface patterns. In this limit, geometric optics candescribe heat radiation leading to the so-called radiositymethod that is widely used for heat phenomena and vi-sual rendering [7]. It assumes diffuse reflections at thesurfaces and hence is an alternate method to ray tracing.The surface is decomposed into patches that are coupledvia a so-called view factor matrix that measures the frac-tion of radiation that travels from one surface patch toanother. Similar methods can be applied to interactivesound propagation in complex environments (urban orindoor environments such as auditoriums) [8]. II. THE MODEL
We consider a geometrically structured two-dimensional surface that is decomposed into smallsurface “patches” given by N mutually joining polygons P j , j = 1 , . . . , N , defined over a planar base plane( xy -plane). The polygons are oriented so that theirsurface normals n j are pointing all into the samehalf-space, the “outside”, (say the positive z -direction)which contains the source of the incoming externalheat flux. For simplicity, we assume further that thepolygon surface normals are either normal or parallel tothe base plane. Each polygon is further characterized a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b by an emissivity (cid:15) j , surface thickness d j , and thermalconductivity κ j . On the “inside” (negative z -direction)of the surface a local equilibrium inside temperature T int j is imposed for each polygon. We assume that thesurface receives a homogeneous radiant flux L from theoutside half-space or “sky”. The goal is to compute theequilibrium temperatures T j on the outside surfaces ofthe polygons assuming that they are insulated againsteach other. These temperatures are determined byequating the internal and external net flux densities foreach polygon. The internal net flux is obtained fromthe stationary heat conduction equation q int j = − κ∂ n T j integrated across the surface thickness d j yielding q int j = ( T j − T int j ) κ j /d j . The external net flux q ext j isobtained as the sum of the incoming fluxes from the sky( L ) and those scattered from all other visible polygonsand the heat flux σ(cid:15) j T j radiated by the surface j where σ is the Stefan-Boltzmann constant.For the simple case of a single planar surface ( j = N =1), the condition q ext1 = q int1 yields( T flat − T int ) κd = (cid:15) ( L − σT ) , (2)which determines the outside surface temperature T flat of the flat surface as function of known parameters.For a general structured surface one has to considermultiple reflections between surface patches that con-tribute to the net external fluxes. To describe this ef-fect, it is assumed that the surface patches are gray diffu-sive emitters, i.e., the emissivity is frequency independentand the radiation density is constant across the surfacepatches and emitted independent of direction. We expectthis to be a good approximation for thermal wavelengthsthat are small compared to the geometric structure of thesurface and hence the size of the patches. Then we canapply to radiosity concept to obtain the external fluxes q ext j [4]. For a given surface patch j , the outgoing radiantflux is given by the sum of emitted thermal radiation andthe reflected incoming radiation, J j = σ(cid:15) j T j + (1 − (cid:15) j ) E j (3)where we used that the reflectivity equals 1 − α j for anopaque surface where α j = (cid:15) j is the absorptivity. Howmuch energy two surface patches exchange via heat trans-fer depends on their size, distance and relative orienta-tion which are encoded in the so called view factor F ij between patches i and j . F ij is a purely geometric quan-tity and does not depend on the wavelength due to theabove assumption of diffusive surfaces. It is defined bythe surface integrals F ij = (cid:90) A i (cid:90) A j cos θ i cos θ j πA i | r ij | dA i dA j (4)where θ i is the angle between the surface patch’s normalvector n i and the distance vector r ij which connects apoint on patch i to a point on patch j , and A i is thesurface area of patch i . The view factor matrix obeys the important reciprocity relation A j F ji = A i F ij andadditivity rule (cid:80) j F ij = 1. With this geometric quantity,the radiative flux received by surface patch j from allother surface patches can be expressed as E j = (cid:80) i F ji J i ,and one can solve Eq. (3) for the vector of outgoing fluxes,yielding J = [ − ( − (cid:15) ) F ] − J , (5)where we combined the fluxes J j from all patches into avector J and the radiation σ(cid:15) j T j into a vector J to usea matrix notation. Here is the identity matrix and (cid:15) the diagonal matrix with elements (cid:15) j . To compute thesurface temperaturs T j we need to compute the net heattransfer to surface patch j which is given by the incidentradiation E j minus the outgoing flux J j , leading to thenet flux q ext j = (cid:80) i F ji J i − J j . In vector notation this netflux becomes q ext = ( F − ) [ − ( − (cid:15) ) F ] − J . (6)In the stationary state, the surface patch temperaturesare then determined by the condition that the net exter-nal flux equals the net internal flux, q ext = q int where q int defines the vector with elements ( T j − T int j ) κ j /d j due to heat conduction across the surface (see above).This condition uniquely fixes the temperatures T j whenall other parameters including the external (“sky”) flux L are known. In the following, technically we include the“sky” as an additional surface so that we have now N + 1surface patches. The corresponding additional matrix el-ements for the view factor matrix F follow from reci-procity and additivity rules, and we include the down-ward radiation L as the ( N + 1) th component in J .Knowing the surface temperatures, a number of inter-esting observables can be obtained. An effective emissiv-ity of the total surface can be defined as the ratio (cid:15) eff = Q/Q bb where Q = [ F [ − ( − (cid:15) ) F ] − J ] j=”sky” is thenet flux towards the “sky” and Q bb = [ F J bb ] j=”sky” isagain a net flux to the “sky” but assuming that all sur-face patches radiate as ideal black bodies, correspondingto J bb = σ [ T , . . . , T N , effective temperature T eff ,as observed from the “sky”, can now be defined as wereall surfaces black bodies at their local temperature, sothat σT = Q bb and Q = σ(cid:15) eff T . We also define thedifference ∆ T = T eff − T flat . III. NUMERICAL IMPLEMENTATION M1 M2 M3 FIG. 1: Surface patch temperature distribution for modelsM1, M2, and M3. Colors represent temperature changes fromminimum (blue) to maximum (red) temperature. For valuessee histograms in Figs. 2 to 4 and Tab. I.
The numerical implementation of the model describedabove follows these steps:1. The surface is decomposed into oriented patcheswhich is done here by triangularization so that theentire surface is composed of planar triangular sur-face elements, see Fig. 1 with their surface nor- mal vector pointing to the “outside” of the surface,i.e., pointing towards the “sky”. For later analy-sis, these elements are grouped into three differentclasses: horizontal “base” patches (b) that are lo-cated within the base plane z = 0, horizontal “top”patches (t) that are located above the base planeand “vertical” patches (v) that are perpendicularthe base plane and connect the patches in class band t.2. Determine for all pairs of patches if the view be-tween them is blocked by other patches. This isdone by testing for potential intersections of the rayconnecting the two centroids of a pair of patchesand all other surface patches. It is sufficient toperform this visibility test for pairs of patches ofthe type ( v, b ), ( v, t ) and ( v, v ) where the first (sec-ond) letter denotes the class of the first (second)patch. For all these combinations potential block-ing patches must be in class v .3. If the view between a pair ( i, j ) of patches is notblocked and the first patch can “see” the outsideof the second, the view factor F ij is computed, us-ing the exact closed form expression described in[9]. This is done for all patch class combinations( v, b ), ( v, t ) and ( v, v ) with the restriction i < j for( v, v ) since the view factors for i > j follow fromreciprocity.4. Construct the total view factor matrix F for allpatches of classes v , b and t and the single en-closing surface describing the “sky”. This is doneby using reciprocity to obtain the matrix elementsfor the patch class combinations ( b, v ) and ( t, v ).The patches of classes b and t cannot see eachother so that the view factor submatrix for theseclasses vanishes. To obtain the view factor for thetransfer from a surface patch i towards the “sky”we use the sum rule (cid:80) j F ij = 1, i.e., F i ”sky” =1 − (cid:80) j ∈{ b,t,v } F ij . The view factor for the transferfrom the “sky” to a patch i follows from reciprocityas F ”sky” i = A i A F i ”sky” where A is the total area ofthe surface.5. The inverse matrix of Eq. (6) can be computedas a truncated geometric series since the emissiv-ities are sufficiently close to unity and the viewfactors F ij < K − ≡ [ − ( − (cid:15) ) F ] − = (cid:80) n c n =0 M n with M = ( − (cid:15) ) F . We find that n c = 6 is suffi-ciently accurate approximation for the parametersused below.6. Finally, we compute the surface patch temperatures T j by an iterative solution of the equilibrium con-dition q ext = q int [see Eq. (6)] for given surfaceemissivities (cid:15) j , downward radiation L , interior tem- model A g A v ¯ F all → sky ¯ F b → sky patches T flat ¯ T ¯ T v ¯ T t ¯ T b σ σ v σ t σ b T eff (cid:15) eff ∆ T M1 50 180 0.7175 0.8037 4140 (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 0 . peratures T int j and effective thermal conductivities κ j /d j . The iteration steps are as follows:(i) Choose initial patch temperatures T ( ν =0) j .(ii) Compute the external flux q ext ( ν =0) =( F − ) K − J ( ν =0)0 with the N + 1dimensional initial vector J ( ν =0)0 =[ L, σ(cid:15) T ( ν =0)1 4 , . . . , σ(cid:15) N T ( ν =0) N ].(iii) Compute the updated patch temperatures T ( ν =1) j from the equation q ext ( ν =0) j =( T ( ν =1) j − T int j ) κ j /d j for j = 1 , . . . , N .(iv) Continue with step (i) to start the next itera-tion step, i.e., q ext ( ν =1) = ( F − ) K − J ( ν =1)0 with the vector J ( ν =1)0 = { L, σ(cid:15) [( T ( ν =1)1 + T ( ν =1)1 ) / , . . . , σ(cid:15) N [( T ( ν =1) N + T ( ν =1) N ) / } .In (iv) and all following iteration steps it is usefulto use the average of the last two iterations for thepatch temperatures, as indicated here, to obtainrapid convergence. Typically, for the models andparameters used below, after about 20 iterations astable solution for the patch temperatures had beenreached (within a relative accuracy of 10 − ). IV. RESULTS
In order to study the influence of the density and shapeof surface patterns on the temperature distribution, wehave considered three different surface structures thatare all periodic in both spatial directions, see Fig. 1.For all surfaces, the dimension of a unit cell given by L x × L y = 20 ×
20 (in arbitrary units). It is assumedthat all spatial dimensions, however, are large comparedto the thermal wavelengths λ T = (cid:126) c/ ( k B T ) of the surfacetemperatures which is in the range of a few microns forthe temperatures considered below. The downward radi-ant flux from the “sky” is set to L = 300W per unit sur-face area, the interior surface temperatures are all set to the temperature T int j ≡ T int = 293 . ◦ K, and all surfacethicknesses d j and thermal conductivities κ j are chosensuch that ratio κ j /d j = 5 . (cid:15) = 0 . (cid:15) = 0 . × × × ×
7, respectively. ModelM2’s unit cell is composed of two rectangular cuboidswith dimensions 6 × × × ×
12, respec-tively. Finally, the unit cell of model M3 is composedof two cylinders of radii r j with dimensions r = 3 × r = 5 ×
12, respectively. The corresponding area A g (per unit cell) of the base plane that is covered bythese elements (cuboids, cylinders) and the area A v (perunit cell) of their vertical surfaces are summarized inTab. I. In that table the total number of surface patchesis also indicated. As we shall see below, other importantgeometric quantities are certain averaged view factors:the average “sky” view ¯ F all → sky = (cid:80) j ∈{ b,t,v } F j ”sky” /N from all surface patches, and the average “sky” view¯ F b → sky = (cid:80) j ∈{ b } F j ”sky” /N b from patches of the baseplane only, where N b is the number of base plane patches.These averages were restricted to the central unit cell toavoid boundary effects and they are also given in Tab. I.Next we analyze the results for the temperature dis-tributions as they follow from the numerical approachoutlined above. As can be seen from Fig. 1, the cold-est patches are those on the top of the structures (classt). Since the top patches of the highest structures donot interact with any other patches, their temperatureequals the temperature T flat of a planar surface which setshence the minimum value for the temperature distribu-tion. Highest temperatures are observed on the verticalsurface patches with an increase in temperature from thetop to the bottom. This pattern results from a decreasedview of open space (“sky”) for vertical patches and re-flections from the base patches close to the bottom ofthe elevated structures. The base patches’ temperaturedecays away from the structures which is clearly visiblefor the low structures of model M1. The non-central unitcells show colder surface patches towards the edges of thesurface due to their proximity to the boundaries whichenables an increased emission of heat.Figures 2 – 4 show histograms for the surface temper-ature distributions of the three models, indicating thenumber of patches at a given temperature. Different col-ors label the three different classes of surface patches:vertical, base, and top patches. To reduce boundary ef-fects, the histograms show the distribution of the cen-ter unit cell. For all models, panels (a) and (b) showthe entire distribution for (cid:15) = 0 . (cid:15) = 0 .
9, respec-tively. Panels (c) and (d) show the distributions for thevertical patches only, again for (cid:15) = 0 . (cid:15) = 0 .
9, re-spectively, with different colors labeling now equidistantheight intervals over the base plane in which the patchesare located. A general feature of all models is that thesurface temperatures increase from top patches over basepatches to vertical patches. It is interesting to note thatonly for model M1 there is a clear separation of baseand vertical temperature ranges whereas for M2 and M3the base temperatures fall into the mid or lower range ofvertical temperatures. Another interesting observationis that the vertical temperature distribution has a singlepeak for models M1 and M3, particularly in the latter,and a two-peak structure for model M2. We interpretthis as a consequence of the proximity of two cuboidsof different height. This view if supported by the vari-ation of the distribution of vertical temperatures withheight, see Fig. 3(c) and (d): Only the peak at smallertemperatures contains patches of the largest height classH4, and hence must represent mainly the taller cuboid.In generel, models M1 and M2 display little overlap be-tween the temperatures corresponding to different heightintervals while model M3 shows less separated tempera-ture ranges for the height intervals. This is presumablyrelated to the continuous range of vertical surface patchorientations for cylinders as compared to cuboids.Table I summarizes various characteristics of the tem-perature distributions. In addition to the quantities T eff ,∆ T and (cid:15) eff defined above, the mean temperature ¯ T ofthe full distribution and the mean temperatures ¯ T j ofthe patch classes j = v, t, b are shown. The measurethe temperature variations across different surface areas,we have also computed the standard deviation σ for thefull distribution and the standard deviations σ j for thedifferent patch classes. Generally, a surface profile withdeeper “canyons” leads a trapping of radiation and hencea larger T eff which measures shape effects. Similarly, theeffective emissivities (cid:15) eff show a larger increase for pro-files with narrow “canyons” since they render the surfacemore black due to the trapping of radiation. A surfacewith a lower bare emissivity ( (cid:15) = 0 .
5) has a larger shapeinduced increase in emissivity as an already highly emis-sive surface ( (cid:15) = 0 .
282 283 284 285 286 287010203040 T ( K ) pa t c he s Surface Temperatures, ϵ all = verticalbasetop T ( K ) pa t c he s Surface Temperatures, ϵ all = T int = verticalbasetop T ( K ) pa t c he s Vertical Surface Temperatures, ϵ all = H1H2H3H4 T ( K ) pa t c he s Vertical Surface Temperatures, ϵ all = H1H2H3H4 T int = T int = T int = T flat T flat (a)(b)(c)(d) FIG. 2: Histograms for surface patch temperatures of thecentral unit cell of model M1: (a) temperatures for the threedifferent patch classes vertical ( v ), base ( b ), and top ( t ) foremissivity (cid:15) = 0 .
9, (b) same as (a) for emissivity (cid:15) = 0 .
5, (c)temperatures for vertical ( v ) patches for emissivity (cid:15) = 0 . H H (cid:15) = 0 .
282 284 286 288 290 292 2940102030405060 T ( K ) pa t c he s Surface Temperatures, ϵ all = verticalbasetop
286 288 290 292010203040506070 T ( K ) pa t c he s Surface Temperatures, ϵ all = verticalbasetop
286 288 290 2920510152025 T ( K ) pa t c he s Vertical Surface Temperatures, ϵ all = H1H2H3H4
287 288 289 290 291 292 293010203040 T ( K ) pa t c he s Vertical Surface Temperatures, ϵ all = H1H2H3H4 T int = T int = T int = T int = T flat T flat (a)(b)(c)(d) FIG. 3: Histograms for surface patch temperatures as inFig. 2 for model M2.
An important problem is the identification of geomet-ric parameters that characterize relevant features of thesurface shape and show a universal relation to certainmoments of the surface temperature distributions. Uni-versal means here that the relation, instead of dependingon particular details of the surface structure, relates tosimple overall features of the surface shape. Potential
282 283 284 285 286 287 288 289020406080 T ( K ) pa t c he s Surface Temperatures, ϵ all = verticalbasetop
285 286 287 288 289 290020406080100120 T ( K ) pa t c he s Surface Temperatures, ϵ all = verticalbasetop
285 286 287 288 28901020304050 T ( K ) pa t c he s Vertical Surface Temperatures, ϵ all = H1H2H3H4 T ( K ) pa t c he s Vertical Surface Temperatures, ϵ all = H1H2H3H4 T flat T flat T int = T int = T int = T int = (a)(b)(c)(d) FIG. 4: Histograms for surface patch temperatures as inFig. 2 for model M3. candidates for such geometric parameters are listed inTab. I: The surface areas A g , A v , and the averaged viewfactors ¯ F all → sky , ¯ F b → sky .According to the Stefan-Boltzmann radiation law, thetotal radiative power emitted by an ideal black body isproportional to its surface area. For non-ideal bodies,the radiative power is reduced by an effective emissivitythat depends in general on material, size and shape ofthe body. Postulating that multiple reflections of heatradiation is of sub-leading order for the surface modelsconsidered here, one can expect that the shape inducedincrease in mean surface temperature ¯ T is proportionalto the increase in surface area due to the surface pat-tern. Fig. 5 shows the dependence of ¯ T on the relativeincrease in surface area (due to vertical patches of totalarea A v ). Indeed, the data are well described by a linearscaling, demonstrating that the detailed shape of surfacestructures is unimportant for the mean temperature.Another geometric quantity that is more sensitive toshape than the overall increase in surface area is the av-eraged open (“sky”) view ¯ F b → sky from the base planepatches. For a planar surface with ¯ T = T flat , the view isunobstructed and hence ¯ F b → sky = 1. Any surface struc-ture reduces ¯ F b → sky and in fact it has been observed ex-perimentally in the context of urban climate that meanair and building surface temperatures tend to increaselinearly with a decrease of the so-called sky-view. Toprobe this relation quantitatively, we show in Fig. 6 themean surface temperature as function of the mean openview factor ¯ F b → sky . Our data for ¯ T show a clear lineardecrease with increasing mean “sky” view, with a uni-versal slope that is independent of the particular surfacepatterns. The slope, however, does depend on the emis-sivity. The total view factor ¯ F all → sky , averaged over allsurface patches (see Tab. I) does not show a universallinear relation across all models.Fig. 1 shows that the temperature distributions havestrong spatial variations. Hence, it is interesting to iden-tify the key geometric parameters that determine the sta-tistical moments of the temperature distributions. Wehave computed the standard deviation σ of the total dis-tribution which is shown in Fig. 7, rescaled by the tem-perature difference ¯ T − T flat . The value of σ increaseswith the emissivity (cid:15) which sets the scale for the typicalsurface temperatures (which are of course also dependenton the heat flux from the interior side of the surface, char-acterized by the temperature T int and heat conductivityof the surface patches.) However, after the rescaling by¯ T − T flat , we observe a convincing collapse of the data fordifferent (cid:15) (see Fig. 7). Interestingly, the shape depen-dence of σ/ ( ¯ T − T flat ) is controlled by the ratio of verticalsurface area A v and base surface area A g covered by ele-vated structures. This ratio measures the aspect ratio ofheight and width of the surface structures, and it showsa linear relation to σ/ ( ¯ T − T flat ). We interpret this ob-servation as follows: by how much the temperature actu-ally varies within the typical range between the minimum T flat and the mean ¯ T is controlled by the homogeneity ofthe heat flux impinging on the surface patches. Tall andthin, antenna like structures (like the cylinders of modelM3) produce a more homogeneous heat flux (due to theirincreased view factors) and hence less temperature varia-tion. This can be observed clearly from the temperaturedistribution on the base plane patches in Fig. 1 which shows least variation for model M3. ▲ ▲▲■ ■■▲ ϵ= ■ ϵ= A v + AA T T f l a t M1 M3 M2
FIG. 5: Mean surface temperature (rescaled by the flatsurface temperature) as function of the relative increase( A v + A ) /A in surface area A = L x L y due to vertical sur-face patches. ▲▲ ▲ ■■ ■ ▲ ϵ= ■ ϵ= F b → sky T T f l a t M1M3M2
FIG. 6: Mean surface temperature (rescaled by the flat sur-face temperature) as function of the mean view factor ¯ F b → sky from base surface patches towards the “sky”. ▲ ▲ ▲■ ■ ■▲ ϵ= ■ ϵ= A v A g σ T - T f l a t M1 M3M2
FIG. 7: Standard deviation σ of the surface temperaturedistribution, rescaled by the difference ¯ T − T flat , as functionof the ratio of vertical surface area A v and surface area A g covered by patterns (cuboids, cylinders). Data collapse isobserved for different emissivities. V. CONCLUSIONS
We have analyzed the influence of geometric surfacepatterns and emissivity on the surface temperaturedistribution, assuming a homogeneous internal tempera-ture and external radiative flux. The surface geometryis assumed to vary on scales large compared to the thermal wavelengths, i.e., the temperatures have tobe sufficiently large. The details of the temperaturedistributions show a rich structure that is dependent onthe detailed surface shape. However, we could identifyparameters that measure relevant overall geometricfeatures which obey universal relations to the mean andstandard deviation of the surface temperature distribu-tions. It would be interesting to probe more geometriesand a larger range of parameters to determine the rangeof validity of these relations. Also, our study should beextended to non-periodic patterns, and random surfaceprofiles. There are a number of interesting conceptionalextensions of the approach presented here. For lowertemperatures, or shorter scale surface patterns, diffrac-tion effects should be added to the radiosity approach.For highly reflective materials, specular reflections areexpected to be important and hence should be includedin the iteraction (view) matrix. Surface geometry isalso expected to modify convective heat transfer whichinfluences surface temperatures. There is a plethora ofpossible applications of our results ranging from heattransfer between structured surfaces to the study ofclimate phenomena.
Acknowledgments