DYNAMIC THERMAL BUILDING ANALYSIS WITH CFD MODELLING RADIATION

Proceedngs: Buldng Smulaton 2007 DYNAMIC THERMAL BUILDING ANALYSIS WITH CFD MODELLING RADIATION Ztzmann T. 1, Pfrommer P. 1, Cook M.J. 2 1 Unversty of Apples Scences Coburg, Germany 2 Insttute of Energy and Sustanable Development, De Montfort Unversty, UK ABSTRACT In an attempt to reduce the hgh computatonal efforts for dynamc thermal smulatons usng Computatonal Flud Dynamcs (CFD) the authors have recently developed an adaptve freeze-flow method (.e. freezng of flow equatons over varable tme perods). Ths paper documents the work that has been carred out to predct the radatve surface heat transfer n dynamc thermal buldng processes usng CFD. The Monte Carlo and Dscrete Transfer radaton models were nvestgated and results compared wth analytcal solutons. The Dscrete Transfer model has shown good performance whereas an unrealstc radaton dstrbuton on the surfaces was observed for the Monte Carlo model. A further nvestgaton of the Dscrete Transfer model for the coolng of a sold wall has shown that the adaptve freeze-flow method s an effcent and accurate way to carry out dynamc thermal CFD smulatons whch nvolve radaton. KEYWORDS Radaton modellng; dynamc CFD; conjugate heat transfer; freeze flow method. INTRODUCTION Zonal programs are commonly used for predctng the dynamc thermal performance of buldngs. However, due to ther assumpton of perfectly mxed thermal condtons and one-dmensonal heat conducton n the buldng fabrc, ths can lead to sgnfcant predcton errors. Although CFD programs potentally lead to greater accuracy as they are based on the Fnte Volume Method (Versteeg and Malalasekera 1995), they are often restrcted to steady-state problems or to only few mnutes of smulaton due to the hgh requrement of computatonal resources (e.g. see Groleau 1997, Haupt 2001 and Farvarolo et al 2005). In an attempt to overcome ths lmtaton, an nvarable freeze-flow method was developed by Somarathne et al (2002) The freezng of the flow equatons for fxed tme perods enabled dynamc CFD smulatons to be completed n about 80% of the tme requred for a full dynamc CFD smulaton n whch all equatons were calculated contnuously at every tme step. Predcton errors were small. An enhanced, adaptve freeze-flow method was developed by Ztzmann et al (2007a) whch freezes the flow equatons for varable tme perods. Ths method further reduced the computatonal effort compared wth a fully dynamc CFD smulaton. However, the method was verfed for conjugate heat transfer processes n whch radatve heat transfer was not consdered. Radatve surface heat transfer often domnates the convectve surface heat transfer n buldngs (e.g. Ghatt and Autf 2002 and Sharma et al 2007). It s thus stll unknown how the method behaves f radaton s consdered. CFD s manly used to predct the ar flow patterns usng steady-state schemes (e.g. Cook et al 2003 and Farvarolo et al 2005). Snce the thermal mass of the enclosure s often not modelled and fxed thermal boundary condtons are assgned to the surfaces to reduce the compuatonal effort, radaton modellng s often neglected. Thus, lttle nformaton has been publshed whch provdes reccomendatons on the use of radaton models and approprate parameter settngs for dynamc buldng smulatons usng CFD. Ths paper documents the work that has been carred out to nvestgate the nfluence of radaton models n CFD for the applcaton to dynamc thermal smulaton. Predctons were verfed by comparng the results wth analytcal solutons for two smple test models. Radatve heat transfer was then mplemented n a room model of conjugate heat transfer for dynamc CFD smulatons to verfy the adaptve freeze-flow method. Smulaton results have been compared wth a fully transent CFD smulaton. In ths work the CFD program ANSYS CFX (ANSYS 2005) has been used. ANALYTICAL METHODS The radatve heat flux, E, emtted from a grey body s derved from the law of Stefan-Boltzmann and s calculated as follows: E = σ T ε (1) where ε s the radatve emssvty, σ s the Stefan- Boltzmann constant and T the surface temperature of the body. The radatve heat exchange between surfaces can be analytcally derved for certan smple cases (see Wagner 1998). Two analytcal cases whch have - 31 -

Proceedngs: Buldng Smulaton 2007 been used for comparson wth numercal solutons n ths work are descrbed as follows. The analytcal solutons are based on the assumpton that the surfaces are grey and dffuse reflectve. Parallel plates (Infnte surfaces) For two parallel nfnte large surfaces, wth rays reflected multple tmes between the surfaces before the reflected radaton ntensty approaches zero, e.g. for ε<<1, Eq. (2) s used for calculatng the radatve heat exchange (Wagner 1998): σ (2) q& k = ( T Tk ) 1 1 + ε ε Enclosure (Fnte surfaces) For the case of fnte surfaces n a realstc room, the total ncdent radaton, B, at a surface s the sum of the emtted and reflected radaton from n partcpatng surfaces (Sharma et al 2007): B = n ( ϕ σ ε T + ϕ (1 ε ) B ) (3) k k k k where φ k s the vew factor between two surfaceelements whch may be dvded nto sub-elements to ncrease accuracy. For smple geometres, typcal vew factors are known, whch accurately consder the spatal dstrbuton of the geometrcal condtons (see for example Wagner 1998). Eq. (3) leads to an equaton-system whch couples all surfaces. An mplct soluton of the system n a matrx becomes dffcult due to the non-lnearty of the temperature terms. If the surface temperatures are known, the system becomes lnear and can be solved wth standard matrx soluton technques gvng the total ncdent radaton, B, at each surface. Due to the law of Krchhoff (ε=α) the radatve heat absorbed at each surface s gven by the dfference between the ncdent and emtted radaton usng Eq. (): q& = ε B ε σ T () k No heat can be stored at adabatc wall condtons and Eq. () becomes zero. The equaton smplfes to Eq. (5): σ T = (5) B Insertng Eq. (5) nto Eq. (3) the emssvtes and temperatures dsappear n the equatons for the adabatc surfaces. For the calculaton of radatve heat transfer only the temperatures of the other, nonadabatc surfaces need to be known whch smplfes the soluton of the matrx. The resultng surface temperatures for the adabatc walls are then determned by nsertng the relevant B n Eq. (5). k k RADIATION MODELS IN CFX In CFX, radaton s represented by partcles whch are tracked through the ar doman usng a raytracng method. The spectral radatve transport equaton used n CFX (ANSYS 2005) s: r r diν (, s) r r = ( Ka ν + Ks ν ) Iν (, s) + Ka ν Ib( ν, T) + ds (6) Ks ν r r + di (, s ) r r ν Φ ( s s ) dω + S π π I b blackbody emsson ntensty [W/m 2 ] I v spectral emsson ntensty [W/m 2 ] poston vector [m] s r drecton vector [m] s path length [m] K a absorpton coeffcent [-] K s scatterng coeffcent [-] V frequency [s -1 ] T local absolute temperature [T] Φ n-scatterng phase functon Ω sold angle [rad] S a source term [W/m 2 ] The formal soluton of the radatve transfer equaton s very tme consumng and acheved n CFX by usng approxmate models for the drectonal and spectral dependences. The spectral approxmaton used n ths work s that the medum whch takes part at radaton heat transfer s non-scatterng and grey (.e. ndependent of the wavelength). Openng boundary condtons are consdered as fully transparent to radaton. Walls are treated as dffuse emttng and reflectng opaque boundares whereas symmetry planes are treated as dffusely emttng and specular reflectng boundares n CFX. Two drectonal radaton models are recommended for optcally thn meda (.e. transparent to radaton at wavelengths n whch the majorty of the heat transfer occurs) n ANSYS (2005). One s the Monte Carlo (MC) and the other s the Dscrete Transfer (DT) radaton model. Non-lneartes n the systems due to scatterng, dffuse reflecton, or temperature dependency of radaton quanttes are overcome by an teratve soluton technque. Monte Carlo (MC) model The MC model treats the radaton feld as a photon gas. A photon s selected from a photon source and stochastcally tracked through the system untl ts weght falls below some mnmum at whch pont t des.' Each tme the photon experences an event, for example a surface ntersecton, scatterng or absorpton, the physcal quantty of the radaton ntensty s updated along the ray. Usng ths method, a complete hstory' of that photon n the system s generated. Many photon hstores need to be generated to acheve good estmates of the physcal quantty. Ths value can be specfed by the user. The - 32 -

Proceedngs: Buldng Smulaton 2007 man computatonal overhead for CFX n generatng a hstory s n trackng the photons across the doman. Dscrete Transfer (DT) model The DT assumes that the spatal radaton gradents are relatvely small and the radaton s emtted sotropcally from the surfaces. The user defned parameter number of rays n CFX determnes the degree of spatal dscretsaton of the hemsphere above each fnte surface element for radaton emsson. Hgher values mean a better representaton of the realty and a hgher accuracy, but at the same tme wll result n a sgnfcant ncrease n computatonal effort. The paths of rays are calculated only once, at the begnnng of the smulaton, and are then stored and re-used whch leads to sgnfcant savngs of computer resources. Due to the hgh computatonal resources requred for the calculaton of the radaton feld for the MC and DT radaton models, t s essental to fnd a trade-off between accuracy and computatonal effort. Ths s obtaned usng a coarser mesh for the radaton feld than for the flow feld assumng that the radaton feld changes at a slower rate than any other transport varables (ANSYS 2005). MODEL DETAILS The quas-2d model used for verfcaton of radaton modellng usng CFX contans a square cavty of X=1000mm and Y=1000mm whch nteracts thermally wth an adjacent nternal wall of X=220mm thckness (see Fgure 1). The hgh and low end z-planes form symmetry boundares. The other surfaces are opaque to radaton. y 1 x 2 3 7 AIR 6 5 Fgure 1: Room model of conjugate heat transfer. The surface numbers assgn the boundary condtons specfc to the cases nvestgated to the surfaces. Verfcaton has been carred out n three steps usng test cases 1, 2 and 3. Case 1 The model n case 1 represents two parallel nfnte surfaces of dfferent temperature. The surface numbers 3 and 5 shown n Fgure 1 contan symmetry condtons. The surfaces 1, 2 and 6 are adabatc and surface contans an sothermal temperature of 20 C. The thermal condton at the nterface surface 7 adjusts to the condtons of the sold wall (c=880 J/(kg K), ρ=2300kg/m 3, λ=1.w/(m K)) and the adjacent ar. The sold has an ntal temperature of 27.5 C and the flud has an ntal temperature of 20 C throughout the doman. Radatve emsson coeffcents of ε=0.9 are consdered at the enclosure surfaces. Convectve heat transfer s neglected n ths model (.e. mass and momentum equatons are not calculated) and only conducton and radaton heat transfer s consdered. Case 2 Ths model represents a room cavty enclosed by fnte walls. The model boundary condtons of case 1 are modfed for case 2 replacng the symmetry boundary condtons of surfaces 3 and 5 by adabatc boundary condtons whch contan a radatve emssvty of ε=0.01. Heat transfer due to radaton and conducton s consdered as for case 1. Case 3 The room model for case 3 contans all aspects of heat transfer (.e. conducton, convecton and radaton). Case 2 was modfed usng radaton emssvty values of ε=0.9 for surfaces 3 and 5 and common brck materal for the sold wall (c=835j/(kg K), ρ=1920kg/m 3, λ= 0.72W/(m K)). The flud doman contans an ntal temperature of 23.75 C. Numercal results of CFX for radatve heat exchange were verfed for cases 1 and 2 usng analytcal solutons for two parallel plates of nfnte sze and for a cavty enclosed by fnte surfaces, respectvely. The radatve surface heat transfer for surfaces and 7 was evaluated for the thermal condtons after a dynamc CFX smulaton of 2h. Although verfcaton could have been carred out usng a steady-state soluton scheme for cases 1 and 2, the transent scheme was used n order to nvestgate the potental applcaton of the radaton models to future research of transent problems n terms of computatonal effort. To consder the conjugate heat transfer between the sold wall and the ar, heat conducton was also calculated. The radaton and energy equatons were calculated every tme step n CFX usng a tme step sze of 60s. The room model depth used n CFX for cases 1 and 2 was Z=500mm. The mesh surface and core elements contaned maxmum edge length scales of 50 mm and 100mm, respectvely. The mesh was refned close to the - 33 -

Proceedngs: Buldng Smulaton 2007 surfaces usng prsm nflaton (frst prsm layer heght 0.1 mm, total number of prsms ). Dynamc predctons for CFX usng the adaptve freeze-flow method (Ztzmann et al 2007a) were verfed usng the room model of case 3. Thus, the radaton and energy equatons were calculated every tme step whle the flow equatons were frozen for varable tme perods and later updated. The tme step szes for the unfrozen and frozen flow perods were 1s and 60s respectevely, adoptng the parameter settngs for the lengths of unfrozen and frozen perods from Ztzmann et al 2007a. The maxmum edge length scale of the mesh n the core of the ar space was reduced to a value of 50mm to predct the ar flow patterns more accurately n case 3. To resolve the wall boundary layer for an accurate predcton of convectve surface heat transfer, the near wall regon was further refned usng more prsm elements (16 prsms) than n case 2. The model depth was reduced to Z=50mm to reduce the addtonal computatonal effort. The dynamc temperature dstrbuton n the room was compared wth that of a tradtonal CFD smulaton n whch all equatons are solved wthout nterrupton for each tme step (1s). Temperatures were evaluated for dfferent montorng ponts (.e. at x/x=0.5 at 10%, 50% and 90% of the cavty heght and at y/y=0.5 n the core and the surface of the sold doman, see Ztzmann et al 2007a for a perod of 12h. The k-ω turbulence model was appled (ANSYS 2005). CFD smulatons were consdered to have converged f a resdual root mean square value of 10 - for all equatons solved was acheved. An average temperature of 25.8 C s predcted for surface 7 after two smulaton hours. The analytcally calculated radatve heat flux for surface 7 for these condtons s 27.7W/m 2 usng Eq. (2). The analytcal soluton compares well wth the numercally predcted average values. However, the DT model s the preferable model snce the MC Model showed spatally unrealstc behavour (compare Fgures 2a and 2b). Furthermore, the smulaton usng the DT model requred less than 1% of the CPU tme requred by the MC model for the predcton of the same average values. a) b) RESULTS AND DISCUSSION The data output nterval for smulatons was 5mn. All CFD smulatons reached convergence as defned by the crteron above. Case 1 Smulatons were carred out usng the MC model usng 1,000,000 and 200,000 number of hstores, and smulatons were carred out usng the DT model usng 8 rays. The predcted maxmum, mnmum and average wall radatve heat fluxes after a smulaton perod of 2h are summarsed n Table 1. For the DT model an average value of 27.7W/m 2 s predcted wthout spatal varaton. For the MC model smlar average values (27.8W/m 2 for 1,000,000 and 27.6W/m2 for 200,000 hstores, respectvely) are predcted. However, the surface radatve heat transfer range spatally consderably for the MC model shown by the mnma and maxma n Table 1. Ths unrealstc behavour s thought to be caused due to an nsuffcent long trackng of radaton partcles. Fgure 2 shows the dstrbuton of radatve heat flux for the MC and the DT model whch reflect the observatons for Table 1. Fgure 2: Case 1: Predcted wall radatve heat flux for surfaces and 7 usng (a) the Monte Carlo model usng 1,000,000 hstores and (b) the Dscrete Transfer model usng 8 rays. Case 2 Predctons of the wall radatve heat flux are compared for the MC model (2,000,000, 1,000,000 and 200,000 hstores) and the DT model (8, 15, 30, 50 and 100 rays). Table 2 shows the predcted wall radatve heat fluxes after a smulaton perod of 2h. The range of values for the MC model usng 2,000,000 hstores was about 30W/m 2 and the - 3 -

Proceedngs: Buldng Smulaton 2007 average values obtaned for surfaces and 7 are 21.W/m 2 and 20.9W/m 2, respectvely. Reduced average heat transfer rates are observed for smaller numbers of hstores. Smlar ranges of radatve heat flux are obtaned for the DT model. However, the maxmum wall radatve flux s about 23 W/m 2 for the DT model usng 30 rays and s smaller than for the MC model. Average values predcted for surfaces and 7 are 21.2 W/m 2 and 20.7W/m 2 usng 30 rays. Small dfferences exst between the average radatve heat flux predctons of DT model smulatons usng less than 30 rays compared wth smulatons usng 30 rays (Δ q& 0.8W/m 2 ). The dfferences n average values between the emtted and absorbed radaton for the MC and DT models mght be the consequence of () numercal naccuraces due to mesh dscretsaton and equaton mbalances and () heat conducton consdered wthn the transent smulaton. The latter case means that the adabatc surfaces adapt the temperatures to the adjacent ar whch can lead to a change of the radatve surface heat transfer at the adabatc surfaces and therefore to a change of the radatve heat transfer at the surfaces and 7 compared wth the case n whch no thermal conducton s ncluded. Fgure 3 shows the dstrbuton of the radatve heat flux at the surfaces and 7. A smlar dstrbuton s mapped for the MC model as for case 1 (Fgure 3a). The soluton showed agan an unrealstc radatve heat transfer dstrbuton at the surfaces whch s thought to be caused by an nsuffcently long trackng of partcles as ndcated above for case 1. However, a tendency towards smaller wall radatve heat fluxes s observed near the top and bottom where the surfaces approach the horzontal adabatc surfaces. Ths behavour s also clearly shown for the DT model (Fgure 3b). Such dstrbuton was expected snce the elements close to the boundary of the surface see the opposte wall wth a smaller vew factor than the elements n the core of the surface. Due to the hgher radatve heat transfer at the core regons (see Fgure 3b), the sold wall (surface 7) cools down faster than n the regons close to the horzontal adabatc surfaces. Temperature dfferences of about 0.1K are observed along surface 7 for a smulaton usng 30 rays. The predcted average surface temperatures obtaned for the MC and the DT model at surface 7 after two hours of smulaton s 26.16 C. Usng Eqs. (3) and () a wall radatve heat flux of 22.3W/m 2 s analytcally calculated. For the analytcal model, the surfaces are not dscretsed. However, the average vew factors used from Wagner (1998) consder mplctly the spatal dstbuton of the local vew factors. a) b) Fgure 3: Predcted wall radatve heat flux for surfaces and 7 usng (a) the Monte Carlo model usng 1,000,000 hstores and (b) the Dscrete Transfer model usng 8 rays. Good agreement between the MC model and the analytcal data was acheved usng 2,000,000 hstores when comparng the average radaton heat exchange between surfaces and 7 (5% dfference). Smulatons usng a smaller number of hstores led to hgher devatons. The average radatve heat transfer predctons for the Dscrete Transfer model also compare favourably wth the analytcal soluton (6% dfference for ray numbers of 30 and more). The small dfferences between the analytcal soluton and the numercal predctons of the MC and DT models are thought to be caused by the followng: Inaccuraces of vew factor values used for the analytcal soluton whch were obtaned from vew-factor dagrams from the lterature. The smplfed use of average surface temperatures for the analytcal soluton. A change of the temperature dfference of 0.15K at the nvestgated temperature level for example - 35 -

Proceedngs: Buldng Smulaton 2007 leads to a change of 0.7W/m 2 of the radatve wall heat exchange for the analytcal calculaton (3% devaton from the soluton obtaned above). Ths demonstrates the senstvty of radaton heat exchange on the boundary temperatures for the analytcal soluton. Takng nto consderaton the possble reasons consdered above, the numercal predctons were consdered to be satsfactory compared wth the analytcal soluton. However, snce the radatve dstrbuton on the surfaces for the MC model s unrealstc and the smulatons requred hgh computatonal resources (see Table 2), ths leads to the concluson that the MC model mght not be sutable for dynamc thermal buldng smulatons. The DT model predcts the radatve dstrbuton well and the smulaton tme requred for the dynamc smulaton s sgnfcantly reduced (2% of CPU tme for the DT model usng 30 rays compared wth the MC model usng 2,000,000 hstores). However, an ncrease of the number of rays leads to a sgnfcant ncrease of computatonal tme for the DT model thus suggestng careful use of ths parameter for tmeeffcent smulatons. Addtonal smulatons usng an emssvty of ε=0.9 nstead of ε=0.01 for the adabatc surfaces 3 and 5 led only to margnal changes n the numercal predctons. Ths corresponds wth the theory that the emsson coeffcent at the adabatc boundares s rrelevant n the analytcal soluton. In contrast, a varaton of the emmssvty at surfaces and 7 led to sgnfcant changes. Case 3 Cases 1 and 2 have demonstrated that the DT model s capable of predctng the surface heat transfer accurately and was therefore consdered to be the prefered radaton model for use n ths work. Ths model was further nvestgated usng case 3. Snce the followng nter-model comparson was conducted wth the same CFD code a number of 8 rays was used nstead of 30 rays to reduce the computatonal effort. Fgure a compares the predcted temperatures at dfferent montor ponts for a smulaton perod of 12h usng the adaptve freeze-flow method and a smulaton for whch the flow patterns are updated every tme step (base case). The observed temperature decrease at all montor ponts ndcates a gradual coolng of the sold wall. The temperature predctons agree well usng the adaptve freeze-flow method wth those of the base case. A margnal maxmum temperature dfference at the end of the smulaton perod of less than 0.03K s observed (see MP 1 at the end of a smulaton perod of 12h). Compared wth the base case, a CPU tme reducton of 8% was possble usng the freeze-flow method. Fgure b shows the temperature predctons obtaned for the same room model wthout radatve surface heat transfer, publshed n Ztzmann et al (2007a). The temperatures n Fgure a show a faster coolng than n Fgure b whch ndcates a notceable nfluence of radaton on the overall surface heat transfer. Furthermore, the ncreasng temperature under-predcton observed usng the adaptve freeze-flow method when radaton was not modelled (ΔT max =0.15K at MP3) s reduced by 80% when the radaton heat transfer s ncluded. Two reasons are thought to be responsble for the reducton n error: The reducton of nfluence of convecton surface heat transfer on the thermal room behavour due to the domnaton of radatve heat transfer as shown by Ztzmann et al 2007b. The compensaton of the over-predcted coolng from convecton due to a coherent decrease of radatve heat transfer (Sharma et al 2007) whch subsequently occurred due to the smaller temperature dfference between surfaces and 7. a) b) Fgure Temperature predctons applyng the adaptve freeze-flow control method to case 3 n whch radaton modellng s (a) ncluded and (b) excluded. - 36 -

Proceedngs: Buldng Smulaton 2007 Further nvestgatons were carred out to nvestgate whether the radaton equaton can be frozen for certan tme perods to reduce the CPU tme. However, for some of the room models nvestgated sgnfcant dscrepances were observed whch led to the concluson that the radaton equaton should be calculated almost every tme step. CONCLUSIONS Radaton modellng n spaces wth hgh thermal mass was verfed for two smple test cases usng CFD. The predcted surface radatve heat transfer was compared wth analytcal solutons. In case 1, the radatve heat exchange occurred between two parallel surfaces of nfnte sze. Case 2 contaned a room enclosure wth two walls of dfferent temperature. The Monte Carlo and the DT radaton model mplemented n CFX were used. The Monte Carlo model showed an unrealstc radaton dstrbuton n all cases. The reason was thought to be the result of an nsuffcent number of hstores used to track the reflected radaton partcles through the doman. However, an ncrease of the number of hstores led to a sgnfcant ncrease n computatonal effort. The unrealstc representaton of radatve surface heat transfer and the hgh computatonal resources requred led to the concluson that the Monte Carlo model mght not be sutable for dynamc thermal buldng smulatons. The Dscrete Transfer model was computatonal cost effcent and predctons for radatve surface heat transfer showed good agreement wth analytcal solutons. As a result from ths study, the Dscrete Transfer model s suggested as the prefered model for dynamc thermal buldng smulatons. In case 3 the dynamc coolng of a sold wall was nvestgated usng the Dscrete Transfer model. CFX predctons usng the adaptve freeze-flow control method have been compared wth a full dynamc CFX smulaton n whch all equatons were solved at every tme-step. The predcted temperatures at dfferent montorng ponts compared well wth the base case. The small predcton errors whch exsted for nvestgatons of Ztzmann et al (2007a) usng the adaptve freeze-flow method n whch radaton was neglected were reduced by 80% when radaton was consdered. The CPU tme was reduced by 8% usng the adaptve freeze-flow control method compared wth the base case. The nvestgaton has demonstrated that radaton heat transfer n buldngs can be predcted well usng the Dscrete Transfer model n CFX. Furthermore, an effcent method was demonstrated for modellng dynamc thermal buldng smulatons n whch all mechansms of heat transfer are consdered usng CFD by applyng the adaptve freeze-flow method. ACKNOWLEDGEMENT Ths report s based on a research project whch was funded by the German Mnstry of Educaton and Research (fundng NO: 179B0). The responsblty for the content les wth the lead author of ths publcaton. NOMENCLATURE B ncdent radaton [W/m 2 ] E emtted radaton [W/m 2 ] I radaton ntensty [W/m 2 ] K coeffcent [-] q& exchanged radatve heat flux [W/m 2 ] poston vector [m] s r dstance vector [m] s path length [m] S radatve source [W/m 2 ] T temperature [K] β angle between surface element normals [rad] Ω sold angle [rad] ε emssvty [-] ϕ vew factor [-] φ n-scatterng phase functon ν frequency [s -1 ] σ Stefan Boltzmann constant (5.67 10-8 W/(m 2 K )) INDICES a absorpton b black body,k,n surface number s specular v frequency [s -1 ] REFERENCES ANSYS 2005. CFX manual. Theory. Verson 10.0. Cook M. J Y. and Hunt G. 2003. "CFD modellng of natural ventlaton: combned wnd and buoyancy forces." Internatonal Journal of Ventlaton 1 (3), pp. 169-179. Farvarolo P. Manz H. 2005. Temperature-drven sngle-sded ventlaton through large rectangular openng, Buldng and Envronment 0, pp 689-699. Ghatt V. Autf S. 2002. Study of convectve heat transfer n a radatvely cooled buldng usng computatonal flud dynamcs, Reno: Solar 2002, Nevada, 6 pages. Groleau D. Marenne C. Raymond F. 1997. Smulaton of the nght coolng effect of the nght tme natural ventlaton: a 3D numercal applcaton to the <Mason Ronde> of Botta, Athens: 18 th AIVC Conference Ventlaton and Coolng (1), Greece. Haupt W. 2001. Zur Smulaton von auftrebsnduzerten Innenraumströmungen. - 37 -

Proceedngs: Buldng Smulaton 2007 Doctor thess. Unverstät Gesamthochschule Kassel. Sharma A. Velusamy K. et al. 2007. Conjugate turbulent natural convecton wth surface radaton n ar flled rectangular enclosures, Internatonal Journal of Heat and Mass Transfer 50, pp 625-639. Somarathne S. Seymour M. Kolokotron M. 2002. Transent soluton methods for dynamc thermal modellng wthn CFD, Internatonal Journal of Ventlaton 1 (2), pp. 11-156. Versteeg H. Malalasekera W. 1995. Computatonal Flud Dynamcs. The Fnte Volume Method. Essex: Pearson Prentce Hall. UK. Wagner W. 1998. Wärmeuebertragung. Würzburg: Vogel. Germany. Ztzmann T. Pfrommer P. and Cook M. 2007a. Dynamsch thermsches CFD-Verfahren mt angepasster Regelungsmethode, Bauphysk 29 (1), pp 12-16. Ztzmann T. Pfrommer P. and Cook M. 2007b. Thermal Mass and Nght-Tme Ventlaton usng Dynamc CFD, Helsnk: Roomvent 2007, submtted. Table 1: Predcted radatve surface heat transfer n W/m 2 for surfaces and 7 usng CFX (MC=Monte Carlo, DT=Dscrete Transfer) (Case 1). RADIATION NUMBER OF q& SURFACE q& SURFACE 7 CPU MODEL HISTORIES/RAYS MIN MAX AVE MIN MAX AVE TIME MC 1,000,000-0.6-19.9-27.8 20.3 38. 27.8 33926 MC 200,000-58.9 2.5-27.7-100.1 60.3 27.5 7171 DT 8-27.7-27.7-27.7 27. 27.8 27.7 280 Table 2: Predcted radatve surface heat transfer n W/m 2 for surfaces and 7 usng CFX (MC=Monte Carlo, DT=Dscrete Transfer) (Case 2). RADIATION NUMBER OF q& SURFACE q& SURFACE 7 CPU MODEL HISTORIES/RAYS MIN MAX AVE MIN MAX AVE TIME MC 2,000,000-33.8-6.0-21. 0 31 20.9 65205 MC 1,000,000-3.8-2.7-21.0 0 35.6 20.5 32736 MC 200,000-6.2 10.9-20.1-18. 56.8 19.9 6996 DT 8-23.2-6.3-20.8 0.1 22.2 20. 251 DT 15-2.3-5. -20.6 0.0 23.8 21.5 15 DT 30-22.9-6. -21.2 0.1 22. 20.7 1169 DT 50-22.9-6.5-21.2 0.1 22. 20.8 2389 DT 100-22.9-6.5-21.2 0.1 22. 20.8 9110-38 -