A Mathematical Model for Infiltration Heat Recovery

Transcription

1 A Mathematcal Model for Infltraton Heat Recovery C. R. Buchanan and M. H. Sherman 1 Energy Performance of Buldngs Group Indoor Envronment Department Envronmental Energy Technologes Dvson Lawrence Berkeley Natonal Laboratory Unversty of Calforna Abstract Infltraton has tradtonally been assumed to affect the energy load of a buldng by an amount equal to the product of the nfltraton flow rate and the sensble enthalpy dfference between nsde and outsde. However, laboratory and smulaton research has ndcated that heat transfer between the nfltratng ar and walls may be substantal, reducng the mpact of nfltraton. In ths paper, two- and three-dmensonal CFD smulatons are used to study the fundamental physcs of the nfltraton heat recovery process and a smple macro-scale mathematcal model for the predcton of a heat recovery factor s developed. CFD results were found to compare well (wthn about 10 percent) wth lmted publshed laboratory data correspondng to one of the scenaros examned. The model, based on the steady-state one-dmensonal convecton-dffuson equaton, provdes a smple analytcal soluton for the heat recovery factor and requres only three nputs: the nfltraton rate, the U- value for the buldng, and estmates of the effectve areas for nfltraton and exfltraton. The most dffcult aspect of usng the model s estmaton of the effectve areas, whch s done here through comparson wth the CFD results. Wth proper nput, the model gves predctons that agree well wth CFD results over a large range of nfltraton rates. Results show that nfltraton heat recovery can be a substantal effect and that the tradtonal method may greatly over-predct the nfltraton energy load, by percent at low leakage rates and by about 20 percent at hgh leakage rates. Ths model for nfltraton heat recovery could easly be ncorporated nto whole-buldng energy analyss programs to help provde mproved predctons of the energy mpact of nfltraton. 1 LBNL-44294: Ths work was supported by the Assstant Secretary for Energy Effcency and Renewable Energy, Offce of Buldng Technology of the U.S. Department of Energy under contract no. DE-AC03-76SF00098.

2 Table of Contents Abstract 1 Table of Contents.2 Nomenclature Introducton 4 2. General Problem Formulaton.5 3. CFD Smulatons.7 4. Results and Dscusson D CFD Smulatons D CFD Smulatons Expermental Comparson of CFD Results Development of the Smplfed Infltraton Heat Recovery Model Convecton-Dffuson Equaton Convecton Terms Conducton Terms Introducton of a Dmensonless Flow Rate Area Ratos n the Heat Recovery Factor Equaton Plots of the Heat Recovery Factor Comparson of Mathematcal Model wth CFD Results Conclusons...32 References..33 Appendx 1: Infltraton Bascs..34 App.1.1: Energy flux through the buldng envelope.34 App.1.2: Conventonal method of accountng for nfltraton energy load 35 App.1.3: Includng Infltraton Heat Recovery n the conventonal method..35 Appendx 2: CFD Detals 36 App.2.1: Wall modelng and boundary condtons 36 App.2.2: Thermal equlbrum n the wall nsulaton App.2.3: Soluton Methodology Appendx 3: Sample CFD Results: Energy Flux through Leakng Walls.43 Appendx 4: A Worked Example Usng the Infltraton Heat Recovery Factor 47 2

3 Nomenclature a = dmensonless flow rate (-) a o = dmensonless flow rate based on total buldng surface area (-) A = buldng envelope total surface area (m 2 ) A = effectve areas for heat recovery model (m 2 ) c p = specfc heat capacty of ar (1006 J/kg K) c ps = specfc heat capacty of nsulaton sold component (1006 J/kgK) c pw = specfc heat capacty of wall sheathng (1200 J/kgK) C 1, C 2, C 3 = constants for heat recovery model varable area rato expresson (-) e = external wall faces for conducton terms n model development f 1 = effectve area rato for nfltratng wall (-) f 2 = effectve area rato for exfltratng wall (-) g = gravty (9.81 m/s 2 ) k = ar thermal conductvty (0.025 W/mK) k eff = effectve thermal conductvty of nsulaton ( W/mK) k s = thermal conductvty of nsulaton sold component (0.041 W/mK) k w = wall sheathng thermal conductvty (0.13 W/mK) L = wall thckness (m) m = nfltraton mass flow rate (kg/s) p = ar pressure (Pa) Pe = Peclet number (-) q o = heat energy conducted through wall n model (W/m 2 ) Q = actual total (conducton and convecton) buldng energy load (W) Q cond = conducton energy flux through envelope n smplfed model (W) Q conv = convecton energy flux through envelope n smplfed model (W) Q nf = actual energy load due to nfltraton (W) Q nfc = conventonal energy load due to nfltraton (W) Q o = pure conducton energy load wth no nfltraton (W) t = tme (s) T = temperature (K) T = nsde ar temperature (298 K) T o = outsde ar temperature (274 K) T s = temperature of nsulaton sold component (K) T w = wall sheathng temperature (K) u = ar flow velocty (m/s) U = wall U-value (W/m 2 ) x = dstance co-ordnate (m) α = nsulaton permeablty (10-8 m 2 ) T = T T o (24 K) ε = nfltraton heat exchange effectveness or heat recovery factor (-) Γ = generc dffuson coeffcent (kg/m s n ths paper) φ = mass fracton of ar n wall nsulaton materal (0.99) µ = ar vscosty (1.72x10-5 kg/ms) ρ = ar densty (kg/m 3 ) ρ s = densty of nsulaton sold component (70 kg/m 3 ) ρ w = wall sheathng densty (544 kg/m 3 ) 3

4 τ = flud stress tensor (N/m 2 ) 1. Introducton Infltraton, accdental ar leakage through buldng envelopes, s a common phenomenon that affects both ndoor ar qualty and buldng energy consumpton. Infltraton can contrbute sgnfcantly to the overall heatng or coolng load of a buldng, but the magntude of the effect depends on a host of factors, ncludng envronmental condtons, buldng desgn and operaton, and constructon qualty. Clardge and Bhattacharyya (8) note that a great deal of work has been devoted to the predcton and measurement of nfltraton rates n buldng systems, but lttle effort has been drected toward determnng the actual energy mpact of nfltraton. Few studes regardng the energy ssues of nfltraton have been found n the lterature. Based on feld measurements taken at 50 resdental buldngs, Caffey (6) concluded that up to 40 percent of the heatng/coolng costs n the homes studed was due to nfltraton. In another study of resdental buldngs, Persly (15) attrbuted about one-thrd of the heatng/coolng requrements to nfltraton. Sherman and Matson (16) examned measured leakage data and found that a hgh fracton of the space condtonng load n U.S. resdental buldngs was due to nfltraton. The results of a recent study (14) of U.S. offce buldngs performed by the Natonal Insttute of Standards and Technology (NIST) estmates that ar leakage accounts for about 15 percent of the heatng load n offce buldngs natonwde and about 1 or 2 percent of the coolng load. By all measures, the mpact of nfltraton can be szeable and should therefore be consdered n calculatons of buldng energy consumpton. Q nfc = mc ( T T ) (1) p o The conventonal method of accountng for the extra load due to nfltraton (explaned n Appendx 1) s to add a smple convectve transport term of the form mc p T to the energy balance for the buldng. For sngle-zone buldng models the conventonal nfltraton load, Q nfc, shown n equaton 1, s the product of the nfltratng ar mass flow rate, the specfc heat capacty of ar, and the temperature dfference between nsde and outsde. Ths relaton does not nclude the effects of mosture n the ar and s strctly vald only f the leakng ar does not nteract thermally wth the buldng walls. In realty, leakng ar exchanges heat wth the walls as t enters and leaves the buldng, whch changes the thermal profle n the walls and warms or cools the nfltratng/exfltratng ar. Ths results n dfferent values for the conducton, nfltraton, and total heat losses than are predcted by the conventonal method (see Appendx 1). Some studes have shown that ths effect could be substantal suggestng that the conventonal method over-predcts the energy mpact of nfltraton (2,4,7,8,11). An mproved predcton of the energy load due to nfltraton can be made by ntroducng a correcton factor, the nfltraton heat exchange effectveness, ε, or the heat recovery factor (defned by equaton 2), nto the expresson for the conventonal load (equaton 1). In equaton 2, Q s the actual total energy load of the buldng wth nfltraton and Q o s the conducton load when there s no nfltraton. Ths heat recovery factor, ntroduced by Clardge and Bhattacharyya (8), accounts for the thermal nteracton between 4

5 leakng ar and buldng walls. The actual nfltraton load, Q nf, s calculated usng the heat recovery factor as shown n equaton 3 (detaled n Appendx 1). Q Q mc T o o ε 1 = 1 (2) p Q Q Q nf C Q ( 1 ε) mcp( T To ) = (1 ε Qnf C = (3) nf ) At ths pont, basc nformaton regardng the physcal detals of the problem (lke general ar flow structure or mportant transport mechansms) or the mportance of certan varables (lke wall desgn, leakage path, bulk ar temperature dfference) s not avalable, therefore, nfltraton heat recovery s not well understood. The purpose of ths study s to nvestgate the heat transfer process between nfltratng ar and room walls and determne ts effect on the energy load conventonally attrbuted to nfltraton. A prmary goal of ths work s to provde a foundaton of knowledge about ths process by whch a fundamental understandng can be developed and a drecton for future work can be determned. Another goal s to determne the rough sze of the nfltraton heat recovery effect for a varety of leakage scenaros. If the effect s not szeable, then there would be no pont n further work. A fnal goal s to develop a smplfed mathematcal model for calculaton of the nfltraton heat recovery factor that s based on the mportant physcs of the process. In ths paper, two- and three-dmensonal computatonal flud dynamcs (CFD) smulatons are used to nvestgate the basc physcs of the nfltraton heat recovery process. We choose to start wth a farly smple physcal representaton (only conducton and convecton are consdered for transport) so that an understandng of the phenomenon can be developed from frst prncples. Addtonal processes, lke turbulence or radaton, can be added progressvely f necessary. Also, a one-dmensonal mathematcal model s developed that can be used to determne the extent of heat transfer between leakng ar and walls, represented quanttatvely as the nfltraton heat recovery factor. Ths macro-scale model, based on the steady-state one-dmensonal convecton-dffuson equaton, provdes a smple analytcal relaton for the heat recovery factor. It requres only three nputs: the nfltraton rate, the U-value for the buldng, and estmates of the effectve areas for nfltraton and exfltraton. Predctons from the model are compared wth results from detaled CFD smulatons and lmted expermental results from the lterature. 2. General Problem Formulaton The cross-secton of a hypothetcal test room under a general nfltraton scenaro s shown n fgure 1. Small holes n the outer sheathng of the buldng envelope (plywood n ths study) allow ar to leak nto the wall cavty and flow through the wall from outsde to nsde for the nfltratng wall and vce-versa for the exfltratng wall. The drvng force for leakage s a pressure dfferental due to wnd and temperature dfferences between nsde and outsde. Fgure 1 shows the nfltraton problem n ts entrety, but n ths study only a lmted porton of ths envronment (the walls and the arspace n the vcnty of the walls) s analyzed 5

6 to help understand the nfltraton heat recovery process. As nfltraton heat recovery s largely a localzed process ths s all that s necessary. Eght wall confguratons, shown n fgures 2 and 3, are examned under varous envronmental condtons. Wall geometres 1, 2, 5, and 8 have nsulaton n the full wall cavty, whle geometres 3 and 4 have empty wall cavtes. Walls 6 and 7 are specal cases wth only half of the wall cavty nsulated. Walls 1, 3, 6, and 7 provde a long flow path for nfltratng ar that could potentally create a dsplacement flow n the wall cavty. Walls 2, 4, and 5 provde a short flow path for nfltratng ar that could solate ar n the top and bottom sectons of the wall. Wall 8 has the possblty for both long and short paths. Leakage rates through the wall are vared and the nsde/outsde temperature dfference s fxed at 24 K for most of the cases. The nfluence of the nsde/outsde temperature dfference s also examned. Fgure 1: Cross-secton of a hypothetcal test room showng the general nfltraton problem (wall geometry 1 shown). The nfltratng and exfltratng walls have a conducton and convecton energy flux, but all other walls have only a conducton flux. The example room shown n fgure 1 could represent a row-house nner unt and s composed of a celng, floor, front wall, and rear wall wth no ar leakage and an nfltratng wall wth a correspondng exfltratng wall both wth ar leakage. Heat conducton occurs through all of the walls but the nfltratng and exfltratng walls also have convectve heat transfer. We beleve the mportant physcs of the nfltraton heat recovery process occurs largely wthn the wall structure and n the vcnty of the wall surfaces (.e., a few centmeters) and the results support ths noton. Snce the purpose of ths study s to understand the mportant physcs of heat recovery, t s only necessary to analyze ths select regon. It s not necessary to represent the detals of the buldng nteror or the entre buldng envelope, therefore, the room nteror s not represented and the buldng envelope s 6

7 separated nto non-nteractng wall elements, whch are examned ndvdually. Informaton from the ndvdual walls s added together to determne the overall mpact for a complete room system. The leakng walls are both of the same geometry type and are matched by ther ar leakage rates. The bulk ar flow wthn the room s not represented, but ths should not be a problem because, as Etherdge (9) notes, the nternal room ar flow has only a secondary effect on nfltraton. The most mportant nfluences on nfltraton are wnd-nduced pressure dfferences and buoyancy of room ar n the vcnty of the wall. It s possble that radaton has some effect on the heat recovery process, but ths topc s not examned here. The mportance of radaton wll be assessed n future work. Fgure 2: Wall geometres 1-4; 1 & 2 are nsulated and 3 & 4 are empty. Fgure 3: Wall geometres 5-8; specal cases. 3. CFD Smulatons The purpose of ths study s to nvestgate the potental heat transfer between nfltratng ar and room walls, quantfy ts extent through a heat recovery factor, and determne how ths process could affect the energy load conventonally attrbuted to nfltraton. Prevous studes have measured lump quanttes expermentally (2,7,8) or 7

8 smulated the system usng smplfed modelng, for example, a prescrbed Nusselt number for the ar-wall heat transfer (11). Despte pror research efforts, the fundamental detals of the process are stll not known and, therefore, t s not well understood. In the frst part of ths study, computatonal flud dynamcs (CFD) smulatons are used to examne the basc physcs of the nfltraton heat recovery process n detal. The ndvdual contrbutons of conducton and convecton to the total heatng load are determned wthout makng fundamental smplfcatons as n prevous work. These components of the total energy flux are used to calculate the nfltraton heat recovery factor. An advantage of CFD smulatons over expermental studes s that detals of the system, lke local flow patterns and thermal profles, can be resolved helpng to provde a better overall understandng of the physcs nvolved. Also, many mportant aspects of the problem that are often dffcult to measure, lke boundary condtons, materal propertes, and flow paths, are prescrbed and can be systematcally vared wth ease. Ths makes correlatons between dfferent varables, lke leakage path length and heat recovery, much easer. Also, these smulatons are quck and nexpensve compared to experments and can be used to help desgn experments. A dsadvantage of such smulatons s that t s dffcult to represent the complexty of the true system, especally the varatons assocated wth constructon qualty and materal propertes. The systems here are dealzed, havng homogeneous materal propertes and deal constructon. However, the results should be representatve and of practcal value wth proper nterpretaton. The walls are modeled as two- and three-dmensonal systems n the CFD smulatons. Flow and energy transport n the ar are determned va the Naver-Stokes and energy equatons, equatons 4-6, respectvely. A lamnar representaton s used for the flow, and solutons show ths to be a vald assumpton, as the hghest calculated Reynolds number nsde or near the wall s only about 2000, based on wall thckness. It s possble that turbulence could have some effect even at these moderately low Reynolds numbers, so ths wll be examned n future work. The plywood sheathng s represented as an mpermeable, sold materal. Energy transport wthn the sheathng s calculated va the conducton equaton, shown n equaton 7. Insulaton, f present n the wall, s represented as a porous materal. Ar flow through the nsulaton s determned va Darcy s Law, equaton 8, a common model for flow through porous meda (5,11). Energy transport through the nsulaton s determned va a modfed form of the energy equaton, as gven by equaton 9. In equaton 9, an effectve conductvty, gven by equaton 10, s used n the conducton flux term and the thermal nerta of the sold component s ncluded n the transent term. A fundamental assumpton n the valdty of equaton 9 for ths applcaton s local thermal equlbrum between the flud and sold phases n the porous meda. Ths assumpton and other detals of the wall modelng are dscussed n Appendx 2. ρ t + ρu x = 0 (4) ρu t ρu u + x j j p = x + τ x j j + ρg (5) 8

9 c p ρt t + c p ρu T x = 2 T k x 2 p + + u t p x (6) T 2 w w ρ wc pw = kw (7) 2 t x T ρu t p = x µ + u α + ρg (8) t ( c ρt + ( 1 φ) c ρ T ) ρu T 2 φ p ps s + c p = k eff + + u (9) 2 x x t x T p p k eff ( φ) k s = φk + 1 (10) Thermal gradents n the system develop due to the dfference between ndoor and outdoor condtons and gve rse to natural convecton. As mentoned prevously, t s mportant to represent the effects of buoyancy on the flow to properly determne nfltraton rates and the heat flux at the wall due to boundary layer formaton, so buoyancy s ncluded n these smulatons. A temperature-dependent emprcal equaton of state for the flud densty, coupled wth the body force term n the flud momentum equaton n the vertcal drecton ntroduces the effects of buoyancy nto the flow. Smulatons are performed for the eght wall geometres shown n fgures 2 and 3 under varous nfltraton rates wth a constant temperature dfference of 24 K between nsde and outsde. Ar leakage s acheved by mposng ether velocty or pressure boundary condtons at locatons far enough from the wall constructon to prevent local dsturbances from developng near the wall. Ths requres there to be some amount of empty space on ether sde of the wall. An empty arspace of about twce the wall thckness s placed on ether sde of the wall, whch allows a natural convecton boundary layer to develop. Temperature boundary condtons are mposed upon the ar on ether sde of the wall, not upon the wall tself, meanng heat transfer through the wall structure s calculated entrely as conjugate heat transfer,.e., no assumptons are made about the heat transfer (see Appendx 2 for detals). Due to the complexty of the flow, t s not possble to acheve a converged soluton, based on the sum of the normalzed resduals, usng the steady-state equatons. Therefore, the tme-dependent equatons are ntegrated n tme untl steady-state s reached. The detals of the soluton methodology are dscussed n Appendx 2. Comparson of results from twodmensonal smulatons usng a coarse computatonal grd (33,000 nodes) and a fne grd (140,000 nodes) for the same wall geometry show that the coarse grd s suffcent to provde a grd-ndependent soluton. All two-dmensonal results presented here are from converged steady-state solutons usng a 33,000-node grd. In the three-dmensonal smulatons, a dfferent grd wth approxmately 100,000 nodes s used. Grd-dependency tests are not performed for the three-dmensonal cases because the requred computatonal resources are not avalable. However, t can be nferred from the correspondng two-dmensonal cases that ths grd resoluton should be roughly suffcent to provde some useful results. Examnaton of the two-dmensonal results shows that about 5 to 10 grd ponts across the thermal and 9

10 momentum boundary layers s enough to provde a grd ndependent soluton. In the threedmensonal smulatons, about 5 grd ponts are used across the boundary layers, whch appears to provde farly good resoluton. Although the three-dmensonal results look good and are from converged solutons, t s not known f the solutons are grd ndependent. Ths should be kept n mnd when examnng the three-dmensonal results. All three-dmensonal results are from converged steady-state solutons usng a grd wth approxmately 100,000 nodes. 4. Results and Dscusson The man pont of nterest s the extra energy load ntroduced by nfltraton because ths wll allow calculaton of the heat recovery factor. Ths s determned by frst calculatng the heat flux through the room walls wth no ar leakage, desgnated as Q o. Then, the energy flux s determned for the same wall types wth ar leakage. The dfference between the two values s the nfltraton-nduced energy load. The convecton and conducton energy flux across the external (outsde) face of each wall s calculated for nfltratng and exfltratng confguratons. Usng the external buldng face for the system control volume boundary s an arbtrary choce. The nteror face could be used as well, however, t s mportant from an organzatonal standpont that the energy accountng be performed at a consstent locaton. Detaled results from the wall 1 confguraton are shown n Appendx 3 to llustrate the trends that occur n the flux components as the leakage rate vares. Also, an example s worked n Appendx 4 usng the wall 1 results to llustrate how the heat recovery nformaton can be used and the potental ncreases n accuracy t could provde for buldng energy load predctons D CFD Smulatons Fgure 4 shows the heat recovery factor for wall geometres 1-4 determned from twodmensonal CFD smulatons. The varable on the horzontal axs of the graph s the dmensonless flow rate (a o ), defned n equaton 11. It s the leakage rate nondmensonalzed by the U-value of the buldng and the specfc heat of ar. It was found to be a useful ndependent varable when comparng the heat recovery for dfferent cases because t collapses the data showng the unversal trends. In all cases, the heat recovery factor approaches a value of one at very low flow rates and decreases wth ncreasng flow rate. Heat transfer s lower at hgh flow rates because there s less tme for energy to be transported from the walls to the nfltratng ar resultng n lower heat recovery. mc p a = (11) U A Two dstnct trends can be seen n fgure 4. One trend s that the walls wth holes n a hgh/low confguraton, walls 1 and 3, have a sgnfcantly hgher heat recovery factor than the walls wth holes that are straght through, but these straght through geometres stll have a sgnfcant heat recovery effect. Ths s partly because the hgh/low confguraton has a longer leakage path and, for a gven flow rate, the ar remans wthn the wall cavty for a longer perod of tme. Ths allows for greater heat transfer and hgher heat recovery compared to the straght through case. 10

11 The other trend s that data ponts for the hgh/low confguratons fall roughly on a sngle trend lne, and the same s true for the straght through confguratons. That s, nsulated (1 & 2) and empty walls (3 & 4) wth the same hole confguraton have about the same heat recovery when plotted aganst the chosen ndependent varable, a o. Ths suggests a unversal behavor that may be applcable to all leakage scenaros. Note that the nsulated and empty walls have dfferent leakage rates for a gven value of a o because ther U-values are dfferent, but they have about the same heat recovery. Ths ndcates that the nondmensonal flow rate, a o, s the proper ndependent varable to use when comparng dfferent cases. The heat recovery factor s calculated for wall 5 usng four dfferent flow rates and s compared to the data for wall 2 n fgure 5. The geometry of wall 5 s the same as wall 2, except the hole n the sheathng contnues all the way through the wall, ncludng the nsulaton, and the leakng ar s separated from the nsulaton by a layer of the sheathng materal. Interestngly, ths change n geometry has lttle effect on the heat recovery. The four ponts calculated for wall 5 fall essentally on the same trend lne as the ponts for wall 2. Ths suggests that the nteror detals of the leakage path do not have a great affect on the heat recovery, just the overall hole geometry s mportant. 1 heat recovery factor wall 1 (2D) wall 2 (2D) wall 3 (2D) wall 4 (2D) dmensonless flow rate Fgure 4: Heat recovery factor determned from 2D CFD smulatons for walls 1-4. Sold symbols show data for walls wth a hgh/low hole confguraton (1 and 3) and hollow symbols show data for walls wth a straght through hole confguraton (2 and 4). Notce the two dstnct trends one for each hole confguraton. 11

12 1 heat recovery factor wall 2 (2D) wall 5 (2D) dmensonless flow rate Fgure 5: Heat recovery factor determned from 2D CFD smulatons for walls 2 and 5. Both walls have a straght through hole confguraton but the detals of the leakage path are dfferent-- see fgures 2 and 3. Both walls have about the same heat recovery. The heat recovery factor s calculated for walls 6 and 7 usng several dfferent flow rates and s compared to the data for wall 1 n fgure 6. The leakage path s the same for all of the walls, a hgh/low hole confguraton, but the layout of the nsulaton n the wall cavty s dfferent. We thought that ths mght have some mpact on the heat recovery, so these specal cases were nvestgated. As fgures 2 and 3 show, the nsulaton flls the entre cavty of wall 1, whle only half the cavty s flled wth walls 6 and 7. Wall 6 has nsulaton on the rght half of the cavty and wall 7 has nsulaton on the bottom half. As fgure 6 shows, ths change n the wall cavty nsulaton layout has no sgnfcant effect on the heat recovery. The ponts calculated for walls 1, 6, and 7 all fall on the same trend lne. 12

13 1 heat recovery factor wall 1 (2D) wall 6 (2D) wall 7 (2D) dmensonless flow rate Fgure 6: Heat recovery factor determned from 2D CFD smulatons for walls 1, 6, and 7. All walls have a hgh/low hole confguraton but the layout of nsulaton n the wall cavty s dfferent for each case-- see fgures 2 and 3. The heat recovery factor for all the walls follows a sngle trend lne. 1 heat recovery factor wall 2 (2D) wall 8 (2D) dmensonless flow rate Fgure 7: Heat recovery factor determned from 2D CFD smulatons for walls 2 and 8. Both walls have nsulated wall cavtes. Wall 2 has a sngle straght through hole confguraton and wall 8 has two straght through holes-- see fgures 2 and 3. 13

14 The heat recovery data for walls 2 and 8 s shown n fgure 7. Wall 8 s smlar to wall 2, havng two straght through holes, but s also smlar to wall 1 wth two hgh/low holes. The heat recovery for wall 8 s closer to that of wall 2, however, suggestng that ths confguraton behaves more lke two straght through holes than two hgh/low holes. For values of a o less than one, the heat recovery factor for wall 8 shows strange behavor, droppng sharply wth small changes n flow rate and seemngly not approachng one for no leakage. Ths strange behavor s due to two-way flow occurrng n one or both of the walls caused by buoyancy nduced pressures that act n dfferent drectons at dfferent holes. For example, n a gven wall ar may flow nto the room through the top hole and out of the room through the bottom hole. Ths allows multple possbltes for flows at ndvdual holes for a gven overall leakage rate or a o. The result s that for a gven overall leakage rate there s not a unque heat recovery factor. At larger leakage rates hgher drvng pressures force the flow drecton to be the same for both sets of holes and the data ponts for wall 8 show the same trend as wall 2. There s lttle nteracton between the two sets of holes,.e., the hgh set and low set, because the nsulaton separatng them provdes a large flow resstance. A wall of ths desgn may not need to be modeled n ts entrety. However, prelmnary studes of ths wall wth an empty cavty show that there s a sgnfcant amount of nteracton between the hgh and low holes, so ths may not be a unversal trat for all such wall desgns. Ths wall geometry shows the potental complexty of real lfe stuatons and deserves further nvestgaton. A fnal pont of nterest s the nfluence of the bulk ar temperature dfference (T T o ) on the heat recovery factor as t may not scale drectly wth ths temperature dfference. CFD smulatons were performed over a range of leakage rates to examne ths possblty. The bulk ar temperature dfference was changed from 24 K to 12 K and 18 K for the nsulated wall wth a hgh/low hole confguraton (wall 1). The heat recovery factors for these cases are compared n fgure 8 over the range of leakage rates. All three cases, wth the 24 K, 12 K, and 18 K dfference, have essentally the same heat recovery values and trends. It seems that the heat recovery scales wth the bulk ar temperature dfference, so ths parameter can be removed from the analyss. 14

15 1 heat recovery facto dt = +24 K dt = +12 K dt = -18 K dmensonless flow rate Fgure 8: Heat recovery factor for wall 1 (nsulated wth a hgh/low hole confguraton) determned from 2D CFD smulatons usng bulk ar temperature dfferences (T T o ) of +24 K, +12 K and -18 K. The bulk ar temperature dfference appears to have no affect on the heat recovery. Note the decreased range n the dmensonless flow rate D CFD Smulatons Three-dmensonal smulatons are performed for walls 1 and 2 and the results are compared to those from two-dmensonal smulatons n fgures 9 and 10. In the 3D smulatons, the hole n the wall sheathng s roughly a square, whle the hole n the 2D cases corresponds to a long slt spannng the wdth of the wall. In all other respects, the 2D and 3D walls are geometrcally the same. Ths geometrc dscrepancy causes dfferences n the ar flow patterns n and around the wall, but t does not appear to have any sgnfcant effect on the heat recovery. The 3D smulatons gve nearly the same values and show almost the same trends for the heat recovery factor for a gven hole confguraton as the 2D smulatons. Therefore, t may be suffcent to use 2D smulatons to study a gven wall geometry, whch would mean a large savngs n tme and effort compared to 3D smulatons. 15

16 1 heat recovery factor wall 1 (2D) wall 1 (3D) dmensonless flow rate Fgure 9: Heat recovery factor determned from 2D and 3D CFD smulatons for wall 1, an nsulated wall wth a hgh/low hole confguraton. Heat recovery values are essentally the same for both cases. 1 heat recovery factor wall 2 (2D) wall 2 (3D) dmensonless flow rate Fgure 10: Heat recovery factor determned from 2D and 3D CFD smulatons for wall 2, an nsulated wall wth a straght through hole confguraton. Heat recovery values are essentally the same for both cases. 16

17 4.3. Expermental Comparson of CFD Results The only expermental data avalable for comparson s that of Clardge and Bhattacharyya (8). Unfortunately, ther experments do not correspond exactly to the work n ths study, some mportant detals of the experments are not gven makng drect comparsons dffcult, and leakage rates were vared only over a narrow range. However, one of ther cases corresponds farly well to one of ours and can be used for comparson. The case of Clardge and Bhattacharyya wth a dffuse nlet and outlet (nlet B4, outlet A) s smlar to walls 1 and 3 n ths study (hgh/low hole confguraton). They calculated a heat recovery factor of about 0.80 when a o s 0.05 and about 0.65 when a o s As fgure 8 shows, CFD results for wall 1 gve a heat recovery factor of 0.90 when a o s 0.05 and about 0.72 when a o s There are no CFD data avalable for wall 3 at these low flow rates, but the trend s the same. Consderng the rough nature of the comparson, the expermental and CFD results show good agreement. There s only about a 10 percent dscrepancy n the heat recovery factor at both ponts. Ths suggests that the CFD results are qualtatvely, and, most lkely, quanttatvely accurate. Also, ths suggests that processes whch occur naturally n the experment but that are not represented n the smulatons (turbulence, radaton) may not have a strong effect on the heat recovery. The CFD technque can, therefore, be consdered a useful tool for study of the nfltraton heat recovery process. The CFD results should provde a sound bass for comparson wth our smplfed nfltraton heat recovery model. 5. Development of the Smplfed Infltraton Heat Recovery Model The objectve of the followng work s to develop a smple macro-scale mathematcal model for the nfltraton heat recovery factor, whch s based on the mportant physcs of the process. The model should provde a smple, yet accurate, means for calculatng the heat recovery factor, ε, under a varety of envronmental condtons, buldng desgns, and leakage scenaros. The startng pont s the steady-state one-dmensonal convecton-dffuson equaton (17), shown n equaton 12. Ths smple representaton s used because we beleve t ncludes the most mportant physcal mechansms and wll help provde nsght nto the heat recovery process more easly than a complex representaton. It s a smplfed form of the general transport equaton, whch appeared earler n the CFD smulatons as the Naver- Stokes and energy equatons (equatons 5,6, and 9). In ths case, the transent and source terms are not ncluded and only one dmenson s consdered. The mmedate purpose of ths model s to gve a rough dea of the sze of the nfltraton heat recovery effect, not ncorporaton nto network codes for dynamc buldng smulaton. If the effect s szeable and the topc merts further nvestgaton, addtons can be made to the model n future work, f necessary, to help provde more accurate, realstc results and to make t sutable for use n network codes. 17

18 5.1. Convecton-Dffuson Equaton The one-dmensonal convecton-dffuson equaton below represents transport by combned convecton and dffuson n a steady flow varyng n one spatal drecton,.e., a one-dmensonal flow. Ths relaton s vald for flows exstng n three-dmensonal space (all real flows), but only those that vary n one drecton, for example, a fully-developed ppe flow. In equaton 12, φ represents any scalar flow varable, e.g., temperature or concentraton, and Γ s the dffuson coeffcent for that varable. An analytcal soluton s gven by equaton 13 for the varable φ as a functon of the length coordnate x for ρ, u, and Γ constant and for prescrbed boundary condtons and x = 0 and L. The subscrpts 0 and L represent the bounds of the doman,.e., the wall thckness, and the parameter Pe s the Peclet number, gven by equaton 14. d dx d dφ ( ρuφ) = Γ (12) dx dx φ( x) φ φ L φ 0 0 x exp( Pe ) 1 = L exp( Pe) 1 (13) ρ Pe = ul Γ (14) x exp( Pe ) 1 T ( x) = T + ( ) 0 T L L T0 (15) exp( Pe) 1 Takng φ to be temperature, equaton 13 can be rearranged to provde a smple relatonshp for the temperature at any locaton between the two bounds, as gven by equaton 15. Ths relatonshp wll provde the bass for the heat recovery model. It s approprate for applcaton to three-dmensonal flows nvolvng complex flud/sold nteractons because the domnant physcal processes of heat recovery (convecton and dffuson) nteract most strongly when they are parallel to one another,.e., thermal energy conductng through a buldng envelope s affected most strongly by leakng ar flowng parallel to the drecton of conducton. We beleve ths characterstc feature wll allow the phenomenon to be modeled effectvely usng a sngle drectonal varable resultng n a one-dmensonal model. At ths pont, Γ s left as a generc dffuson coeffcent. Later n the model development t wll be gven a prescrbed value, va nput, so that t takes on an effectve value to represent the composte wall structure. The next step s to apply equaton 15 to a generc buldng envelope under arbtrary condtons to determne the energy flux through the walls. The external envelope of the 18

19 buldng used for ths analyss s shown n fgure 11 along wth the relevant varables. Ths smplfed envelope represents the envelope shown n fgure 1 wth some changes to allow development of the model. The entre buldng envelope has been separated nto two knds of sectons, those that are affected by nfltraton and those that are not. Areas of the envelope that are affected by leakng ar are represented by A1 and A2 (nfltratng and exfltratng, respectvely) and have both a convecton and conducton energy flux. These sectons represent parts of the wall n the vcnty of the leakng hole whch undergo thermal changes due to nfltraton and could potentally represent the entre nfltratng or exfltratng walls. Areas that are not affected by leakng ar are represented by A3 and A4 and have only conducton heat transfer. In the context of fgure 1, these sectons represent the floor, celng, front and back walls, and possbly part of the nfltratng or exfltratng walls. Snce the analyss s one-dmensonal,.e., of a sngle spatal varable, only two buldng walls, those on left and rght, are needed for the model development. In fgure 11, there are no walls on the top, bottom, front, and back because they have been ncorporated nto the left and rght walls. The dashed lnes do not represent walls but just show that the left and rght walls are connected to form a closed system. The external face of the walls on the left and rght form the control volume boundary of the system. Areas of the envelope that are affected by leakng ar are represented n the model as a flud wth the composte propertes the flud/sold/porous wall system. That s, the governng equaton used n the model to descrbe wall sectons affected by nfltraton, the convectondffuson equaton, descrbes a flud, but s used here to represent the composte wall system. Sectons of the wall that are not affected by leakng ar are treated as a sold materal and are represented wth the conducton equaton. The sx ndvdual flux components shown n fgure 11 wll be used to determne the nfltraton heat recovery factor. An mportant mplcaton of the modelng procedure s that detals of the system that affect the heat recovery (for example, boundary layers, thermal gradents n the wall, constructon detals) are lumped and evenly dstrbuted over the entre effectve area. The nfluence of these detals s ncorporated nto the model n an effectve manner through proper choce of the effectve areas A1 and A2 (descrbed later). Once ths s done t s dffcult, f not mpossble, to make comparsons between the physcs of the model and the detals of the real system that the model represents. The purpose of the model s to provde select quanttatve nformaton about the system, n ths case the nfltraton heat recovery factor. It cannot provde detals about the system, lke boundary layer nformaton, whch have been lumped together n the modelng process. 19

20 Fgure 11: Energy flux through the smplfed buldng envelope used n the model development. Sectons A1 and A2 are affected by leakng ar and have both convecton and conducton terms. Sectons A3 and A4 are not affected and have only a conducton term. The outer faces of the left and rght walls form the system control volume boundary. Fgure 12: Arflow and thermal profle n the nfltratng (left) and exfltratng (rght) walls, sectons A1 and A2, respectvely. Ar temperatures at the nternal and external boundares are T and T o, respectvely. There are no boundary layers shown n the dagram on the nner and outer wall surfaces, but the nfluence of the boundary layer s ncorporated nto the model n an effectve manner through proper choce of the effectve areas A1 and A2. 20

21 5.2. Convecton Terms The total energy flux through the buldng walls has two components, one due to convecton (Q conv ) and another due to conducton (Q cond ). For ths ntal study, radaton wll not be ncluded but may be consdered n future work. Each component wll be determned across the control volume boundary va equaton 15. There s a convectve flux only through the two effectve areas A1 and A2. These areas do not correspond to the physcal area of a hole n the envelope, but to the surface area of the envelope that s affected by nfltratng ar. Thus, they are effectve areas. The actual values for these effectve areas, the only parameters not drectly nput nto the model, wll be estmated through comparson wth CFD data. Q conv = Q Q (16) conv2 conv1 Qconv = mcpt( x = L) mcpt( x = 0) (17) Q conv = mc T mc T (18) p o p o Q =0 (19) conv The net convectve energy flux across the effectve areas, A1 and A2, s determned n equatons Equaton 16 states that the net convectve flux s the flux at the exfltratng wall mnus the flux at the nfltratng wall, as shown n fgure 11. Note, n ths analyss, energy flow out of the buldng s consdered postve. The ar temperatures and mass flow rates at the control volume boundares,.e., the external face of the buldng envelope, are used n equaton 17. The temperatures at the external boundares, shown n fgure 12, are by defnton the outsde ar temperature, as equaton 18 reflects. Conceptually, there are no boundary layers at the wall surfaces, but ther nfluence s ncorporated nto the model n an effectve manner through proper choce of the effectve areas A1 and A2. Fnally, equaton 19 gves the nterestng result that the net convectve flux across the chosen control volume boundary s zero. Therefore, the total effectve energy flux due to nfltraton n the effectve areas wll be ncorporated nto the conducton terms Conducton Terms The total conductve energy flux across the control volume boundary s gven by equaton 20. It states that the flux s equal to the dot product of the temperature gradent at the control volume boundary (the external wall faces, denoted by the subscrpt e) and an outward-pontng normal vector multpled by the actve area and the wall thermal conductvty summed over the entre boundary. Note that the thermal conductvtes n the followng equatons are generc quanttes whose values wll be determned n an effectve manner for the entre wall constructon. Equaton 21 gves the expanded form of equaton 20 when appled to the buldng envelope shown n fgure 11. Note that the normal vectors for envelope areas 1 and 3 pont n the negatve x-drecton and those for areas 2 and 4 pont n the postve x-drecton resultng n mxed sgns for the terms n equaton

22 Q k A ( T nˆ) (20) e, = 4 cond = 1 Q cond dt dt = k1a1 k2a2 + k3a3 k4a4 (21) dx e,1 dx e,2 dx e,3 dx e,4 dt dt The gradent terms n equaton 21 are evaluated usng the soluton to the convectondffuson equaton, gven by equaton 15. Frst, the dervatve of temperature s taken wth respect to the length coordnate x and s shown n equaton 22. Usng the approprate boundary values, as shown n fgure 12, the gradent terms are evaluated at the nfltratng and exfltratng walls (A1 and A2) and are gven by equatons 23 and 24, respectvely. Ths leaves only terms for the nactve areas of the envelope, A3 and A4, to be evaluated. Snce these areas are not affected by nfltratng ar, ther conductve flux and the gradent terms reman constant. The product of the gradent term and thermal conductvty n areas 3 and 4 of the envelope wll be represented by a constant as gven by equatons 25 and 26, respectvely. Fnally, the evaluated terms shown n equatons are placed back nto equaton 21 to gve a new relaton for the total conductve flux, equaton 27. dt dx Pe e = ( TL T0 ) Pe L e x Pe L 1 (22) dt dx Pe1 1 = ( T To ) Pe1 L e e, 1 1 (23) dt dx Pe2 e = ( To T ) Pe L e Pe2 2 e, 2 1 (24) dt q 0,3 = k 3 (25) dx e,3 q k dt 0,4 = 4 (26) dx e,4 Q Pe 1 Pe e = A4q 0,4 (27) L e 1 L e 1 Pe2 1 2 cond k1 A1( T To ) + k 2A2 ( T To ) + A3q Pe 0,3 + 1 Pe2 22

23 The defnton for the heat recovery factor s restated n equaton 28 wth the total energy flux separated nto ts convectve and conductve components. The term Q o represents the total energy flux across the buldng envelope when there s no nfltraton,.e., pure conducton. For the envelope n ths analyss, Q o s represented as the sum of four constant terms as shown n equaton 29. Each term n equaton 29 accounts for the conductve flux across a partcular secton n the envelope. Usng equatons 19, 27, and 29 the numerator n equaton 28 can be rewrtten as shown n equaton 30. ( Qconv + Qcond ) Q0 ε = 1 (28) mc & ( T T ) p o Q + 0 = A1q0,1 + A2q0,2 + A3q0,3 A4q0,4 (29) Q conv + Q cond Q 0 = k A ( T 1 1 Pe1 To ) L e Pe 1 + k 1 1 (30) 2 A ( T 2 Pe2 To ) L e e Pe2 Pe2 1 A q 1 0,1 A q 2 0,2 At ths pont, the constant terms used to account for conducton through the envelope wll be examned. In realty, heat conducton through a buldng wall s dependant on a host of factors ncludng constructon detals of the wall, thermal propertes of the wall materals, envronmental condtons, and ar flow n the vcnty of the wall. However, these detals cannot be ncorporated nto ths one-dmensonal model, so a smplfed approach s taken followng conventonal practce n buldng energy smulaton. It s assumed that there s a lnear temperature profle through the wall and the heat flux obeys Fourer s law for heat conducton(13), equaton 20. Wth these assumptons, the conducton heat flux through the wall wth no nfltraton, q o, s expressed n equatons 31 and 32. q dt = k1 = k1 dx T To L 0,1 (31) e,1 q dt T = k2 = k2 dx T L o 0,2 (32) e,2 The nformaton from equatons can be nserted nto equaton 28 to provde a new relatonshp for the heat recovery factor, gven by equaton 33. Note, ths expresson shows that the heat recovery for the entre buldng envelope s dependent only on what happens n the actve areas. Of course, a fundamental assumpton of ths model s that the envelope can be dvded nto separate, non-nteractng regons, some beng affected by nfltraton and some not. The heat recovery factor s a functon of the wall thermal 23

24 conductvty, actve area, wall thckness, and Peclet number for both the nfltratng and exfltratng walls. Pe2 Pe1 Pe2e k1 A1 + k A ( k A k A ) Pe 2 2 Pe e 1 e 1 ε = 1 (33) mc L p 5.4. Introducton of a Dmensonless Flow Rate The fnal step s to recast the Peclet number n terms of quanttes that are meanngful for ths applcaton. Ths s an mportant part of the modelng process, whch can be consdered a modelng step n tself. Equaton 34 shows the defnton of the Peclet number. The velocty, u, n equaton 34 s replaced by the relaton shown n equaton 35 where m s the mass flow rate of leakng ar, ρ s the ar densty, and A s the actve area. Next, the thermal conductvty, k, s replaced by the U-value for the wall va equaton 36, whch makes use of the relaton for conducton gven by equaton 31. Ths step replaces the mcro-scale materal property, k, wth the macro-scale effectve U-value for the composte wall system. The Peclet number s transformed nto a dmensonless flow rate, a, defned by Clardge and Bhattacharyya (8), usng effectve values characterstc to the system as shown n equaton 37. ρul Pe = (34) k / c p u m = (35) ρa U qo, = T = k L (36) mc p a = (37) U A Usng the relatons n equatons 33-37, a new expresson s created for the heat recovery factor and s gven by equaton 38. Equaton 38 shows that the heat recovery factor s a functon only of the dmensonless flow rate, a, for the nfltratng and exfltratng walls. When presented n ths form the symmetry between nfltraton and exfltraton s apparent. It s assumed that n practce the leakage rate and the U-value can be measured or estmated. 24

25 That leaves only the effectve area to be determned before the heat recovery factor can be calculated. Ths wll be done here through comparson wth CFD smulaton results ε = (38) e + + a 1 a2 1 e 1 a a Area Ratos n the Heat Recovery Factor Equaton One last change wll be made n the expresson for the heat recovery factor, a smplfcaton for ease of use. A value s determned for the non-dmensonal flow rate for the entre structure, denoted as a o, usng the overall U-value and the total surface area of the buldng, denoted as A, as shown n equaton 39. Ths s convenent because the UA value for a buldng s a famlar quantty that can be measured or estmated. Both a 1 and a 2 can now be expressed n terms of a o and the unknown effectve areas can be extracted nto separate parameters as area ratos weghted by the U-values, as equatons 40 and 41 show. The fnal form of the expresson for the heat recovery factor s gven by equaton 42. It shows that the heat recovery factor for a gven buldng depends on the UA of the buldng and the nfltraton rate, whch appear as a o, and the effectve areas for nfltraton and exfltraton, whch appear as the effectve area ratos, f 1 and f 2. mc a = p o UA (39) f ao U1A1 = (40) a UA 1 = 1 f ao U2A2 = (41) a UA 2 = f 1 + f 2 ε = + (42) ao / f1 ao / f2 e 1 e 1 a o 5.6. Plots of the Heat Recovery Factor Some plots of the heat recovery factor, calculated wth equaton 42, are shown n fgures 13 and 14. These plots show the range of potental values for the heat recovery factor. The curves shown n fgure 13 each have dfferent values for the area ratos, as shown n the legend, wth the ndvdual area ratos (f 1 and f 2 ) beng equal for a gven curve. Note, the maxmum theoretcal value possble for the sum of the area ratos s1.0, whch 25

26 corresponds to 100 percent of the buldng envelope beng actve n the nfltraton process. The curve wth each rato equal to 0.5 corresponds to ths stuaton. Fgure 13 shows that the heat recovery factor s hgh at small nfltraton rates and drops as the nfltraton rate ncreases. The transt tme of ar leakng through the wall decreases as the flow rate ncreases, allowng less tme for heat transfer to occur between the leakng ar and wall structure. Ths reduces heat recovery. Also, the plot shows that the heat recovery drops wth decreasng area ratos. The rato corresponds to the fracton of the wall that nteracts wth leakng ar. When only a small fracton of the envelope nteracts wth leakng ar, ts mpact s small and the resultng heat recovery s low, as the case wth f = shows. When the rato s large, the heat recovery s hgher. Fgure 14 shows the heat recovery factor curves for four cases. Two of the cases have equal values for the ndvdual area ratos, f 1 and f 2, and are taken from fgure 13. The other two have unequal values for the ndvdual ratos, but the sum of the ratos s equal to that of the correspondng curve wth equal area ratos. The top two curves n fgure 14 have a total effectve area of 100 percent, and the bottom two curves have a total effectve area of 20 percent. Actual values used for the ratos are shown n the legend. The same overall behavor n the heat recovery factor s seen n all the curves. However, at low nfltraton rates, there are small dfferences between curves that have the same total value for the area ratos but dfferent ndvdual values. In both comparsons, the cases wth unequal ratos have a lower heat recovery than the correspondng case wth equal ratos. These results ndcate that for a gven leakage rate heat recovery s hghest when the effectve area s large and the ndvdual areas (nfltratng and exfltratng) are equal heat recovery factor f = 0.5 f = 0.25 f = 0.1 f = dmensonless flow rate Fgure 13: Heat recovery factor calculated wth the smplfed model usng equal area ratos (f 1 =f 2 =f). Note that the heat recovery drops wth decreasng area ratos and ncreasng leakage rate. 26

27 1 heat recovery factor f1=0.5, f2=0.5 f1=0.2, f2=0.8 f1=0.1, f2=0.1 f1=0.01, f2= dmensonless flow rate Fgure 14: Heat recovery factor calculated wth the smplfed model usng unequal area ratos. For a gven sum of area ratos (f 1 +f 2 ) the heat recovery s hghest when the ndvdual ratos are equal Comparson of Mathematcal Model wth CFD Results Predctons of the heat recovery factor made wth the mathematcal model, equaton 42, are now compared to values determned from CFD smulatons. In these comparsons, the area ratos n the model, f 1 and f 2, are adjusted to provde the best overall agreement as determned by a least-squares ft. The ratos are gven equal values here because the nfltratng and exfltratng walls used n the CFD smulatons were symmetrc. Fgure 15 shows heat recovery values for the walls wth a hgh/low hole confguraton determned from 2D and 3D CFD smulatons and the model. The value for the area rato was adjusted to gve the best agreement, whch, for ths case, was found to be 0.33, or 33 percent of the wall area. The model predctons show farly good agreement wth CFD results, but do not match exactly. Better agreement would occur f the model predctons were slghtly lower at small leakage rates and slghtly hgher at large leakage rates. 27

28 1 heat recovery factor wall 1 (2D) wall 3 (2D) wall 1 (3D) model dmensonless flow rate Fgure 15: Heat recovery factor for walls wth a hgh/low hole confguraton (nsulated and empty wall cavtes) determned from CFD smulatons and the smplfed model (equaton 42) usng a constant area rato (f=0.33). Fgure 16 shows heat recovery values for the walls wth a straght through hole confguraton determned from 2D and 3D CFD smulatons and the model. For ths case, the best agreement was acheved wth a value of 0.18 for the area rato, or 18 percent of the wall area. Agreement s good, but, as before, agreement would be better f the model predctons were slghtly lower at small leakage rates and slghtly hgher at large leakage rates. 28

29 1 heat recovery factor wall 2 (2D) wall 4 (2D) wall 2 (3D) model dmensonless flow rate Fgure 16: Heat recovery factor for walls wth a straght through hole confguraton (nsulated and empty wall cavtes) determned from CFD smulatons and the analytcal model (equaton 42) usng a constant area rato (f=0.18). The results n fgures 15 and 16 suggest that our smple model does not capture the full physcs of the problem. Whle the trend s generally correct, the model predctons decrease faster at hgh flow rates than the CFD data does. Some of the CFD results suggest that part of the heat recovery occurs n the thermal boundary layers on each sde of the wall and part occurs wthn the wall. For example, cool nfltratng ar falls along the nner surface of the wall reducng the conductve heat loss. Also, some of the ar sucked nto the leak from the external boundary layer wll be at a hgher temperature (a smaller overall temperature dfference) and wll mtgate the nfltraton load. Ths effect can be seen by lookng at the ar flow velocty vectors from one of the two-dmensonal CFD smulatons shown n fgure 17. Ths effect would mply that even wth no heat exchange wthn the wall tself there would be some heat recovery. We expect ths effect (suckng n the thermal boundary layer) to be the domnant mechansm for heat recovery at hgher leakage rates. Addtonally, non-unform ar flows wthn the wall could enhance ths effect (e.g., through nteracton wth convectve cells). We beleve these effects contrbute to heat recovery, but they are not explctly ncluded n our model. Therefore, we consder the possblty of a varable area rato n whch the area s augmented to nclude the effects of the boundary layer and other convectve nteractons. The varable area rato model must meet certan physcal constrants. When there s very lttle ar leakage, the area rato and the boundary layer effect should be small. Also, there should be a postve defnte ncrease n the magntude of the area rato wth ncreasng leakage to an asymptote that must never be greater than one. 29

30 Fgure 17. Ar flow velocty vectors from a two-dmensonal CFD smulaton of the wall wth a straght through leakage path (wall 2). It shows how the external boundary layer can be sucked nto the leak and nfltratng ar can be held near the nteror wall surface. To test the effcacy of a varable area rato representaton calculatons are performed to determne the rato magntude needed to make the analytcal model predctons of the heat recovery factor agree exactly wth those determned from CFD. Ths s done at every CFD data pont for empty and nsulated walls wth both hgh/low and straght through leakage confguratons. Results for the hgh/low confguraton are shown n fgure 18 and results for the straght through confguraton are shown n fgure 19. In both cases, the area rato starts off small, close to zero, at low leakage rates, as expected. As the leakage rate ncreases, the area rato ncreases and appears to level off at hgher leakage rates. The area rato for the hgh/low confguraton ncreases faster than that for the straght through and levels off at a larger value. Agan, ths s the expected behavor based on physcal constrants. There are some outlyng ponts n fgures 18 and 19, whch correspond to the 3D data. We are not certan why these ponts are outlyng from the general trend, but t could be due to numercal error n the calculatons related to nsuffcent grd resoluton. Ths analyss suggests that a varable representaton for the area rato based on leakage rate could ncrease the accuracy of the analytcal model. The exact mathematcal form for ths relaton and the physcal bass for the form are now under nvestgaton. 30

31 area rato wall 1 (2D) wall 3 (2D) wall 1 (3D) dmensonless flow rate Fgure 18: Area rato for walls wth a hgh/low hole confguraton (nsulated and empty wall cavtes) determned by matchng the analytcal model (equaton 42) predctons to CFD heat recovery factors at ndvdual CFD data ponts. 0.4 area rato 0.2 wall 4 (2D) wall 2 (2D) wall 2 (3D) dmensonless flow rate Fgure 19: Area rato for walls wth a straght through hole confguraton (nsulated and empty wall cavtes) determned by matchng the analytcal model (equaton 42) predctons to CFD heat recovery factor at ndvdual CFD data ponts. 31

32 6. Conclusons In ths study, CFD smulatons were used to examne the fundamental physcs of the nfltraton heat recovery process. Results were found to compare well (wthn about 10 percent) wth lmted publshed laboratory data correspondng to one of the leakage scenaros examned. Also, a smple, robust macro-scale mathematcal model was developed that can accurately predct the heat recovery factor for ar nfltratng through buldng walls when suppled wth the proper nput. The requred nputs are the buldng U-value, the leakage rate, and an effectve area for nfltraton and exfltraton. The effectve areas were determned here for specfc leakage geometres through comparson wth CFD data. The leakage geometres examned may be fundamental cases that can be combned to create most other leakage scenaros. Over the wde range of arflows consdered, model predctons agree farly well wth CFD data. However, t appears that better agreement can be attaned by usng a varable relaton, based on leakage rate, for the effectve area. The exact mathematcal form for ths relaton and the physcal bass for the form are now under nvestgaton. These results show the potental mportance of nfltraton heat recovery. The extent of heat recovery was found to be dependent on the leakage path geometry, nfltraton flow rate, and wall constructon. In some cases wth low nfltraton rates and long leakage paths, the heat recovery can be substantal, well over 80 percent. In these cases, the classcal method would over-predct the extra heatng load due to nfltraton (see appendx 4 for a detaled example). Accordng to these fndngs, under typcal leakage condtons for most resdental buldngs (a o 1) the heat recovery could be around 40 percent. Even at nfltraton rates that would be consdered very hgh for resdental buldngs (a o = 5) the heat recovery s stll szeable at about 20 percent. In ths stuaton, the conventonal predcton for the nfltraton load would be n error by 20 percent. All possbltes have not been examned n ths study, but t s clear that some modfcaton could be made to the conventonal method for predcton of nfltraton energy loads to ncrease ts accuracy. The model for nfltraton heat recovery presented n ths paper could easly be ncorporated nto whole-buldng energy analyss programs to provde mproved predctons for the energy mpact of steady-state nfltraton. An mportant mplcaton of the modelng procedure s that detals of the system that affect heat recovery (for example, boundary layers, thermal gradents n the wall, constructon detals) are lumped and evenly dstrbuted over the entre effectve area. The nfluence of these detals s ncorporated nto the model n an effectve manner through proper choce of the effectve areas. Once ths s done t s dffcult, f not mpossble, to make detaled comparsons between the physcs of the model and the realty that the model represents. The purpose of the model s to provde select quanttatve macro-scale nformaton about the system, n ths case the nfltraton heat recovery factor. It cannot provde detaled nformaton about the system, lke boundary layer structure, whch have been lumped together n the modelng process. 32

33 References 1) Baker, P. H., Sharples, S., Ward, I. C. (1987) Ar flow through cracks, Buldng and Envronment, Vol. 22, No. 4, pp ) Bhattacharyya, S., Clardge, D. E. (1995) The energy mpact of ar leakage through nsulated walls, Transactons of the ASME, Vol. 112, pp ) Brunsell, J. T. (1995) The ndoor ar qualty and the ventlaton performance of four occuped resdental buldngs wth dynamc nsulaton, 16 th AIVC Conference: Implementng the results of ventlaton research, Palm Sprngs, USA, September 18-22, Proceedngs Vol. 2, pp ) Buchanan, C.R., Sherman, M. H. (1998) Smulaton of Infltraton Heat Recovery, 19 th AIVC Annual Conference, Oslo, Norway, September, ) Burns, P. J., Chow, L. C., Ten, C. L. (1977) Convecton n a vertcal slot flled wth porous nsulaton, Int. J. Heat Mass Transfer, Vol. 20, pp ) Caffey, G. E. (1979) Resdental ar nfltraton, ASHRAE Trans., Vol. 85, pp ) Clardge, D. E., Lu, M. (1996) The measured energy mpact of nfltraton n an outdoor test cell, Transactons of the ASME, Vol. 118, pp ) Clardge, D. E., Bhattacharyya, S. (1990) The measured mpact of nfltraton n a test cell, J. Solar Energy Engneerng, Vol. 117, pp ) Etherdge, D. W. (1988) Modellng of ar nfltraton n sngle- and mult-cell buldngs, Energy and Buldngs, Vol. 10, pp ) Jensen, L. (1993) Energy mpact of ventlaton and dynamc nsulaton, 14 th AIVC Conference: Energy mpact of ventlaton and ar nfltraton, Copenhagen, September 21-23, Proceedngs, pp ) Kohonen, R., Vrtanen, M. (1987) Thermal couplng of leakage flow and heatng load of buldngs, ASHRAE Trans., Vol. 93, pp ) Lddament, M. W. (1987) Power law rules-- OK?, Ar Infltraton Revew, Vol. 8, No. 2, pp ) Mlls, A. F. (1992) Heat Transfer. Irwn, Burr Rdge, Illnos 14) NIST estmates natonwde energy mpact of ar leakage n U. S. buldngs (1996) J. Research of NIST, Vol. 101, No. 3, p ) Persly, A. (1982) Understandng ar nfltraton n homes, Report PU/CEES No. 129, Prnceton Unversty Center for Energy and Envronmental Studes, February, p ) Sherman, M., Matson, N. (1993) Ventlaton-Energy labltes n U.S. dwellngs, 14 th AVIC Conference: Energy mpact of ventlaton and ar nfltraton, Copenhagen, September 21-23, Proceedngs, pp

34 17) Versteeg, H. K., Malalasakera, W. (1995) Introducton to Computatonal Flud Dynamcs. Longman Scentfc & Techncal, New York. 18) Vrtanen, M., Hemonen, I., Kohonen, R. (1992) Applcaton of the transfer functon approach n the thermal analyss of dynamc wall structures, ASHRAE/DOE/BTECC Conference: Performance of the Exteror Envelopes of Buldngs, December 7-10, Clearwater Beach, Florda, USA, Proceedngs. Appendx 1: Infltraton Bascs App.1.1: Energy flux through the buldng envelope In ths analyss two modes of energy transfer through the buldng envelope are consdered: conducton, manly through the walls, and convecton, va nfltratng ar. These are shown graphcally n Fgure 1.App. In realty, energy transport also occurs through radaton, but we thnk nfltraton has lttle mpact on ths load and vce versa. Ths hypothess wll be tested n future work. Also, convectve transport occurs va the HVAC system ntake and exhaust, but ths has no drect effect on nfltraton heat recovery so t wll not be consdered here. The nfltraton load and conducton load nteract strongly and, therefore, are the most mportant quanttes to study n order to understand the nfltraton heat recovery phenomenon. Fgure 1.App: Cross secton of a generc buldng envelope showng the conventonal conducton and nfltraton energy load terms. Q o s the conducton energy load wth no nfltraton and mc p (T T o ) s the nfltraton energy load, where T s the ndoor ar temperature, T o s the outdoor ar temperature, and m s the leakage rate. Ar leakng nto the buldng s sad to be nfltratng and ar leakng out of the buldng exfltratng. 34

35 App.1.2: Conventonal method of accountng for nfltraton energy load In the absence of nfltraton through the buldng envelope, the energy load on the buldng, not consderng radaton and HVAC ntake/exhaust, s that purely from conducton, denoted as Q o. When there s nfltraton the conventonal method of accountng for the extra energy load, Q nfc, s to add a smple convectve term to the energy balance for the buldng that s based on the leakage rate and the ndoor and outdoor ar temperatures. The conventonal relatonshp used to calculate the total energy load wth nfltraton, denoted as Q C, s based on the system shown n fgure 1.App and s gven by equaton 1.App. Ths relatonshp assumes there s no nteracton (heat transfer, mosture deposton, etc.) between the leakng ar and buldng walls. Q C = Q + Q = Q + mc ( T T ) (1.App) o nf C o p o App.1.3: Includng Infltraton Heat Recovery n the conventonal method In realty, there s heat transfer between the buldng walls and leakng ar as s enters (nfltraton) and exts (exfltraton) the buldng envelope. The result s that the actual values for the conducton, nfltraton, and total loads are dfferent from that predcted by equaton 1.App. Ths case s shown graphcally n fgure 2.App. wth the approprate varables. The relatonshp for the actual total energy load s gven by equaton 2.App., n whch Q o *, Q nf, and Q are the actual conducton, nfltraton, and total loads, respectvely. Q * o * o = Q + Q = Q + mc ( T T ) 1 (2.App) nf p 2 In the case of a warm nsde and cool outsde (T >T o ), heat transfer wthn and near the surface of the buldng envelope walls would cause nfltratng ar to enter the buldng at a temperature greater than that outsde and exfltratng ar to ext at a temperature less than that nsde. The actual temperatures of nfltratng and exfltratng ar (T 1 and T 2, respectvely, taken at some consstent system boundary) are not known and are dependent on a host of factors, but ths mples that, n general, the nfltraton energy load s less than that predcted by the conventonal method. The effect on the conducton load s not known, however. Snce the process s asymmetrc n ths study (conducton decreased by nfltraton and ncreased by exfltraton), t s thought that the net effect on the conducton load s small. Ths suggests that the actual total energy load s less than that predcted by the conventonal method,.e., Q<Q C. In realty, the nfltratng and exfltratng leakage paths wll most lkely not be completely symmetrc and the actual effect on the conducton load wll depend on the leakage characterstcs of the specfc structure. Ths topc wll be explored n future work. Q = Q + 1 ε ) mc ( T T ) (3.App) o ( p o In lght of the stuaton,.e., an unknown actual conducton load thought to be close to the conducton load wth no nfltraton and a reduced nfltraton load wth unknown actual 35

36 ar temperatures, an obvous and smple method of accountng for the effect of nfltraton heat recovery s to nsert a reducton factor nto the conventonal relatonshp for the total buldng energy load, equaton 1.App., so that t provdes the actual total energy load. The reducton factor s multpled by the nfltraton term, snce ths term s known to be reduced by heat recovery, and the conducton term s represented by the conducton load wth no nfltraton. Equaton 3.App utlzes ths accountng method to gve the actual total energy load. The quantty ε n equaton 3.App s called the nfltraton heat recovery factor and accounts for the net effect of heat transfer between leakng and buldng walls. Note, the values of the ndvdual terms n equaton 3.App probably do not accurately predct the actual values but the sum total s correct f ε s properly chosen. Ths s a convenent and useful method because the parameters n equaton 3.App (Q o, m, T, T o ) should all be known. Ths reduces the problem to smply determnng ε, as compared to determnng the actual values of the ndvdual terms, whch could be very dffcult. Fgure 2.App: Cross secton of a generc buldng envelope showng the actual conducton and nfltraton energy load terms. Q * o s the actual conducton energy load wth nfltraton and mc p (T 2 T 1 ) s the actual nfltraton energy load, where T 2 s the exfltratng ar temperature, T 1 s the nfltratng ar temperature (taken at the system control volume boundary), and m s the leakage rate. Note that for T >T o, n general T 1 >T o due to warmng and T 2 <T due to coolng and the relaton between Q * o and Q o s unknown but they are thought to be nearly equal n sze. Appendx 2: CFD Detals App. 2.1: Wall modelng and boundary condtons In the CFD smulatons, several mathematcal equatons are used wthn the computatonal doman to represent dfferent components of the leakng buldng wall system 36

37 under nvestgaton. A schematc showng a cross-secton of the system s gven n fgure 3.App. for a sngle wall (nfltratng or exfltratng) wth an nsulated wall cavty. Ths schematc llustrates how varous equatons and boundary condtons are combned spatally to create the leakng buldng wall system. Note, results from two or more walls are added together to provde nformaton for a complete room system. The outer border of the system supples boundary condtons (temperature, pressure, velocty) for the smulaton. As fgure 3.App shows, the left and rght sdes of the system boundary are composed of a wall wth zero flow velocty and fxed thermal boundary condtons and an nlet or outlet wth fxed thermal and pressure (or velocty) boundary condtons. The regon wthn the boundary s comprsed entrely of lve cells (control volumes). Calculatons are performed for each lve cell usng the approprate governng equatons to determne numercal values for the relevant varables n that cell. In cells that represent ar, the Naver-Stokes and energy equatons (eq. 4-6) are the governng equatons. Cells that represent the wall sheathng are governed by the conducton equaton (eq. 7). Cells that represent nsulaton are governed by Darcy s Law and a modfed energy equaton (eq. 8-9). Ths nformaton s shown graphcally n fgure 3.App. Fgure 3.App: Schematc of a leakng buldng wall cross-secton showng the boundary condtons and governng equatons used n the CFD smulatons. The lve control volumes (those enclosed by the outer boundary) represent ar, nsulaton, and the wall sheathng. They 37

38 are governed by the equatons as shown n the fgure. The outer boundary s composed of a wall supplyng wall boundary condtons (zero flow velocty and a dfferent fxed temperature on each sde of the leakng wall) wth an nlet on one sde of the leakng wall and an outlet on the other. The nlet and outlet supply thermal and pressure (or velocty) boundary condtons. App. 2.2: Thermal equlbrum n the wall nsulaton A fundamental assumpton n usng a sngle flud dynamcs equaton (equaton 8) to represent energy transport n the wall nsulaton s local thermal equlbrum between the flud and sold phases (ar and glass fbers) of the nsulaton. If the two phases are not n thermal equlbrum then two separate energy equatons must be used, one for the flud phase and one for the sold phase. In ths secton, an analyss s performed usng a representatve system to determne f our ar/nsulaton system s n thermal equlbrum. In the representatve system ar flows nto a porous meda (the nsulaton) at a velocty and temperature of U and T a, respectvely. The sold part of the porous meda (glass fbers) mantans a constant temperature of T s. As ar flows nto the porous meda t exchanges heat wth glass fbers and the ar temperature changes untl t eventually reaches the glass fber temperature. At ths pont, they are n thermal equlbrum. For ths analyss, we wll consder the flud and sold to be n equlbrum when there s a temperature dfference of 0.1 percent or less between the phases. dt ρ c p V = h As ( Ts T ) (4.App) dt Re Pr Nu = (5.App) D 2 3 [ 1+ ( 0.4 / Pr) ] 1 4 h = NuD k / D (6.App) In ths analyss, a control mass of ar s followed as t flows through the porous meda. The temperature of the ar s calculated versus tme, or correspondngly penetraton dstance nto the porous meda, va equaton 4.App. Equaton 4.App s merely a statement of the Frst Law of Thermodynamcs; the control mass enthalpy tme rate of change (lhs) s equal to the rate of heat transfer (rhs). In equaton 4.App, V s the control mass volume, h s the heat transfer coeffcent, A s s the fber surface area, and T s ar temperature. The heat transfer coeffcent for cross-flow over a cylnder s used n these calculatons as ths flow geometry most closely resembles our case (the glass fbers are essentally crcular cylnders). Equatons 5.App and 6.App gve the relatons used to determne the heat transfer coeffcent based on Reynolds number, Prandtl number, and fber dameter (D). The control mass volume s arbtrarly set to 1 m 3. In our system, the nsulaton s 99 percent ar by volume and 1 percent glass fbers. The fber surface area s determned from the volume fracton and by settng the fber dameter to 10 mcrons (a typcal dameter for glass fber nsulaton) and assumng the fbers are 1 meter n length. 38

39 The tme t takes ncomng ar to reach thermal equlbrum and the correspondng penetraton dstance nto the porous meda are calculated wth equatons 4.App to 6.App. The ar temperature versus penetraton dstance nto the porous meda (nsulaton) s shown n fgure 4.App. In ths case, there s a nomnal non-dmensonal leakage rate (a o ) of one (correspondng to a flow velocty of about 0.08 m/s) and the bulk ar and glass fber temperatures are 274 and 298 Kelvn, respectvely. The fgure shows that as the ar flows deeper nto the nsulaton ts temperature ncreases and eventually reaches the glass temperature at a dstance of about 1.5x10-5 m. The ar reaches thermal equlbrum wth the glass fbers very quckly. Compared to the smallest dmenson of the wall nsulaton (0.1 m) ths dstance s neglgble. Therefore, t s approprate to use a sngle equaton to represent the energy transport n the wall nsulaton T (K) ar glass dstance (m) Fgure 4.App: Ar and glass fber temperature versus penetraton dstance nto the nsulaton. The glass fber temperature s held constant at 298 K and the ncomng ar temperature s 274 K. As the ar flows deeper nto the nsulaton ts temperature ncreases and eventually reaches the glass fber temperature at a dstance of about 1.5x10-5 m. Further calculatons are performed to determne the equlbrum dstance for the nsulaton at varous flow rates and for another porous meda, a packed bed composed of sphercal glass balls (2 cm da.). The non-dmensonal flow rate s vared from about 0.25 to 2.25 to see how the equlbrum dstance changes wth flow velocty. Fgure 5.App shows that the equlbrum dstance ncreases almost lnearly wth flow rate. Note that the vertcal scale showng dstance s logarthmc. Even at the hghest leakage rates (around 2 n most buldngs) the equlbrum dstance for the nsulaton would be very small compared to the wall dmensons, so a sngle equaton for energy transport n the porous meda would be approprate. The stuaton s dfferent for the packed bed, however. The equlbrum dstance for the packed bed s roughly three orders of magntude hgher than that for the nsulaton. Ths s manly due to the much lower surface area of the sold phase n the packed bed, whch results n less heat transfer. In ths case, the equlbrum dstance s on the order of the wall 39

40 thckness, so the ar and sold phase would not have enough tme to reach thermal equlbrum. Therefore, two equatons would have to be used to represent energy transport n the packed bed: one for the sold phase and one for the flud phase. 1.00E+00 equlbrum dstance (m) 1.00E E E E E-05 fbers packed bed 1.00E dmensonless flow rate Fgure 5.App: Equlbrum dstance versus leakage rate for two dfferent types of porous materal: glass fbers and a packed bed of glass spheres. The glass fber materal (nsulaton) has a much shorter equlbrum dstance than the packed bed suggestng that a sngle equaton can be used for energy transport n the fbers, whle two equatons would have to be used for energy transport n the packed bed. Fnally, a macro-scale analyss s performed to determne how the glass fber and ar temperatures vary through the entre depth of a leakng wall. Ths s a one-dmensonal analyss n whch t s assumed that only the mode of heat transfer s from the glass fbers to ar. In realty, ths s not the case, but ths analyss s good for llustratve purposes. 2 dt d T ρ c p VA = ka + hasurf ( Ts T ) 7.App 2 dx dx 2 d Ts ks As ha ( T T ) = 0 2 surf s 8.App dx The governng equatons for the ar/fber system are shown n equatons 7.App and 8.App. Equaton 7.App s the energy equaton for ar and 8.App s the energy equaton for glass fbers. In the above equatons, V s the leakng ar velocty, A s the cross-sectonal area of ar, h s the heat transfer coeffcent (see equaton 6.App), A surf s the glass fber surface area (determned earler n ths secton), T s s the fber temperature, T s the ar temperature, k s s the glass thermal conductvty, A s s the total cross-sectonal area of glass fbers, and x s the dstance nto the wall. A smple analyss, n whch the Peclet number (Pe = ρvlc p /k) s calculated usng characterstc values, shows that convecton s the domnant mechansm for 40

41 energy transport n the ar. Ths allows us to drop the conducton term n equaton 7.App (second-order term) n ths analyss. The governng equatons are solved usng standard technques of ODE s under the followng boundary condtons: T(0) = T out, T s (0) = T out, and T s (L) = T n, where T out s the outsde temperature (274 K) and T n s the nsde temperature (298 K). The standard soluton for T and T s s gven below n equatons 9.App and 10.App. In the equatons, C 1, C 2, and C 3 are constants, r 1, r 2, and r 3 are roots of the auxlary equaton, and λ s a rato of certan problem parameters,.e., ρvac p /ha surf. The actual values for the constants, the roots, and λ vary wth the assgned parameter values n the equatons. Fgure 6.App shows the ar and glass fber temperatures through the wall for a gven, characterstc set of parameter values. The two curves essentally overlap, as the temperatures are nearly equal. The same behavor occurrs wth all realstc varatons n the parameters, ndcatng that the assumpton of thermal equlbrum between the ar and glass fbers s vald. r1 x r21x r3x T ( x) C1 e + C2 e + C3 e = 9.App T s dt ( x) = T + λ 10.App dx temperature (K) ar fbers dstance (m) Fgure 6.App: Ar and glass fber temperatures versus dstance through the leakng wall for a chosen set of system parameters. There are two curves plotted, but only one s apparent because the ar and fber temperatures are nearly dentcal. Ths further supports the assumpton of thermal equlbrum between the ar and glass fbers. 41

42 App. 2.3: Soluton Methodology Although we want steady-state values for our analyss, t was not possble to acheve a converged soluton to the steady-state equatons (.e., no tme dependent term). The soluton ether dverged or would not converge to the set lmts when solvng the steady-state equatons. The convergence crtera are that the sum of the normalzed resduals are less than or equal to than 10-3 for the contnuty and momentum equatons and 10-6 for the energy equaton. These are the standard convergence crtera suggested n the Fluent user gudes. In an attempt to acheve a converged soluton, the transent equatons were solved and allowed to proceed n tme untl a steady-state soluton was reached. Ths would have worked well, but t would have taken too long. Usng a tme-step of 0.1 sec provded a converged soluton at each tme step and proceeded ncely toward steady-state, but t would have taken days to complete a sngle smulaton. To speed up the soluton process, a method was developed n whch the tme step was gradually ncreased from 0.1 sec to 60 sec and then gradually decreased agan to 0.1 sec. The tme step sze was changed gradually because t was found that sudden changes n the tme step could result n dvergence of the soluton. By ncreasng the tme step to a large sze of 60 sec, the flow and thermal felds can develop much more quckly (n wall clock tme) whch helps to reduce the overall smulaton tme. Unfortunately, wth the 60 sec tme step the soluton dd not converge, so t was necessary to reduce the tme step back down to 0.1 sec to provde a converged soluton for the fnal results. Snce the soluton process uses an teratve method, the fnal results at the end of the smulaton are all that really matters. All of the prevous values calculated durng the soluton process, whether converged or not, can be consdered mproved ntal guesses n the teratve process that lead to the fnal soluton. One addtonal technque was used to speed up the soluton process. At several ponts n the smulaton, when the 60 sec tme step was n use, the soluton was stopped and the equatons were decoupled. The contnuty and momentum equatons were turned off (the flow feld was frozen n tme) and the energy equaton was allowed to proceed n tme untl a steady-state soluton (based on the ntermedate frozen flow feld) was reached. Then, the contnuty and momentum equatons were turned back on and the soluton proceeded wth the equatons coupled. After a certan number of tme steps have passed, the equatons were decoupled agan and the process repeated. Ths helped speed up the soluton because the thermal feld developed slowly, relatve to the flow feld, due to the nsulatng effects of the wall. In ths way, the thermal feld was allowed to jump ahead of the flow feld and very quckly reach an ntermedate soluton. Then, once the flow equatons were turned back on the flow and thermal felds would nteract and come back nto equlbrum. Ths entre method s a farly common technque for speedng up solutons of complex CFD problems and s dscussed n the Fluent users gude (Fluent 4.4 user gude, Volume 3, Chapter 16, pages 44-49). The soluton process used n ths work s detaled below. Soluton Process: 1) Provde ntal guess (ntal condtons) for flow and thermal felds. All veloctes are set to zero. The left half of the doman s set to the nsde temperature (or outsde dependng on whether the wall s nfltratng or exfltratng) and the rght half s set to the outsde temperature (or nsde temp.) 2) The soluton s begun wth a 0.1 sec tme step and proceeds for 5 tme steps. 42

43 3) The tme step sze s changed to 1 sec and the soluton proceeds for another 5 tme steps. 4) The tme step sze s changed to 10 sec and the soluton proceeds for another 5 tme steps. 5) The tme step sze s changed to 60 sec and the soluton proceeds for another 60 tme steps. 6) The flow equatons are turned off and the energy equaton s solved to steady-state usng the ntermedate flow feld. 7) The flow equatons are turned back on and the soluton proceeds for another 60 tme steps wth a tme step sze of 60 sec. 8) Steps 6 and 7 are repeated twce. 9) The tme step sze s reduced to 10 sec and the soluton proceeds for 20 tme steps. 10) The flow equatons are turned off and the energy equaton s solved to steady-state usng the ntermedate flow feld. 11) The flow equatons are turned back on and the solutons proceeds for another 20 tme steps. 12) Steps 10 and 11 are repeated once. 13) The tme step s reduced to 1 sec and the soluton proceeds for 60 tme steps. 14) The flow equatons are turned off and the energy equaton s solved to steady-state usng the ntermedate flow feld. 15) The flow equatons are turned back on. 16) The tme step s reduced to 0.1 sec and the soluton proceeds for 100 tme steps. 17) The data s wrtten to a fle. Appendx 3: Sample CFD Results: Energy Flux through Leakng Walls In the framework of ths study, a buldng envelope secton wth no ar leakage has only one component n the energy flux through the wall, pure conducton, denoted as Q o. An envelope secton whch experences ar leakage has two components n the energy flux, that due to conducton (Q * o) and that due to convecton or nfltraton (Q nf ). In calculaton of the heat recovery factor, the non-leakng elements subtract out of the equaton (see equaton 2), so only the leakng walls need to be consdered. Therefore, n ths study a complete room s constructed by combnng two leakng walls, one nfltratng and one exfltratng. Ths s llustrated n fgure 7.App. The nfltratng and exfltratng walls are smulated ndependently and the four flux terms (two for each wall) are added together to determne the heat recovery factor for a gven leakage rate. In each case, the conducton and convecton flux component s determned at the external (outsde) face of the wall as shown n fgure 7.App. 43

44 Fgure 7.App: Convecton (Q nf ) and conducton (Q * o) energy flux components through nfltratng (left) and exfltratng (rght) walls wth a h/low hole confguraton (wall 1). The drectons of leakng ar (convecton) and conducton are ant-parallel n the nfltratng case and parallel n the exfltratng case. The control volume boundary s the external face of the walls,.e., the left face of the nfltratng wall and rght face of the exfltratng wall. As the leakage rate vares, the conducton and convecton flux components change n both the nfltratng and exfltratng walls. The total energy flux through the boundary of the room system changes, also. Ths s because leakng ar alters the thermal profles n and near the wall changng the overall heat transfer. Ths s llustrated n fgure 8.App usng CFD results from an nfltratng wall 1 confguraton. Fgure 8.App shows that parts of the wall that are far away from the holes n the sheathng (the mddle secton of the wall n ths case) have a nearly lnear temperature profle through the wall. Ths s smlar to a wall wth no leakage. Leakng ar does not seem to have a sgnfcant effect n ths regon. In the vcnty of the holes, leakng ar changes the thermal profles n the wall and sgnfcantly mpacts the heat transfer through ths regon. 44