A SIMPLE METHOD TO INCORPORATE THERMAL BRIDGE EFFECTS INTO DYNAMIC HEAT LOAD CALCULATION PROGRAMS. Akihiro Nagata

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1 Nnth Internatonal IPSA Conference Montréal, Canada Augut 5-8, 005 A SIMPLE MEHOD O INCORPORAE HERMAL RIDGE EFFECS INO DYNAMIC HEA LOAD CALCULAION PROGRAMS Akhro Nagata Faculty of Urban Envronental Scence okyo Metropoltan Unverty okyo 9-097, Japan ASRAC he a of th tudy to develop a ple ethod to predct heat flow and nu urface teperature n dynac heat load calculaton progra. A copoton of two lnear flow coponent aued ntead of an eleent ncludng a theral brdge. A fractonal area of the theral brdge part deterned by the axu and the average theral tranttance. We alo dcu a ethod to conttute a one-denonal agnary coponent of the theral brdge. INRODUCION here are any tude to predct the teady-tate perforance of theral brdge, whle few tude addreed the unteady-tate perforance (urch, D.M., See, J.E., Walton, G.N., and Lctra,.A., 99). Recently, however, theral brdge effect have becoe ncorporated nto dynac heat load calculaton progra (for exaple, EnergyPlu, 004 ). Although they addre only whole heat flow, the nu urface teperature alo portant ue n conderng theral brdge effect. In th paper, we propoe a ple ethod to conder both heat flow and nu urface teperature. heral brdge gve re to coplex two or three denonal heat flow, whch calculated by a two or three denonal fnte dfference ethod (FDM) n the Laplace tranfor doan. A copoton of two lnear flow coponent aued ntead of an eleent ncludng a theral brdge. One a theral brdge part of the eleent and the other a general part. he axu and nu of the dtrbuton of the theral tranttance correpond to the theral tranttance of the theral brdge coponent and that of the general coponent, repectvely. he fractonal area of the theral brdge coponent deterned to preerve the whole theral tranttance. h the fundaental dea of the propong ethod to conder both heat flow and nu urface teperature. he arguent n the above teady tate extended to an unteady tate. A for the tranttance and adttance tranfer functon, the lar relaton are uppoed. If a heat load calculaton progra baed on the repone factor ethod, what neceary only the Laplace tranfor nveron nuercally. If, however, a progra baed on FDM etc., t neceary to conttute a one denonal agnary coponent. herefore, a ethod to conttute the -layer odel alo propoed. h ethod ple and deterntc unlke the teratve ethod adopted by EnergyPlu. It appled to a flat-urface theral brdge and gve a reaonably good approxaton. HERMAL RANSMIANCE In ISO 468, the calculaton of the tranon heat tranfer coeffcent L nclude the contrbuton due to theral brdge, accordng to Equaton (). () L = AU + Ψ l + χ where, k k j k j U : the theral tranttance of eleent [W/( K)] A : the area of eleent [ ] Ψ k : the lnear theral tranttance of lnear theral brdge k [W/(K)] l k : the length of lnear theral brdge k [] χ j : the pont theral tranttance of the pont theral brdge j [W/K] Default value of lnear theral tranttance Ψ for varou type of theral brdge are alo preented n ISO 468. In Japan, the theral tranttance of a coponent ncludng theral brdge frt etated by parallel-path ethod and the effect of the theral brdge corrected by ultplyng the average theral tranttance U by the "theral brdge factor" β to eet the actual theral tranttance U. hu, U = βu () and default value of theral brdge factor β are gven n "Expoton of the Energy Conevaton Standard for Redental uldng"

2 Although thee two approache treat only whole heat flow rate, the nu urface teperature alo portant ue n conderng the theral brdge effect. In th paper, we propoe a ple ethod to conder both heat flow and nu urface teperature. A copoton of two lnear flow coponent aued ntead of an eleent ncludng a theral brdge. One coponent a theral brdge part of the eleent wth a fractonal area f and the other coponent a general part wth a fractonal area f. he axu and nu of the dtrbuton of the theral tranttance correpond to the theral tranttance of the theral brdge coponent U b and that of the general coponent U g, repectvely. he theral tranttance U then obtaned by the followng equaton. general part theral brdge part ISO 468 lnear theral tranttance theral tranttance of general part Expoton of the Energy Conevaton Standard for Redental uldng ( Japan) area-averaged theral tranttance general part actual theral tranttance dtrbuton actual theral tranttance dtrbuton U = ( f) U g + fu b () Actually, U, U b and U g are calculated nuercally by the fnte dfference ethod or the fnte eleent ethod, and thee wll deterne the fractonal area f. hu, U Ug f = (4) U U b g he theral tranttance drbuton are appeared on both nteror and exteror urface. he fractonal area hould be deterned by the dtrbuton of the nteror urface. If both nteror and exteror axu theral tranttance are conerved, each urface generally ha a dfferent fractonal area and t poble to replace a theral brdge wth a ple lnear flow coponent. h the fundaental dea of the propong ethod. Fgure how the concept of above three ethod. SAE-SPACE REPRESENAION If we dcrtze the pace ung the fnte dfference ethod or the fnte eleent ethod, the lnear te nvarant ultdenonal heat tranfer yte generally repreented a the followng decrptor for. C () t = K() t + K () t (5) Ω Ω Ω Ω Ω q () t = K () t + K () t C () t (6) where, Ω Ω Ω () t : nternal teperature vector at te t () t :boundary teperature vector at te t q () t :boundary heat flux vector at te t C Ω :nternal heat capacty atrx Propong ethod n th paper C :boundary heat capacty atrx K Ω, K, K Ω : heat tranfer atrx and a dot ean te dervatve. In cae of C 0, the above repreentaton not a foral decrtor for but we leave C n the repreentaton. Snce there ay be a node wthout a heat capacty, t not obvou that C Ω regular (the nvere atx C Ω ext). Equaton (5) and (6) are uually olved by ung a fnte te dfference approach, but oete olved by ung an analytcal te ntegraton approach. he Laplace tranfor of equaton (5) and (6) are C () = K () + K () (7) Ω Ω Ω Ω Ω q () = K () + K () C () (8) Ω Ω where the Laplace paraeter. Input-output for q () = G() () (9) where theral tranttance of general part area adjutent Fgure Concept of the odel () = ( ) + G KΩ CΩ KΩ KΩ K C G () called the tranfer functon atrx and G KΩKΩ KΩ K theral tranttance of theral brdge actual theral tranttance dtrbuton = + ()

3 = () G KΩKΩ CΩKΩ KΩ C nce ( ) C K = K K C K + () Ω Ω Ω Ω Ω Ω he Fourer tranfor can be forally obtaned by ubttutng nto ω, where the agnary unt and ω the angular frequency. If there only two boundare, one nteror and the other exteror, () and q () n equaton (9) are () () () = e (4) q () () () = q qe (5) where and q are the nternal envronental teperature and the nteror urface heat flux, e and q e are the external envronental teperature and the exteror urface heat flux, repectvely. When the gn potve for the outward heat flux, the tranfer functon atrx pecfcally GA() G () G () = G() GAe() (6) where GA(), GAe() and G () are called the nteror adttance tranfer functon, the exteror adttance tranfer functon and the tranttance tranfer functon, repectvely. SYSEM REDUCION AND RESPONSE APPROXIMAION In the feld of yte theory, odel reducton ethod of a lnear te nvarant yte have been alot etablhed (Obnata, G., Anderon,.D.O. 000). Aong the varou ethod, odal truncaton and balanced realzaton are ajor ethod. Modal truncaton a ethod to preerve egenvalue of C Ω KΩ up to oe order and to cut off the hgher order. It clac and wdely ued epecally n the vbraton analy. alanced realzaton a ethod to buld a yte that the obervablty graan and controllablty graan ay becoe the ae dagonal atrx. It ued n SISLEY oftware (Déqué, F., Noel, J. and Roux, J.J., 00) whch calculate dynac properte of a theral brdge. Approxate accuracy of both odel by ple truncaton good for hgh frequency but not good for low frequency, and teady-tate repone are uually unpreerved. It, however, poble to preerve teady-tate repone wth a ngular perturbaton ethod. A a ethod baed on a repone approxaton, Padé approxaton well known. Padé approxaton a way to approxate a tranfer functon by a ratonal polynoal, whch few ter of aylor' ere concde wth thoe of the orgnal functon. he collocaton ethod (Schapery, R.A., 96), oete called the leat-quare ethod, a ajor technque of the Laplace tranfor nveron. Accordng to th technque, t neceary to calculate only few nuercal value of a tranfer functon for real potve value of the Laplace paraeter. A ratonal polynoal atfed thee collocated value can be obtaned by olvng a yte of lnear equaton. he collocaton ethod wa appled to approxate repone of planar wall (Matuo, Y., 97), and t wa alo appled to earth contact wall (Nagata, A. and Matuo, Y., 990) and lnear heat and oture tranfer proble (Nagata, A. and Matuo, Y., 99). A nuber of the Laplace paraeter deterne the order of reduced yte. he et of the Laplace paraeter n hour k k = 0and = 0, = 0,,, (7) whch uually enough for buldng heat tranfer proble. In th tudy, we ue the collocaton ethod for obtanng arroxate repone of theral brdge. DYNAMIC PROPERY OF HERMAL RIDGE he arguent n the teady tate extended to an unteady tate. he fractonal area of a theral brdge fxed at the value of the teady tate ( = 0). h ean that the approxate accurary of the dynac property of the theral brdge part becoe worth but t poble to preerve the whole heat flux. A for the tranttance tranfer functon G (), a lar relaton to equaton () uppoed. hu G () = ( f) G () + fg () (8) g b where Gg() and Gb() are the tranttance tranfer functon of the general coponent and that of the theral brdge coponent, repectvely. When G () and Gg() are obtaned fro the FDM calculaton, then the tranttance tranfer functon of the theral brdge coponent, Gb(), derved fro G() Gg() Gb() = Gg() + (9) f he adttance tranfer functon of the theral brdge coponent alo derved n a lar way. GA() GAg() GAb() = GAg () + (0) f

4 where GA(), GAg () and GAb() are the adttance tranfer functon of the whole coponent, the general coponent and the theral brdge coponent, repectvely. Althogh there are nteror and exteror adttance tranfer functon, we need nteror one for a heat load calculaton. If a heat load calculaton progra baed on the repone factor ethod, what neceary only the Laplace tranfor nveron nuercally by the collocaton ethod. If, however, a progra baed on FDM etc., t neceary to conttute a one denonal agnary coponent. IMAGINARY COMPONEN In th tudy, ple and deterntc ethod to conttute the -layer odel and the -layer odel are alo propoed. Here we have already obtaned the tranttance and adttance tranfer functon. oth nteror and exteror adttance tranfer functon are needed to conttute a one denonal agnary coponent. he agnary coponent odel atfed the followng condton. a) the theral tranttance (total retance) conerved. b) the total capacty conerved. c) each layer ha no negatve retance and capacty. d) the boundary layer of both urface are eparated fro the body (bound node are put on the urface). Careful attenton hould be gven to the urface coeffcent of heat tranfer (urface retance). If they vary, the fractonal area of the theral brdge coponent alo vare. A the nteror urface coeffcent of heat tranfer becoe large, the fractonal area of the theral brdge becoe large and the theral tranttance of the theral brdge coponent becoe all. herefore, the nteror urface coeffcent of heat tranfer hould be all for conderng rk factor of nu teeprature and hold be large for conderng that of whole heat flow rate. he tranfer functon atrx wthout urface boundary layer GA() G () G () GAe() G G h = + G () () 0 h GAe e where A() () 0 () h : nteror urface coeffcent of heat tranfer h e : exteror urface coeffcent of heat tranfer GA () : nteror adttance tranfer funton atrx wthout urface boundary layer GAe( ) : exteror adttance tranfer funton atrx wthout urface boundary layer G () : tranttance tranfer funton atrx wthout urface boundary layer We expand the tranfer functon atrx wothout urface boudary layer n a aylor' ere GA() G () G () GAe() () G A G = R + + G G Ae where R the total retance wthout urface boundary layer and / R = G = G = G () A Ae he quantty of torage heat n the coponent when nteror urface teperature equal and external urface teperature equal 0, C = G G (4) A and the torage heat n the revere cae C = G G (5) e Ae he u of C and C e the heat capacty of the coponent. C = C + Ce (6) = G + G G A Ae Frt we defne the followng two paraeter. 4 G rs = + (0 r S ) (7) C c G A G Ae = C + 4 G ( c ) (8) For conttutng a -layer odel (Fgure-), R devded n the rato r : r, and C devded n the rato c : c. r and c are deterned by the frt order Padé approxaton. he tranfer functon of the -layer odel can be derved analytcally. hu C G A = c( r) + ( r) (9)

5 C G Ae = ( c)( + r) + r (0) C G = [ c(r ) + ( r)( + r)] () 6 then rs r = + () 4cr S R r R ( r ) C c C ( c ) Fgure- -layer odel e e rs c = + + crs () 4cr S hee oluton, however, reach nfnty at c = 0. Snce the coponent yetrcal at c = 0, t preferable that r and c are /. hey are, therfore, rs r = + crs (4) 4 5 rs c = ax 0, n + crs (5) 4 For conttutng a -layer odel (Fgure-), R devded n the rato r : r r : r, and C devded n the rato c : c c : c. Fro the reult that the tranfer functon of -layer odel are derved analytcally and expanded n aylor' ere, we obtan the followng relaton. e e R r R ( r r ) R r C c C ( c c ) C c Fgure- -layer odel rs = ( c )( + r r ) ( )( c ) + r r ( cr cr) ( c c ) rr (6) cr = ( c)( r) ( c)( r) (7) S whch gve rs + crs( r ) c = r + crs 4( r ) rs crs( r ) c = r crs 4( r ) (8) (9) A for the theral retance allocaton, we adopt rs r = ( crs) (40) rs r = ( + crs) (4) whch atfy c = c = r = r = / at r = / and c = 0 (a ngle layer wall). S NUMERICAL EXAMPLE A ple exaple of a flat-urface theral brdge (Fgure-4) hown n th ecton. Fgure-5 how heral tranttance U [W/ K] Fgure-4 Exaple of a theral brdge approxaton wdth: wdth: x [] Fgure-5 heral tranttance dtrbuton and t approxaton - 8 -

6 theral tranttance dtrbuton along the nteror urface whch are calculated by FDM progra. he ptche of a theral brdge are et and. Approxaton of both cae are alot ae and the theral brdge coponent yply eparated fro the general coponent. Fgure-6 how tranfer functon of the theral brdge coponent and at the center of the theral brdge n the Laplace tranfor doan. h reult ean that dynac properte of the theral brdge coponent derved fro the propoed ethod n th paper approxate thoe at the center of the theral brdge. hree-layer odel gve a reaonably good approxaton n th exaple (Fgure-7). he retance of the layer are 0.065, 0., 0.08 [ K/W] fro nteror to exteror, repectvely and the heat capacte of the layer are , , [J/ K], repectvely. CONCLUSION h paper propoe a ple ethod to predct both heat flow and nu urface teperature. REFERENCES urch, D.M., See, J.E., Walton, G.N., and Lctra,.A., 99. Dynac evaluaton of theral brdged n a typcal offce buldng, ASHRAE ran. 98(), EnergyPlu, 004. Engneerng Docuentaton: he Reference to EnergyPlu Calculaton. uldng Sulaton Group. Lawrence erkeley Natonal Laboratory. erkeley, Calforna. Obnata, G., Anderon,.D.O Model Reducton for Control Syte Degn, Sprnger-Verlag. Qu, Z., 004. Model Order Reducton echnque, Sprnger-Verlag. Inttute for uldng Envronent and Energy Conervaton. 00. Expoton of the Energy Conevaton Standard for Redental uldng (n Japanee) ISO 468: 999. heral brdge n buldng contructon - Lnear theral tranttance - Splfed ethod and default value Schapery, R.A., 96. Approxate Method of ranfor Inveron for Vcoelatc Stre Analy, Proc. of the Fourth U.S. Natonal Congre on Appled Mechanc, Vol., Déqué, F., Noel, J. and Roux, J.J., 00. SISLEY : An open tool for tranent-tate two-denonal heat tranfer, Proc. of uldng Sulaton '0, ranfer functon[w/ K] ranfer functon of the theral brdge coponent ranfer functon at the center of theral brdge G Ab () G b () E-8 E-6 E-4 E- E+0 Laplace paraeter[ ] Fgure-6 ranfer functon of the theral brdge (FDM reult) ranfer functon[w/ K] FDM -layer odel G Ab () G b () E-8 E-6 E-4 E- E+0 Laplace paraeter[ ] Fgure-7 ranfer functon of the theral brdge (FDM v. -layer odel) Matuo, Y., 97. Approxaton ethod and t applcaton to heat tranfer of buldng, Suare of echncal Paper of Annual Meetng, A.I.J., -4 (n Japanee). Nagata, A. and Matuo, Y., 990. Approxaton ethod for heat tranfer analy of e-nfnte old, Suare of echncal Paper of Annual Meetng, A.I.J., D, (n Japanee). Nagata, A. and Matuo, Y., 99. Repone analy of wall conderng ultaneou heat and oture tranfer (part ) Approxaton by collocaton ethod, Suare of echncal Paper of Annual Meetng, A.I.J., D, (n Japanee)