IN-SITU THERMAL RESISTANCE EVALUATION OF WALLS USING AN ITERATIVE DYNAMIC MODEL

Transcription

1 In. Journal for Housing Science, Vol.38, No.1 pp.1-12, 214 Pulished in he Unied Saes IN-SITU THERMAL RESISTANCE EVALUATION OF WALLS USING AN ITERATIVE DYNAMIC MODEL Anónio Jóse Barreo TADEU and Nuno SIMÕES Universiy of Coimra, Deparmen of Civil Engineering (DEC) Coimra, Porugal Inês SIMÕES and Filipe PEDRO Insiuo de Invesigação em Ciências da Consrução (ITeCons) Coimra, Porugal Leopold ŠKERGET Universiy of Marior, Faculy of Mechanical Engineering Marior, Slovenia ABSTRACT This paper proposes and validaes a numerical ieraive model o evaluae he hermal resisance of mulilayer sysems (walls) when in a dynamic sae. Firs, he validaion is performed numerically, hen he second sep of validaion uses he emperaure and hea flux values recorded during experimenal ess performed in a ho ox chamer. These validaions involve comparing he resuls oained wih hose expeced, given he hermal properies of each maerial and hickness of each wall layer. The paper firs presens he analyical soluion for simulaing hea ransfer y conducion in he frequency domain, hrough he mulilayer sysem. This is generaed y imposing emperaures on he exernal surfaces, when he hermal properies of he maerials are known. The model is hen modified y assuming he wall is composed of a single layer wih unknown hermal properies. The emperaures and hea fluxes, provided earlier y he analyical model and imposed on he exernal surfaces, lead o a nonlinear sysem ha can e solved for he unknown hermal properies. I is solved y implemening an ieraive approach ased on he Newon-Raphson mehod /1/1-12, 214 Copyrigh 213 IAHS

2 2 Tadeu, N. Simões, I. Simões, Pedro and Škerge Afer he validaion of he proposed model, his is used o evaluae he hermal resisance of a mulilayered wall sueced o real condiions. Key words: Green s Funcion Formulaion, Frequency Domain, Mulilayer Walls, Thermal Resisance, Ieraive Dynamic Model. Inroducion The numer of sudies on esimaing energy use for heaing and cooling spaces in exising uildings has een seadily increasing as a resul of people's growing expecaions regarding energy consumpion reducion [1]. In EU counries, he required energy cerificaions [Direcive 21/31/UE, Direcive 22/91/EC] have played a par in he recen increases oserved in he numer of energy audis. Since he energy performance of a uilding ends o depend on he envelope performance, i is very imporan o have accurae informaion aou is hermal performance o define measures o increase he energy performance of exising uildings. Overall hea ransmission is frequenly ased on he hermal ransmiance, or U-value, of each elemen in he uilding envelope [ISO 13789:27, ISO 1379:28]. Mos of he ime here is no informaion aou he composiion of he envelope of exising uildings. Two possiiliies may e explored in hese circumsances: o apply desrucive ess o idenify he maerials and heir hickness, or o perform direc measuremens of he hea flow [3]. The U-value can e found y measuring oh he hea flow (in accordance wih ISO 9869:19974) hrough he uilding elemen and he emperaure on oh sides of i. If he sysem is under seady-sae condiions, he U-value can e very accurae. However, since oudoor condiions are always changing, is no usual o find seadysae condiions during in-siu measuremens. Two main approaches can hen e applied: record he hea flow rae and emperaures over a long period in such way ha allows he good esimaion of equivalen seady-sae ehavior, or apply a dynamic model o ake ino accoun he surface emperaure and hea flow rae variaions. In he firs, simpler, approach, he resuls may e quie inaccurae unless he sorage effec caused y hermal mass (ineria) is negligile for he hea flow raes in quesion. Changes in hea flow direcion will also lead o imprecise measuremens. Laureni e al. [4] presen a mahemaical model for calculaing he hermal resisance of a wall hrough he dynamic analysis of in-siu daa, oh hea flux and surface emperaure measuremens. The proposed mehod models he ransien hermal response of he wall hrough a linear relaion wih consan parameers ha links he insananeous hea flux a he inner surface of he wall o he emperaure difference eween he surfaces of he same wall a he same insan. Cucumo e al. [5] proposed a mehod for he experimenal deerminaion of he in siu uilding wall conducance, ased on oh inside and ouside hea fluxes and surface emperaures. This mehod was applied o a es wall of an exernal esing saion a he Universiy of Calaria a differen imes; he resuls agreed wih values oained y means of he progressive mehod [6]. The proposed mehod provides an equivalen conduciviy. Wang e al.

3 In-Siu Thermal Resisance Evaluaion of Walls 3 [7] esalished and validaed a ransien hea ransfer model of a wall y applying he finie difference mehod. Addiionally, an in-siu measuremen program using a hea flow meer was underaken, paying special aenion o he wind velociy. The proposed mehod, he mean mehod and he dynamic analysis mehod proposed y ISO 9869:1994 were implemened on he es wall under differen wind velociies. The wall hermal resisance value oained y he proposed mehod was shown o e in eer agreemen wih ha oained for a seady sae. Our paper proposes and validaes a numerical ieraive model o evaluae he hermal resisance of mulilayer sysems (walls) when in a dynamic sae. I firs descries a dynamic model for simulaing hea diffusion hrough mulilayer sysems in he frequency domain, generaed y imposing emperaure variaion on he exernal surfaces. Then, a dynamic ieraive model is implemened for evaluaing he hermal resisance of a mulilayer sysem, ased on he Newon-Raphson mehod, where he exernal hea fluxes and emperaures are prescried. Afer he validaion of hese models, he dynamic ieraive model is used o evaluae he hermal resisance of mulilayer sysems sueced o hea diffusion in a conrolled environmen, using a ho-ox, and in an in-siu environmen. Differen experimenal measuremens are presened and he expeced hermal resisance is compared wih numerically compued values. Dynamic Model for a Mulilayer Sysem Consider a muli-layered sysem uil from a se of m plane layers of infinie exen, as shown in Figure 1. This sysem is sueced o emperaures and a he op and oom exernal surfaces. The layers are assumed o e infinie in he x and z direcions. The hermal maerial properies and hickness of various layers may differ. The ransien hea ransfer y conducion in each layer is expressed y he equaion 2 2 ( y ) T(, y) ρ c ( T(, y) ) λ = (1) in which T (, y ) is ime, is emperaure, λ idenifies he layer, conduciviy, ρ c is he densiy and is he specific hea. is he hermal The soluion is defined in he frequency domain afer a Fourier ransformaion is applied o equaion [1]: ( ( ) ) 2 y 2 i ω α 2 + ˆ T( ω, y) = (2) where i= 1 α = λ, ( ρ c) frequency. is he hermal diffusiviy of he layer, and ω is he

4 4 Tadeu, N. Simões, I. Simões, Pedro and Škerge The oal hea field is achieved y adding he surface erms arising wihin each layer and a each inerface, as required saisfying he oundary condiions a he inerfaces, i.e. coninuiy of emperaures and normal flows eween layers. For he layer, he hea surface erms on he upper and lower inerfaces can e expressed as ( ν ) T% ( ω, y) = E E A 1 1 ( ν ) T% ( ω, y) = E E A 2 2 (3) (4) E = 1 where A λ, 1 y hl l= 1 E 1 e =, y hl l= 1 E 2 e = h and is he hickness of he layer l l. are a priori unknown poenial ampliudes. A sysem of 2m equaions is derived, ensuring he coninuiy of emperaures and hea flows along he m 1 inerfaces eween layers, and y imposing he emperaures $ and $ a he op and oom exernal surfaces. $ and $ are oained y Fourier ransformaion of and in he ime domain. All he erms are organized according o he form A and 1h1 1 e h1 e 1... $ 1h1 e A n A n =... mhm... 1 e A nm mhm 1 e A... nm 3 m λmνm $ mhm e 1... m m m m (5) The resoluion of his sysem gives he ampliude of he surface erms a each inerface, leading o he following emperaure and hea flux fields a layer, (( ) ( ) ) T% ( ω, y) = E E ν A + E ν A 1 2 if 1 h < y< h l l l= 1 l= 1 (6) T( ω, y) λ % = i E A + y ( 1 E 2A ) if 1 (7) hl < y< hl l= 1 l= 1.

5 In-Siu Thermal Resisance Evaluaion of Walls 5 h 1 Inerface 1 Medium 1 θ = x x Inerface 2 Medium 2 Inerface 3 h m Inerface m Medium m y Inerface m+1 θ = FIG 1. Geomery of he prolem. Ieraive Dynamic Model o Evaluae Thermal Resisance This secion descries how he hermal resisance of a mulilayer sysem is evaluaed. For his purpose, he model descried aove is replaced y a single-layer wall ounded y wo hin air layers o accoun for he surfaces hermal resisance (see Figure 2). The hickness of he wall is he same as he overall hickness of he mulilayer sysem. Is hermal properies ( λ v = ρ c and ) are a priori unknown However, i assumed ha in addiion o $ and $ on he exernal surfaces (inerfaces 1 and 4) eing known in he frequency domain, he hea fluxes ( q and q ) on he wall surfaces are also known (inerfaces 2 and 3). This resuls in a nonlinear sysem 1h1 1 e h1 2h2 e 1 1 e $ e 1 1 e A 1h1 2h A 1 2h2 3h 3 e 1 1 e A 2 = 2h2 3h 3 e 1 1 e A A 3 $ 3h 3 e 1 A 3 q q 1h1 ie i 3h3 i ie [8] This sysem is solved y employing an ieraive approach ased on he Newon- Raphson mehod (Newon s mehod for he common eigenvecor prolem). This requires defining a marix of he firs derivaives of he sysem of equaions in equaion [8], ha is no presened in his paper.

6 6 Tadeu, N. Simões, I. Simões, Pedro and Škerge The ieraive process sars wih an iniial guess ha is updaed in every ieraion unil convergence is reached. If he soluion sars eing unsale a new guess is inroduced and he process coninues unil convergence is reached. The hermal resisance of he sysem corresponds o ha o he saic response, ha is, for null frequency. Inerface 1 Inerface 2 Medium 1 θ = $ o h 1 x Medium 2 h 2 y Inerface 3 Inerface 4 Medium 3 θ = $ o h 3 FIG 2. Geomery of he prolem used for he applicaion of ieraive dynamic model. Analyical Verificaion of he Model Two consrucion sysem walls are sudied, as shown in Figure 3. These sysems are composed of differen maerials, viz., concree, radiional morar and radiional plaser. The hermal insulaion is provided y exruded polysyrene (XPS). Tale 1 liss he hermal properies of hese maerials. In all compuaions hin air layers are simulaed on he ouer surfaces o accoun for convecion and radiaion phenomena. I was imposed a sinusoidal variaion in emperaure on he ouer surface (he surface lef of each uilding sysem shown in Figure 3): he iniial emperaure is assumed o e 2ºC and i flucuaes y 2ºC in each 24-hour period. The oher surface is sueced o a consan emperaure of 2ºC. The analyical compuaions were performed in he frequency domain for frequencies ranging from. Hz -3 o Hz, wih a frequency incremen of analysis window of approximaely 16 days and 13 hours. Tale 2 liss he imposed emperaures $ and $ Hz, in a full, he compued q q and using he dynamic model for he mulilayer sysem (he direc prolem) ha are used as inpu daa in he ieraive dynamic model for hermal resisance evaluaion for a null frequency (saic response). In addiion, i includes equivalen hermal conduciviy λ model given y he ieraive model and ha expeced assuming he exisence of a permanen hea flow rae (saic response), m 1 m 1hi λequiv hi λ = i= 2 i= 2 i.

7 In-Siu Thermal Resisance Evaluaion of Walls 7 Maerial Tale 1. Maerials hermal properies. Conduciviy λ Mass Densiy -3 ρ (kg.m ) c Specific Hea (J.kg.ºC ) Thermal Diffusiviy K 2 1 (m. s ) (1) Concree e-7 Exruded (3) polysyrene e-7 (XPS) (4) Tradiional morar e-8 (5) Tradiional plaser e-7 Air (ouer exposed surface) e-5 Air (inner exposed surface) e-5 Example 1 Example 2 1 Concree. 3 - Thermal insulaion (exruded polysyrene). 4 Tradiional morar. 5 Tradiional plaser. Noe: hicknesses in meer FIG 3. Wall sysems sudied: composiion and dimensional characerisics of he soluions. Tale 2. Thermal resisance evaluaion for a null frequency (saic response). Case ex = = (ºC) (ºC) (W.m -2-2 q ) q (W.m ) λ equiv λ model The resuls show a very good agreemen eween he hermal resisance evaluaion given y he ieraive model and he expeced value.

8 8 Tadeu, N. Simões, I. Simões, Pedro and Škerge Experimenal Validaion of he Model Using a Ho Box (Unseady Sae Hea Flow) The proposed ieraive dynamic model is verified using oh he numerical resuls and he daa recorded in a ho ox. The es specimen is placed eween wo climaic chamers ha simulae he indoor and oudoor environmens. The numerical verificaion uses he resuls oained from one caliraion panel used as es specimens. This caliraion panel was chosen as es samples ecause heir hermal performances had een previously evaluaed y he Naional Physical Laoraory, England. The es sample is sueced o oh seady and unseady hea flow condiions. The unseady hea flow was generaed y keeping a consan emperaure in one of he climaic chamers (2 C) while he emperaure in he oher varied cyclically. Each cycle has a period of 24 h and an ampliude emperaure of 1ºC. Thus, he emperaure flucuaes eween -1ºC and 1 ºC (see Figure 4) o simulae he oudoor environmen. 1 Temperaure (ºC) Time (h) FIG 4. Temperaure variaion simulaing an oudoor environmen during a dynamic ho ox es. The es sample comprises wo glass panels, each 4 mm hick sandwiching a layer of expanded polysyrene (EPS) aou 5 mm hick. Thus, his sample has a oal hickness of 58 mm. According he caliraion repor of Naional Physical Laoraory, he hermal conducance of he es sample is defined y equaion [9]. K1 =.25 T (9) In his equaion, T is he emperaure in ºC. The equivalen hermal conduciviy of he specimen ( λ ) equiv. real is deermined y equaion [1]. λ. = e 1 equiv real ( K ) i (1) where e is he oal hickness of he specimen, in meers.

9 In-Siu Thermal Resisance Evaluaion of Walls 9 To accoun for convecion and radiaion phenomena he surfaces hermal resisance is modelled as hin air layers on he ouer surfaces. A layer of air 1 mm hick is assumed on each ouer surface (layers 1 and 3 in Figure 2) whose hermal properies are presened in Tale 1. The equipmen used o record he emperaures and hea fluxes on he es sample surfaces inside he ho ox, is a ransverse gradien hea flux sensor (TRSYS1 from Hukseflux). The emperaures and hea fluxes recorded on he exernal wall surfaces were used in he ieraive dynamic model o evaluae he equivalen hermal conduciviy ( λ ) mod el of he es specimen, assuming he exisence of a single-layer wall. The oained resuls were compared wih he acual hermal conduciviy ( λ ) equiv. real. The equivalen hermal conduciviy was evaluaed over hree periods: 24 h, 36 h and 72 h of measuremens. I was hus possile o see how imporan he duraion of he daa acquisiion is for he esimaion of he hermal conduciviy of he es specimen. The analyical compuaions were performed in he frequency domain for differen frequency ranges and differen frequency incremens, for he full analysis window The frequency incremens used were Hz, Hz and Hz, respecively. Figure 5 shows he emperaure and hea flux recorded over hree days when es samples were sueced o unseady sae condiions. The hea fluxes exhii similar cycles for he unseady sae hea flow condiions Temperaure (ºC) 1 Hea fluxes (W.m -2 ) Time (h) a) ) Time (h) FIG 5. Resuls of unseady sae es: a) emperaures; ) hea fluxes. ( ) equiv real el equiv real λ λ mod el. The hermal conduciviies, and equiv real, and he relaive error ( λ λ ) λ. mod. are given in Tale 3, for hree differen measuremen duraions: 24 h, 36 h and 72 h. The resuls show he exisence of a good agreemen eween he values deermined y he proposed mehodology and hose given in he caliraion repor.

10 1 Tadeu, N. Simões, I. Simões, Pedro and Škerge Tale 3. Equivalen hermal conduciviy oained under unseady sae es condiions. Duraion of Relaive λmodel λ equiv real Tes Sample Recorded Error Daa (%) 1 24 h h h In-Siu Thermal Resisance Evaluaion This secion illusraes he applicaion of he proposed algorihm o daa recorded using a real dynamic hea ransfer phenomenon hroughou a wall. The wall sudied faces he wes direcion. The wall is composed, from he ouside o he inside, of hese layers:.25 m concree,.4 m air caviy,.3 m exruded polysyrene (XPS), horizonally perforaed clay ricks,.11 m hick, and.2 m radiional morar. According o he hermal conduciviy/resisance and hickness of each layer, indicaed in Tale 4, he equivalen hermal conduciviy of his wall when sueced.31w.m o seady sae condiions is.ºc. Maerial Tale 4. Thermal properies of he wall sudied Thickness (m) Conduciviy λ Thermal Resisance 2 1 (m.ºc.w ) Concree Air caviy Exruded polysyrene (XPS) Horizonally perforaed clay ricks Tradiional morar TOTAL Figure 6 illusraes he emperaure and hea flux measuremens during he real dynamic hea ransfer. The lines wih marks represen he oudoor daa while he solid line illusraes he indoor measuremens. The hermal conduciviies oained wih he proposed ieraive model, considering differen periods of daa acquisiion, are presened in Tale 5. Comparing he heoreical and numerical values, he smalles error is oained for a las measuremen day, while he larges error is given y he model ha uses 58 h of measuremen. Analysis of Tale 5 indicaes ha he equivalen hermal conduciviy of he wall is ( ).298 ±.14 W.m.ºC. I can e concluded ha in general he oained values are accepale even when relaively shor periods of measuremen are considered.

11 In-Siu Thermal Resisance Evaluaion of Walls Temperaure (ºC) Hea fluxes (W/m 2 ) Time (h) Time (h) a) ) FIG 6. Resuls of in-siu measuremens over 7 days: a) emperaures; ) hea fluxes. Tale 5. Equivalen hermal conduciviy oained under real dynamic es condiions. Duraion of λmodel λequiv real Relaive Error Recorded Daa (%) Las 12 h Las 24 h h Conclusion This paper has proposed an analyical formulaion o simulae hea ransfer hrough mulilayer sysems. The proposed mehod is ased on a dynamic ieraive model for evaluaing he hermal resisance of a mulilayer sysem, ased on he Newon- Raphson mehod, when he exernal hea fluxes and emperaures are known. Firs, he mehod was presened and verified numerically using four wall sysems. Then, he model was validaed y using emperaures and hea fluxes recorded during experimenal ess in a ho ox. The hermal performance samples used in validaions were known eforehand. The comparison has revealed a good agreemen eween he resuls. Finally, he hermal resisance of an exernal wall exposed o real climaic condiions is evaluaed using he mehod, and he equivalen hermal conduciviy was calculaed successfully. References [1] Nemry, F.; Uihlein, A.; Colodel, C.M.; Wezel, C.; Braune, A.; Wisock, B.; Hasan, I.; Kreißig, J.; Gallon, N.; Niemeier, S. and Frech, Y. Opions o Reduce he Environmenal Impacs of Residenial Buildings in he European Union Poenial and Coss, in Energy and Buildings, Vol. 42/7 (21), pp [2] Desogus, G.; Mura, S. and Ricciu, R. Comparing Differen Approaches o in Siu Measuremen of Building Componens Thermal Resisance, in Energy and Buildings, Vol. 43/1 (211), pp

12 12 Tadeu, N. Simões, I. Simões, Pedro and Škerge [3] Alaici, R., and Tonelli, A.M. Infrared Thermovision Technique for he Assessmen of Thermal Transmiance Value of Opaque Building Elemens on Sie, in Energy and Buildings, Vol. 42/11 (21), pp [4] Roule, C.; Gass, J. and Marcus, I. In Siu U-Value Measuremen: Reliale Resuls in Shorer Time y Dynamic Inerpreaion of he Measured Daa, in ASHRAE Trans, Vol. 18 (1987), [5] Laureni, L.; Marcoullio, F. and De Mone, F. Deerminaion of he Thermal Resisance of Walls Through a Dynamic Analysis of in-siu Daa, in Inernaional Journal of Thermal Sciences, Vol. 43/3 (24), pp [6] Cucumo, M.; De Rosa, A.; Ferraro, V.; Kaliakasos, D. and Marinelli, V. A Mehod for he Experimenal Evaluaion in Siu of he Wall Conducance, in Energy and Buildings, Vol. 38/3 (26), pp [7] Repor on suppor o CEN TC89/WG8 Updae of ISO 9869 o EN12494 In-Siu Measuremen of he Thermal Resisance and Thermal Transmiance y PASLINK EEIG; edied (1995) y J.J. BLOEM. Join Research Cenre Insiue for Sysems Engineering and Informaics. [8] Wang, F.; Wang, D.; Wang, X. and Yao, J. A Daa Analysis Mehod for Deecing Wall Thermal Resisance Considering Wind Velociy in Siu, in Energy and Buildings, Vol. 42/1 (21), pp