Thermal conductance measurement of windows: An innovative radiative method

Experimental Thermal and Fluid Science 32 (2008) 1731–1739

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Thermal conductance measurement of windows: An innovative radiative method F. Arpino, G. Buonanno, G. Giovinco * Department of Mechanics, Structures and Environment, University of Cassino, Via G. Di Biasio 43, 03043 Cassino (FR), Italy

a r t i c l e

i n f o

Article history: Received 15 January 2008 Received in revised form 25 June 2008 Accepted 25 June 2008

Keywords: Thermal conductance Window thermal properties Radiative heat transfer Building energy saving

a b s t r a c t Heat transfer through window surfaces is one of the most important contributions to energy losses in buildings. Therefore, great efforts are made to design new window frames and glass assemblies with low thermal conductance. At the same time, it is also necessary to develop accurate measurement techniques in thermal characterisation of the above-mentioned building components. In this paper the authors show an innovative measurement method mainly based on radiative heat transfer (instead of the traditional convective one) which allows window thermal conductance measurements with corresponding uncertainty budget evaluation. The authors used the 3D finite volume software FLUENTÒ to design the experimental apparatus. The numerical results have been employed for the system optimisation and metrological characterisation. Ó 2008 Elsevier Inc. All rights reserved.

1. Introduction In accordance with the Kyoto protocol, the European Community promotes actions in the sustainable building sector with the intent to: (1) reduce carbon dioxide emissions, (2) minimise fuel energy depletion and (3) prevent indoor pollution. To this viewpoint the Energy Performance of Building Directive (EPBD) (2002/91/CE of 16/12/2002 [1]) introduces improvements for building energy performance within the Community, taking into account outdoor climatic and local conditions, as well as indoor climate requirements and cost-effectiveness. In Italy the EPBD was assimilated by the legislative degree n. 192/05 [2] and its added integration [3]. In particular, the requirements introduced by the EPBD concern: (1) the general structure of a methodology to calculate building integrated energy performance; (2) minimum requirements to energy performance of new buildings and large existing buildings subjected to major renovation; (3) energy certification of buildings; and (4) regular inspection of boilers and of air-conditioning systems and in addition an assessment of heating installations in which the boilers are more than 15 years old. As regards the minimum requirements for the energy performance of buildings, very severe limits are imposed to the thermal transmittance of building envelopes whose windows have a determinant role. Available approach to the evaluation of thermal transmittance of windows can be: (1) the simplified numerical method [4] and (2) the experimental one [5]. * Corresponding author. Tel.: +39 0776 299 3744; fax: +39 0776 299 3393. E-mail address: [email protected] (G. Giovinco). 0894-1777/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2008.06.010

The simplified model in [4] rarely gives low uncertainty values for complex geometry estimated parameters whereas the measurement method reported in [5] presents several disadvantages such as the procedure complexity (a control on convective heat transfer is necessary) and the complex routine for uncertainty evaluation. In the present paper, the authors introduce an innovative measurement methodology which leads to the determination of window thermal conductance, getting the measurement independent on the thermal convective coefficients by conjugating the use of radiative heat transfer and the vacuum technology.

2. Traditional experimental methods The hot box method [5] is one of the traditional experimental methods. It can be divided into calibrated hot box method and guarded hot box method. The first version is mainly adopted for research purposes and the second one is mainly adopted in US and in EU for more practical tests. In the hot-box method the overall heat transfer through inhomogeneous structures of large dimensions (as building components) is usually measured in steady state conditions. In particular, this technique allows both thermal transmittance and thermal conductance evaluation. The measurement principle is quite simple: a large specimen panel is instrumented on each side with temperature contact sensors arranged to obtain a representative temperature distribution across the surface of the sample. The sample is then placed between hot and cold chambers operating at controlled fixed temperatures, humidity and air flow conditions.

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Nomenclature c C d E F g h Ic k L q_ Q_ R T u U v x

specific heat, J/(kg K) thermal conductance, W/(m2 K) minimum distance between aluminium plate and window surface, m total specific energy, J/kg view factor gravity acceleration, m/s2 convective heat transfer coefficient, W/(m2 K) compatibility index thermal conductivity, W/(m K) distance, m heat flux, W/m2 heat transfer rate, W volumetric heat source due to radiation, W/m3 temperature, K standard uncertainty thermal transmittance, W/(m2 K) velocity, m/s physical quantity

Greek symbols a absorption coefficient h time, s q density, kg/m3

Temperature sensors are also placed in air, approximately at positions opposite respect to the ones placed on the specimen surface, in order to obtain the corresponding air temperatures. The surface chamber temperature is finally measured to evaluate radiant thermal exchange with the specimen. In the guarded hot-box, an internal metering box that covers a representative area of the panel is surrounding by an outer guard box (Fig. 1). Temperature and air-flow conditions in the guard box are set to obtain negligible heat transfer through the metering box walls. The heat transfer rate Q_ s which goes through the sample is given by:

Q_ s ¼ Q_ m  Q_ ml  Q_ sp  Q_ tb

ð1Þ

e /

r s

s000

thermal emissivity relative humidity Stefan-Boltzman’s costant, W/m2 K4 thermal transmission coefficient fluid stress tensor, Pa

Subscripts a air c measurement cup down lower measurement system m measured ml losses through the metering box o lens p aluminium plate ps painted surface r reflected s sample sp losses through the panel that surroundings the sample tb losses through the edges of the surrounding insulating panel up upper measurement system w window 1 environment

where: Q_ m is measured in the metering box, Q_ ml represents the heat losses through the metering box, Q_ sp are the heat losses through the panel that surroundings the sample (calculated from the known thermal conductivity of the surrounding panel material and measured temperatures on surrounding panel surfaces, by using 1D heat transfer hypothesis), while Q_ tb represents the heat losses through the edges of the surrounding insulating panel (thermal bridge). In the calibrated hot-box there is no inner metering box. The outer walls of the hot chamber are consequently made with very thick insulating material. The name ‘‘calibrated” comes out from the design feature of the box to subtract all losses other than heat flow through the test specimen during the calibration procedure which consists in

Fig. 1. Guarded hot box.

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several measurement of the homogenous surrounding panel (the same one used in sample measurements but without opening for test specimen). During calibration runs temperature differences between warm chamber and surroundings or warm and cold chambers are kept at the minimum value (but not at the same time). Hot-box methods assure an integral measurement of the heat flow across the sample. Alternatively, local heat flow measurements over the sample can be performed and, through interpolations/extrapolations, heat flow across the whole sample can be evaluated. The above mentioned method requires several temperature measurements and very accurate air temperature and velocity measurements. Moreover the experimental apparatus can be extremely expensive and the calibration procedure is very complex.

sure lower than 1000 Pa) and the use of radiative shields avoids radiative ‘‘leakages” along the boundaries. Moreover, in order to reduce thermal bridge contributions due to the supports, the measurement apparatus was bounded with a 10 cm thick polystyrene. The dimensions of the vacuum chamber (1500  1000  1000 mm3) and of the aluminium plates (1250  750 mm2) are wide enough to allow the measure of the thermal conductance of several window configurations, also on the basis of geometrical and thermal symmetries. By neglecting the cross-sectional area variation the following equations can be obtained: Thermal balance between facing plates pi and pj

q_ pi pj

     4 4  ka;p p   r  T  T p p i j   ¼  i j  T pi  T pj þ 1e  1epj pi  Lpi pj 1  þ þ  F p p  ep ep j

3. Proposed experimental methodology The proposed methodology consists in measuring window thermal conductance on the basis of one-dimensional steady state comparative method in a vacuum chamber. Energy is supplied at the top by means of electrical heaters and it is withdrawn using a water cooling system at the bottom (Fig. 2). The heat flow is measured through a thermal radiative and conductive heat transfer balance between the different aluminium plates (p1–4) and the upper and lower face of the window (w1,2). The authors point out that convection is negligible because of the low air pressure in the measurement chamber (absolute pres-

j

i

ð2Þ

j

Thermal balance between plate p and window face w

q_ pw

     r  T 4p  T 4w  ka;pw    ¼   T p  T w þ 1ep 1e 1  þ ew w þ F pw   Lpw ep

ð3Þ

The view factors F p1 p2 ; F p2 w1 ; F w2 p3 and F p1 p2 are evaluated in accordance with [6] as a function of the minimum distance d between the facing plates and the plates and the sample, necessary for the evaluation of the view factors in the case of non planar surfaces against planar ones. In particular, for two identical, parallel, directly opposed rectangles, in accordance with Fig. 3:

F 12

8 9 h i1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 ð1þX 2 Þð1þY 2 Þ 1 pffiffiffiffiffiffiffiffi X > > tan þ X 1 þ Y ln < = 1þX 2 þY 2 2 1þY 2 ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi > pXY > 2 Y >  X tan1 X  Y tan1 Y > : þY 1 þ X tan1 pffiffiffiffiffiffiffiffi ; 2 1þX

ð4Þ where

X ¼ a=c

and Y ¼ b=c

ð5Þ

The temperatures T p14 of the isothermal plates were measured through miniaturised Pt100 resistance thermometers inserted in 2 mm diameter holes at the centre of each 10 mm thick plate. The resistance temperature detectors were calibrated using a Pt25 transfer standard directly traceable to the National Institute in Metrological Research (INRIM) and a thermostatic bath at the Laboratory for Industrial Measurements (LAMI) at the University of Cassino. The calibration curves were obtained on the basis of 7 calibration points in the range 0–100 °C using the generalized least squares technique [7]. The combined standard uncertainty was

Fig. 2. Experimental apparatus.

Fig. 3. View factor scheme for two identical, parallel, and opposed rectangles.

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equal to 0.06 C in the above mentioned temperature range. The acquisition data system, interfaced by means of a RS-232 C connector to a personal computer, allows a scan interval equal to 1 minute. As regards the distances Lp2 w1 and Lw2 p3 , they are obtained through the measurement of the profile and the measurement of the minimum distance between the window surface and the corresponding plates. Hence, the uncertainty is obtained by the statistical sum of the above mentioned contributions. Moreover, all the surfaces of the window are covered by a paint of known thermal emissivity (the same paint used for the isothermal aluminium plates). Details on thermal emissivity measurement procedure are described in Appendix Therefore:

ep1 ¼ ep2 ¼ ep3 ¼ ep4 ¼ ew1 ¼ ew2 ¼ e

ð6Þ

Because of heat flow continuity, the temperatures T w1 and T w2 are obtained by equating Eqs. (2)–(3). The thermal conductivity of the air between the plates and the windows was taken at the mean temperature value. In particular its dependence upon temperature is [8,9]

ka =ðW m1 K1 Þ ¼ 1:8939E  12  ðT=KÞ4

In particular:

u2 ðC a Þ ¼

2

 u2 ðT w Þ þ



2

 u2 ðT f Þ    oC a a þ2  qwf  dwf  oT  oC  uðT w Þ  uðT f Þ oT f w  2  2 a þ okoC  u2 ðka;bc Þ þ okoCa;cad  u2 ðka;cd Þ ð12Þ a;bc  2  2  2 a  u2 ðLcd Þ þ oCoea  u2 ðeÞ þ oLoCbac  u2 ðLbc Þ þ oLoCcd  2  2 a þ oFoCbac  u2 ðF bc Þ þ oFoCcd  u2 ðF cd Þ oC a oT w



oC a oT f

where (a, b, c, d) are equal to (up, p1, p2, w1) and (down, p3, p4, w2) alternatively. Moreover, w and f are equal to (p1, p2, p3, p4) with:

dwf ¼

1=2 for w 6¼ f 0

ð13Þ

for w ¼ f

As regards the correlation factor qw-f, it is evaluated as:

qwf ¼

þ 2:6357E  09  ðT=KÞ3  1:4176E



^2 u uðT w Þ  uðT f Þ

ð14Þ

2

 06  ðT=KÞ þ 4:1761E  04  ðT=KÞ  2:7388E  02

ð7Þ

and:

 T p þ T pj ka;pi pj ¼ ka T ¼ i 2  Tp þ Tw ka;pw ¼ ka T ¼ 2

ð8Þ

The standard relative uncertainty of air thermal conductivity is equal to about 2.5% [9]. As regards the effects of pressure on air thermal conductivity, they were neglected by the authors. In particular, for distances between isothermal surfaces lower than 20 cm, for temperatures lower than 373 K and for an absolute pressure equal to 1000 Pa, a percentage variation lower than – 0.5% is observed respect to the reference value at 101325 Pa [10]. Once the two window temperatures T w1 and T w2 were evaluated, two thermal balances across the window can be applied:

q_ p1 p2 ¼ C up  ðT w1  T w2 Þ

ð9Þ

and

q_ p3 p4 ¼ C down  ðT w1  T w2 Þ

ð10Þ

The thermal conductance values measured through the upper parallel plates, Cup, and the lower ones, Cdown, may differ since thermal leakages may occur. Hence, a mean value can be taken:

C¼

C up þ C down 2

ð11Þ

evaluated by means of Eqs. (2), (3), (9), and (10), and as a function of T p14 ; e; ka;pi pj ; ka;pw ; Lpi pj ; Lpw ; dpi pj ; dpw . It must be pointed out that the present methodology does not require several temperature measurements and complex calibration procedures as the hot-box method. Since the estimation of the measured thermal conductance depends on different parameters as reported in Eqs. (2), (3), the corresponding standard uncertainty, u(C), is evaluated by considering the standard uncertainties of the input quantities, (oC/oxi)u(xi), where (oC/oxi) are the sensitivity coefficients of each parameter or physical quantity xi [11].

^ is the standard uncertainty related to the calibration chain where u (standard and thermostatic bath) and to the data-logger, and it represents the contribution common to all temperature sensors, since they where calibrated at the same time and they are connected to ^ ¼ 0:037 C. the same data-logger. In particular, u As regards the partial derivatives, they where evaluated through the classical finite difference procedure:

oC ¼ oxi

    C xi þ uðx2 i Þ  C xi  uðx2 i Þ

ð15Þ

uðxi Þ

Finally:

u2 ðCÞ ¼

u2 ðC up Þ þ u2 ðC down Þ þ u2method 4

ð16Þ

where umethod is the method uncertainty, numerically evaluated through the Finite Volumes commercial software FLUENTÒ and it is equal to about 2% of the measured conductance value. The final design of the experimental apparatus required a simulation process that was carried out by using the Finite Volumes commercial software FLUENTÒ. Thermal convection, conduction and radiation are simulated and, in particular, the last one was simulated through the Discrete Ordinate Model (DOM) [12]. Moreover, the authors used in the simulation a window schematised in Fig. 4a, with a thermal conductance equal to 5.47 W/(m2 K) evaluated by a numerical simulation of the heat transfer through the window once its temperatures of the two surfaces were imposed (only conductive problem). In particular the following equations were used: Continuity equation

oq þ r  ðqvÞ ¼ 0 Air o# oq ¼ 0 Solid o#

ð17Þ ð18Þ

Momentum equations

oqv þ r  ðqvvÞ ¼ rp þ r  ðs@ Þ þ qg o# v ¼ 0 Solid

Air

ð19Þ ð20Þ

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Fig. 5. Numerical temperature distribution in stationary conditions.

Fig. 4. (a) Scheme of the window used in the numerical simulation (b) Scheme of the calculative domain.

Energy equation

oðqEÞ þ r  ðvðqE þ pÞÞ ¼ r  ðkrTÞ þ R Air o# oðqEÞ ¼ r  ðkrTÞ Solid o#

ð21Þ ð22Þ

As regards the last term R in Eq. (21), the radiative heat transfer in a participating grey medium is given by [12] on the basis of the Discrete Ordinate Radiative Model. The boundary condition adopted, with reference to Fig. 4b, are:?

ksteel 

oT ¼ h  ðT  T 1 Þ on AB [ BC [ CD [ DA on

ð23Þ

where n is the unit vector normal to the surface and h = 5.0 W/(m2 K) and T1 = 300 K.

q_ ¼ hlocal  ðT wall  T fluid Þ þ q_ radiative at solid—fluid interfaces

The temperature uncertainty takes into account calibration, uniformity and stability contributions. The difference between the results obtained by the model, 5.58 W m2 K1, and the window value 5.47 W/(m2 K) is about 2%, while the obtained uncertainty is about 2.2%, and, consequently, a very good agreement is obtained. To investigate the time required to reach stationary conditions, unsteady numerical simulations were carried out and the results are reported in Fig. 6. The steady temperature distribution is reached in about 11 h (variations lower than 103 K for both T p1  T p2 and T p3  T p4 ). This value is comparable to the time required for the measurement of the effective thermal conductivity through the experimental apparatus (conductivimeter) [13]. The long time required to reach stationary conditions, in particular, is the weakest point of the proposed methodology. Once the prototype was built, the authors experimentally validated the proposed measurement method, the authors measured the thermal conductance of a polystyrene and a glass panel, whose thermal conductivity was previously determined through the experimental apparatus, conductivimeter, reported in [13]. The experimental uncertainty propagations are reported in Table 2.

Table 1 Numerical determination of window thermal conductance Parameter x

u(x)

Percentage contribution to u2(C)

e = 0.95 T p1 = 343 K T p2 = 334.99 K T p3 = 303.42 K T p4 = 293 K Lp1 p2 = 2.0E-02 m Lp3 p4 = 2.0E-02 m Lp2 w1 = 3.4E-02 m Lw2 p3 = 2.6E-02 m ka;p1 p2 = 2.89E-02 W/(m K) ka;p3 p4 = 2.60E-02 W/(m K) ka;p2 w1 = 2.83E-02 W/(m K) ka;w2 p3 = 2.68E-02 W/(m K) dp2 w1 = 5.0E-03 m dw2 p3 = 5.0E-03 m Correlation between T p1 and T p2 Correlation between T p3 and T p4 Method C ¼ 5:58 mW2 K

0.01 0.08 K 0.06 K 0.06 K 0.06 K 1.4E-04 m 1.4E-04 m 1.4E-04 m 1.4E-04 m 7.28E-04 W/(m 6.53E-04 W/(m 7.09E-04 W/(m 6.69E-04 W/(m 1.4E-04 m 1.4E-04 m – – –

19.9 20.5 22.7 15.3 5.55 0.346 0.524 4.7E 5.4E 2.59 3.91 0.442 0.905 8.36E-05 2.92E-03 –48.9 –27.8 83.9 uðCÞ ¼ 0:12 mW2 K

ð24Þ

The stationary temperature distribution in the chamber and across the window is reported in Fig. 5. The above mentioned results were obtained through a structured mesh adapted through the ‘‘Hanging node” technique, leading to 250,500 cells, 507,029 faces and 256,492 nodes. In particular, the gradient method on static temperature was applied until a maximum change in temperature across the numerical domain was less than 0.05 K. The numerically estimated thermal leakages are equal to about 7% respect to q_ p1 p2 . The authors also estimated the uncertainty budget on the basis of the numerical results. The obtained numerical results and the corresponding uncertainty propagations are reported in Table 1. It must be pointed out that the uncertainties of the parameters are typical values that can be obtained during experimental measurements in laboratory.

K) K) K) K)

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350

340

340

330

330

320

320

T/K

T/K

Tp1 Tp1 Tp2 Tp3 Tp4

Tp2 Tp3

310

Tp4 300

310

290

300

280 290 0

12

6

18

0

2

4

6

24

8

10

12

14

Time/h

Time/h Fig. 7. Experimental time evolution for temperature of aluminium plates during polystyrene panel measurement.

Fig. 6. Numerical time evolution for temperature of aluminium plates.

As regards the steady state, it is reached in about 13 hours (variations lower than 103 K for both T p1  T p2 and T p3  T p4 ), as reported in Fig. 7. The corresponding values obtained from the conductivimeter are: C ¼ 1:07 mW2 K and uðCÞ ¼ 0:05 mW2 K; C ¼ 63 mW2 K and uðCÞ ¼ 3:1 mW2 K for polystyrene and glass panel, respectively. Therefore, a perfect compatibility with the ones obtained from the present apparatus is shown. Infact:

jC present method  C conductivimeter j Ic ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 u ðC present method Þ þ u2 ðC conductivimeter Þ2 1:07  0:98 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:84 < 1 2 0:052 þ 0:022

ð25Þ

The authors also measured the thermal conductance of a single glazed window with an aluminium frame obtaining the results

reported in Table 2. In particular, a thermal transmittance value U ¼ 5:3 mW2 K was obtained from the C value by considering the two additional resistance (internal and external) as in [4], showing a good agreement with the typical values [4].

4. Conclusions In the present paper the authors show an innovative measurement methodology for thermal conductance of windows. This methodology allows an accurate uncertainty evaluation and it does not require a complex procedure as the hot box method. Moreover, it gives directly the value of the thermal conductance and not the thermal transmittance, avoiding convective contribution. The experimental results were compared to the ones obtained for a reference material, previously characterised by means of the experimental apparatus reported in [13], giving a very good agreement.

Table 2 Experimental determination of thermal conductance uncertainty Parameter x

Polystirene panel

Glass panel

Single glazed window with aluminium frame

u(x)

Percentage contribution to u2(C)

u(x)

Percentage contribution to u2(C)

u(x)

Percentage contribution to u2(C)

e T p1 T p2 T p3 T p4 Lp1 p2 Lp3 p4 Lp2 —w1 Lw2 p3

0.01 0.08 K 0.06 K 0.06 K 0.06 K 1.4E-04 m 1.4E-04 m 1.4E-04 m 1.4E-04 m

2.57E+01 3.32E+01 2.53E+01 1.58E+01 1.01E+01 1.11E-02 2.55E-03 6.57E-04 6.57E-04

0.01 0.08 K 0.06 K 0.06 K 0.06 K 1.4E-04 1.4E-04 1.4E-04 1.4E-04

3.19E+00 4.72E+01 8.29E+01 5.23E+01 1.03E+01 2.56E+00 5.67E+00 6.89E-01 6.90E-01

0.01 0.08 K 0.06 K 0.06 K 0.06 K 1.4E-04 1.4E-04 1.4E-04 1.4E-04

4.07E+01 1.13E+03 2.17E+03 2.06E+03 5.80E+02 1.85E+01 1.38E+02 1.38E+00 2.84E-02

ka;p1 p2 ka;p3 —p4 ka;p2 —w1 ka;w2 —p3 dp2 —w1 dw2 p3 Correlation between T p1 and T p2 Correlation between T p3 and T p4 Method Results

7.11E-04 W/(m 6.43E-04 W/(m 7.00E-04 W/(m 6.49E-04 W/(m 1.4E-04 m 1.4E-04 m – – – C ¼ 0:98 mW2 K

5.69E-02 2.07E-02 2.39E-03 2.39E-03 7.75E-07 3.69E-04 6.59E+01 3.79E+01 9.36E+01 uðCÞ ¼ 0:02 mW2 K

7.71E-04 W/(m 6.93E-04 W/(m 7.34E-04 W/(m 6.99E-04 W/(m 1.4E-04 m 1.4E-04 m – – – C ¼ 65:9 mW2 K

1.69E+01 3.46E+01 1.66E+01 2.04E+01 1.05E-03 6.65E-02 1.43E+02 7.06E+01 1.92E+01 uðCÞ ¼ 3:0 mW2 K

6.69E-04 W/(m 6.46E-04 W/(m 6.79E-04 W/(m 6.62E-04 W/(m 1.4E-04 m 1.4E-04 m – – – C ¼ 58:4 mW2 K

K) K) K) K)

m m m m K) K) K) K)

m m m m K) K) K) K)

1.34E+02 4.43E+02 1.12E+02 7.48E+01 4.07E-02 3.40E-01 3.57E+03 3.32E+03 9.62E+01 uðCÞ ¼ 1:2 mW2 K

F. Arpino et al. / Experimental Thermal and Fluid Science 32 (2008) 1731–1739

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Future improvements will be made by the authors in order to reduce measurement uncertainty. Acknowledgments This paper was developed within the ‘‘Numerical and experimental methods for the energetic characterisation of building innovative components” project financed by MIUR (N. 2005095104_004/2005). Appendix In order to measure the thermal emissivity of the paint used in thermal conductance measurement, a copper plate is covered by the paint whose thermal emissivity has to be determined (Fig. 8). Then, an aluminium measurement cover (Fig. 9) is placed over the copper plate in order to obtain a cavity with known temperatures of the surfaces. The cavity surfaces temperatures have to be measured in order to control the radiative heat exchange involving the surface covered by the paint In order to take into account eventual temperature gradients across the cover, three calibrated type K thermocouples are placed on the inner part of the cover (Fig. 9). A calibrated type K thermocouple is also used to measure the copper plate temperature. When stationary conditions are reached all the temperatures are measured and the thermal imagine of the plate is taken by an IR camera (Fig. 10). In order to use the IR camera, the data reported in Table 3 are needed. It must be pointed out that the value of the painted surface set on the IR camera is not the real one which, on the contrary, is measured in accordance with the procedure shown hereafter. The emissivity value, in fact, is obtained by equating the heat flow obtained through the set emissivity value eps,IR and the surface temperature measured by the IR camera Tps,IR with the actual one [8]:

eps ¼ eps;IR 

T 4ps;IR  T 4r

Fig. 9. Measurement cover with type K thermocouple at the inner side.

ð26Þ

T 4ps  T 4r

as regards the value of Tr, it is evaluated from the temperature distribution along the inner surface of the measurement cover. The inner temperature Tc is evaluated as the mean value of the three temperature measured by the three thermocouples T1,2,3:

Fig. 10. IR camera positioning.

Table 3 Data required by the IR camera D (m) Tr (°C) Ta (°C) /a (%) To (°C)

so eps

Tc ¼

Fig. 8. Painted surface.

T1 þ T2 þ T3 3

Distance between camera lens and the painted surface Reflected temperature (i.e. temperature of the cavity surrounding the painted surface) Atmospheric temperature (i.e. temperature of the air between the camera lens and the painted surface) Relative humidity of the air between the camera lens and the painted surface Temperature of the camera lens Thermal transmission coefficient of the camera lens Painted surface emissivity

ð27Þ

The value of the reflected temperature would be equal to the one of the measurement cover if the cover surface would be much more greater than the one of the painted surface. Therefore, the value of Tr can be evaluated by means of Tc by equating the heat flow between the painted surface and the measured cover with the one between the painted surface and the cavity at a temperature Tr [8]:

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r  eps  Aps  ðT 4ps  T 4r Þ ¼

r  ðT 4ps  T 4c Þ 1eps eps Aps

1

þ F psc Aps þ

ð28Þ

1ec ec Ac

In particular: 2

u ðeps Þ ¼

where Fs-c = 1 is the view factor between the painted surface and the measurement cover [6]. From Eq. (28):

 2  2 oe oe  u2 ðDÞ þ oTpsa  u2 ðT a Þ þ o/psa  u2 ð/a Þ  2  2  2 oe oe oe þ oTpsr  u2 ðT r Þ þ oTpso  u2 ðT o Þ þ ospso  u2 ðso Þ  2 oe þ oTpsps  u2 ðT ps Þ

eps 2 oD

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 13ffi u u 4 4 Tc 1 u 1 4 A5 Tr ¼ t þ T 4ps  @eps  Aps  1eps 1 c eps  Aps 1eps þ 1 þ 1ec þ þ 1e F psc Aps F psc Aps ep Aps ec Ac eps Aps ec Ac

ð30Þ

ð29Þ

s

Since Tr depends on eps, it must be iteratively determined through the procedure reported in Fig. 11. As the thermal emissivity depends on the parameters reported in Table 3, its uncertainty u(eps) is evaluated through the contribution (oeps/oxi)u(xi), where (oeps/oxi) are the sensitivity coefficients of each parameter xi [11].

As regards D, Ta and ua, they are required by the camera in order to consider the radiative adsorption of the air between the camera lens and the painted surface. However, for air temperatures values (lower than 60 C) and for the relative humidity values (lower than 60%) measured and for distances lower than 1 m, these contributions are negligible. Moreover, when he thermal equilibrium is reached:

Tr ¼ To

START

ð31Þ

Finally, a value of so = 1 is assumed. Under these hypothesis:

u2 ðeps Þ ¼

 2  2 oeps oeps  u2 ðT r Þ þ  u2 ðT ps Þ oT r oT ps

ð32Þ

The sensitivity coefficients are evaluated through finite differences as:

i=0

    uðxi Þ uðxi Þ oeps eps xi þ 2  eps xi  2 ¼ oxi uðxi Þ

ð33Þ

Remembering Eq. (29), u({T_r}) is given by:

Tri = Tc

u2 ðT r Þ ¼

ε ps from eq. (17)



2  2  2 oT r oT r oT r u2 ðeps Þ þ  u2 ðec Þ þ  u2 ðAps Þ oeps oec oAps  2  2 oT r oT r  u2 ðT ps Þ þ  u2 ðT c Þ þ oT ps oT c

ð34Þ

with:

    uðxi Þ uðxi Þ oT r T r xi þ 2  T r xi  2 ¼ oxi uðxi Þ

ð35Þ

and through Eqs. (26), (32):

Tr from eq. (20) u2 ðeps Þ ¼

Tri − Tri −1 ≤ 10−3 False

 2  2  2 oeps oeps oeps  u2 ðec Þ þ  u2 ðAps Þ þ oec oAps oT ps  2  2 oeps oeps  u2 ðT ps Þ þ  u2 ðT c Þ þ  u2 ðAc Þ oT c oAc  2 oeps þ  u2 ðT ps;IR Þ oT ps;IR

ð36Þ

As regards u(Tc), remembering Eq. (27):

True u2 ðT c Þ ¼

END

Fig. 11. Iterative procedure for Tr evaluation.



maxðT 1 ; T 2 ; T 3 Þ  minðT 1 ; T 2 ; T 3 Þ pffiffiffi 2 3

þ ðmaxðuðT 1 Þ; uðT 2 Þ; uðT 1 ÞÞÞ2

2 ð37Þ

given by calibration and uniformity contribution, being stability uncertainty negligible.

F. Arpino et al. / Experimental Thermal and Fluid Science 32 (2008) 1731–1739

References [1] Directive 2002/91/CE of the European Parliament and of the Council, 16/12/ 2002. [2] Decreto Legislativo 19 agosto 2005, n. 192, Attuazione della direttiva 2002/91/ CE relativa al rendimento energetico nell’edilizia. [3] Decreto Legislativo 29 dicembre 2006, n. 311, Disposizioni correttive ed integrative al decreto legislativo 19 agosto 2005, n. 192, recante attuazione della direttiva 2002/91/CE, relativa al rendimento energetico in edilizia. [4] UNI EN ISO 10077-1:2002, Thermal performance of windows, doors and shutters – Calculation of thermal transmittance – Simplified method. [5] UNI EN ISO 12567-1:2002, Thermal performance of windows and doors – determination of thermal transmittance by hot box method – Complete windows and doors. [6] J.R. Howell, A catalog of Radiation Heat Transfer Configuration Factors, second ed., .

1739

[7] A.B. Forbes, Generalised regression problems in metrology, Numerical Algorithms 5 (1993) 523–533. [8] F. Kreith, M.S. Bohn, Principles of Heat Transfer, Thomson-Engineering Ed., sixth ed., 2000. [9] E.W. Lemmon, R.T. Jacobsen, Viscosity and thermal conductivity equations for nitrogen oxygen, argon, and air, Int. J. Thermophys. 25 (1) (2004) 21–69. [10] J.A. Potkay, G.R. Lambertus, R.D. Sacks, K.D. Wise, A low-power temperatureand pressure-programmable uGC column, in: Solid-State Sensors, Actuators and Microsystems Workshop (Hilton Head), Hilton Head Island, SC, USA, June 2006, pp. 144–147. [11] ISO GUIDE, Guide to the Expression of Uncertainty in Measurement, Geneve, 1993. [12] Fluent, 2003, Fluent 6.1User’s Guide, Fluent Inc. [13] G. Buonanno, A. Carotenuto, G. Giovinco, N. Massarotti, Experimental and theoretical modeling of the effective thermal conductivity of rough steel spheroid packed beds, J. Heat Transfer 125 (4) (2003) 693–702.