Building and Environment 39 (2004) 1049–1053www.elsevier.com/locate/buildenv

Natural convection at an indoor glazing surface

Cristian Cuevas, Adelqui Fissore∗

Departamento de Ingenier��a Mec�anica, Universidad de Concepci�on, Casilla 53-C, correo 3 Concepci�on, Concepci�on, Chile

Received 10 January 2000; received in revised form 14 September 2000; accepted 19 January 2004

Abstract

An experimental study was made to determine correlations that allow the calculation of heat transferred by convection through thewindow. Three con1gurations were studied: a hot plate; a cold plate and a window with a single-step frame placed on the wall of a room.We obtained a correlation that can be used to calculate the convection heat transfer through the window. The new correlation in the hotplate con1guration di5ers by 14.5% from the ASHRAE correlation for laminar free convection on a vertical surface, is 27.5% from thecold plate and is 12% from the single step-frame.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Building simulation; Window heat transfer coe:cient; Experimental

1. Introduction

The heat transfer through fenestration plays an importantrole in energy balance in buildings and this is mainly so inwinter when there is a high temperature di5erence betweenthe inner air of the room and the outside air.In the literature, there are few reports that study the heat

transfer by convection through the window. Mainly, thereare reports for calculating the heat transfer coe:cients forwalls [1,2] and for some very speci1c conditions, as radia-tors and heated walls [3]. Also, there is a scale model [4],which presents problems due to recirculations producing analteration in the air speeds. Another alternative to calculatethe heat loss through the window is to use the correlationsfor vertical ?at plate for which well-known solutions exist[5,6]. Some researchers have made numerical simulationsfor a window [7–9] and a wall of a room [10,11]. Correla-tions found in the literature signi1cantly di5er among them.In this report, three correlations are developed for calcu-

lating the heat transfer coe:cients in a vertical plate placedon a wall of a room. The results are presented through cor-relations based on the Nusselt and Grashof numbers.

∗ Corresponding author. Fax: +56-41-25-11-42.E-mail address: a1ssore@udec.cl (A. Fissore).

2. Experimental apparatus

Experiments were made on a wall of a climate chamber,with dimensions 4:5 m × 3:2 m × 2:20 m. (length × width× height). The window was simulated by a vertical ?atplate made of aluminum with thickness 6 mm, height 1 mand width 0:85 m. The plate was polished to give a lowemissivity. A cross sectional view of the experimental modeland climate chamber is shown in Fig. 1.The temperature was measured with 24-gauge copper-

constantan thermocouples and with a data acquisition sys-tem. On the test plate 12 thermocouples were placed. Oneach wall of the test room one thermocouple was placedand another thermocouple was placed in the center of thetest room for measuring the air temperature. The back sur-face of the test plate was insulated with polystyrene oflow density (14:5 kg= m3). The climate chamber used inthe experiments was placed inside another climate cham-ber, which eliminates the e5ect of solar radiation and otherdisturbances.For the experiments, the tested plate was heated or cooled

and was then exposed to the air and walls of the test room.For heating the plate another plate (pre-heated to 80◦C) ofthe same dimension as the tested plate was used. Both platesremain together for some minutes (see Fig. 2). After thatthe hot plate is removed from the climate chamber and the

0360-1323/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.buildenv.2004.01.028

1050 C. Cuevas, A. Fissore / Building and Environment 39 (2004) 1049–1053

Nomenclature

A area, m2

Cp speci1c heat, J=kg KF12 view factorGr Grashof numberg gravitational acceleration, m=s2

k thermal conductivity, W/mKL characteristic length, mNu Nusselt numberQ heat ?ux, WT temperature, Kt time, s.

Greek symbols

J error, di5erence� volumetric thermal expansion coe:cient, 1/K� emissivity

� kinematic viscosity, m2=s� density, kg=m3

� Stefan–Boltzmann constant, W=m2K4

Subscripts

cond conductionconv convectioni initialf 1nalmrt mean radiant temperaturep platerad radiationrandom random errors surfacesystematic systematic error∞ room air

tested plate exchanges heat with the chamber in transientregime (see Fig. 1).For cooling the tested plate we use a similar procedure,

but in this case, we used congealed gel behind the movableplate.

3. Calculation procedure

Heat transfer coe:cients were determined from the ther-mal balance on the plate in transient state. It is given by

Qconv + Qrad + Qcond = mcpdTdt

: (1)

The heat transfer by radiation and convection are evaluatedby

Qrad =�(T 4p − T 4mrt)

1− �pAp�p

+1

ApF12+1− �sAs�s

; (2)

Qconv = hAp(Tp − T∞): (3)

Τ

Tp

0.4 m.

4.5 m.

2.2

m.

A

B

Qr

Qc

(a) (b)

∞

Fig. 1. (a) Climate chamber; (b) experimental model.

The conduction heat transfer was calculated with theone-dimensional 1nite di5erence method. This heat wasconsidered only for the uncertainty analysis.If we take a time interval Jt and integrate Eq. (1),

we obtain, for the heat transfer coe:cient, the followingexpression:

Lh=mcp

Ap( LT∞ − LT p)( LT fp − LT ip)

Jt

− �( LT 4p − LT 4mrt)(1− �pAp�p

+1

ApF12+1− �sAs�s

)Ap( LT∞ − LT p)

: (4)

Fig. 2. Heating test plate.

C. Cuevas, A. Fissore / Building and Environment 39 (2004) 1049–1053 1051

After that, we can determinate the Nusselt and Grashofnumbers:

Nu=hLk

; Gr =g�(Tp − T∞)L3

�2; (5)

where air properties are calculated to 1lm temperature.Finally, correlations are given by

Nu= 0:68 + a Grn: (6)

The constant 0.68 was taken from the results obtained byChurchill [5] for conduction when Gr → 0.

4. Con�gurations tested

The geometry of the two con1gurations tested are shownin Fig. 3. First a simple plate con1ned in a room was tested,Fig. 3 (a). In this case, the plate was heated or cooled fordetermining the heat transfer coe:cient. The second casetested was a single-step frame con1guration. In this case,the in?uence of the single-step frame on the heat transfer inthe glazing surface was studied.

5. Uncertainty analysis

The analysis recommended by ASHRAE Guideline [12]was used for determining the uncertainty of each experiment.So the uncertainty for Nusselt number is given by

JNu=

√√√√ m∑i=1

(@Nu@Vi

JVi

)2; (7)

where the Vi are them variables on which the Nusselt numberdepends. For this analysis, we must estimate the uncertaintyof each one of these variables.The data acquisition system and calibrated thermocouples

have a total uncertainty of ±0:15◦C. The emissivity of theplate was measured with an uncertainty of ±0:03. We usethe following procedure to measure emissivity: the surfaceis heated until 70◦C, and then we measure the temperatureof the surface using an infra-red thermometer (selling �=1)and a contact thermometer. Other surrounding temperaturesare also measured by a contact thermometer. We calculatethe emissivity of the surface, solving the balance equationof the surface and the sensor.The calibration of the thermocouple was performed com-

paring the signal of the thermocouple with a precision glassthermometer.For the air temperature, the error of the radiation with the

shielded foil was added. This error varies between 0:1◦Cand 0:2◦C, depending on the speci1c test.

100

10

10

10

0

(a) (b)

Fig. 3. Con1gurations tested (units are in centimeters).

For the correlation, the con1dence interval is estimatedby the Eq. (8)

JNucorrelation=

√(tJNurandom1:96

√n

)2+(

tJNusystematic1:96

)2;

(8)

where t is a parameter of student’s t-distribution and n thenumber of experimental data points.Here we separated the random and systematic errors. A

systematic error is, for example, the error in the Cp of theplate. In fact, we use only one standard value for the Cp.The actual value of the Cp of the plate is a constant valueand di5ers by a constant factor from the used value. Thedi5erence between the used value and the actual one is im-possible to know, but it is possible to estimate. Then, this isa systematic error present in all the results. The di5erencebetween a systematic error and a random one is that the ran-dom error goes down with the number of data points and thesystematic error does not. Then the two must be consideredin di5erent ways in Eq. (8). For the present test, the system-atic error was estimated in about 18% of the total error.The uncertainty is determined with 95% of probability.

6. Results and discussion

The correlations obtained with their intervals of con1-dence and the uncertainty of each point are given in the nextsection.

6.1. Hot plate con5guration

In this case, the plate exchanged heat with the room byconvection and radiation. Table 1 shows the correlation ob-tained. For this case 27 values of the Nusselt number wereobtained, and they are shown in the Fig. 4 with their uncer-tainties and con1dence interval. For the low Grashof num-bers big uncertainties were obtained, but the dispersion doesnot increase. It indicates that the error is overpredicted inthis zone.

1052 C. Cuevas, A. Fissore / Building and Environment 39 (2004) 1049–1053

Table 1Correlation and con1dence interval for hot plate con1guration

Correlation Nu = 0:68 + 0:578Gr0:25

Con1dence interval ±4:539 (17:8%)

2x108 5x108 109 2x109 3x10920

50

100

200

300

Gr

Nu

Nu = 0.68 + 0.578 Gr0.25

Fig. 4. Nusselt for hot plate con1guration.

2x108 5x108 10920

50

100

200

300

Gr

Nu

Nu = 0.68 + 0.644 Gr0.25

Fig. 5. Nusselt for cold plate con1guration.

Table 2Correlation and con1dence interval for cold plate con1guration

Correlation Nu = 0:68 + 0:644Gr0:25

Con1dence interval ±6:832 (20:6%)

6.2. Cold plate con5guration

In this case, the heat was transferred from the room to theplate. Twelve values for the Nusselt numbers were obtained.Fig. 5 shows the values and the correlation obtained withtheir con1dence interval. In Table 2 we show the correlationobtained and con1dence interval, with a 95% probability.

2x108 5x108 109 2x10920

50

100

200

300

Gr

Nu

Nu = 0.68 + 0.565*Gr0.25

Cold plate

Hot plate

Fig. 6. Nusselt for single-step con1guration.

Table 3Correlation and con1dence interval for single-step con1guration

Correlation Nu = 0:68 + 0:565Gr0:25

Con1dence interval ±5:945 (17:8%)

6.3. Single-step frame con5guration

This case is similar to a real windowwithout a curtain, andwas developed with hot plate and cold plate for analyzing thee5ect of the two cases. We obtained 12 values with singlestep frame con1guration, which are shown in the Fig. 6.Table 3 shows the correlation obtained with their con1denceinterval.In this case, the experimental data obtained with hot and

cold plate con1guration are correlated for the same equa-tion. The statistical analysis shows that both cases are sim-ilar. This is because the frame attenuates the e5ect of roomrecirculations that are present when there are no frames.

7. Conclusions

Correlations for the calculation of the convective heattransfer coe:cient in a glazing surface have been devel-oped. These models are valid for natural convection and forGrashof numbers between 3× 108 and 2× 109. Results canbe used for estimating the convective heat loss in a win-dow. The radiation must be added for estimating the totalheat loss. The con1dence interval for the hot plate con1gu-ration and single-step frame con1guration is 17.8% and forcold plate con1guration is 20.6%. In the results it is possi-ble to observe that the uncertainty is overpredicted and thecon1dence interval can be lower.

Acknowledgements

The authors acknowledge the contribution of CONICYTChile and DirecciPon de InvestigaciPon of the Universidad ofConcepciPon.

C. Cuevas, A. Fissore / Building and Environment 39 (2004) 1049–1053 1053

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