International Journal of Heat and Mass Transfer 55 (2012) 1226–1235

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International Journal of Heat and Mass Transfer

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Heat transfer through a double-glazed unit with an internal louvered blind:Determination of the thermal transmittance using a biquadratic equation

M. Karmele Urbikain ⇑, José M. SalaDepartment of Thermal Engineering, School of Engineering of Bilbao, The University of the Basque Country, Alameda Urquijo s/n, 48013 Bilbao, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 January 2011Accepted 1 September 2011Available online 8 November 2011

Keywords:Thermal transmittanceU-factorHeat-transfer coefficientBlindDouble-glazed unitWindow

0017-9310/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2011.09.032

⇑ Corresponding author.E-mail address: mirenkarmele.urbicain@ehu.es (M

The aim of this research is to quantify the influence of a louvered blind in a double-glazed unit duringnight-time conditions. First, an analytical study of free convection was conducted to obtain a set of cor-relations for the Nusselt numbers of the cavity. Second, a parametric study was performed to calculatethe total heat transfer (convective and long-wave radiative) during night-time conditions. The analysisaccounted for aspect ratio, blind thermal conductivity, surface emissivity and slat angle. Using these data,a biquadratic equation was developed to calculate the U-factor of a double-glazed unit with an internallouvered blind in terms of the U-factor of the unit without the blind, slat surface emissivity and slat angle.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Windows can contribute to heat loss during winter and solarheat gain during summer, but they also have the potential to re-duce energy consumption in buildings. Different studies havequantified the heat loss and gain that is related to windows inbuildings and classified these according to their thermal and solarparameters [1–3]. The use of shading devices is an effective way tocontrol solar heat gain. A venetian blind that is located adjacent tothe indoor surface of a window or inside the double glazing pro-vides a flexible element but complicates the calculation of the ther-mal transmittance (or U-value) and the solar heat-gain coefficient(or g-value). This study focuses on the calculation of the thermaltransmittance of a double-glazed window with an internal lou-vered blind.

Different approaches to the problem have been attempted. Phillipset al. [4] carried out a free-convective study for a louvered blind lo-cated adjacent to an isothermal surface and considered the con-duction along the slats and the radiative exchange. This studywas performed in night-time conditions, which means that theeffects of incident solar radiation were not included. Finite elementmethods were used to solve the problem, and the results werecompared with data obtained using a Mach–Zehnder interferome-ter. For the purely convective solution, the temperatures of theblind agreed closely with the experimental results reported byMachin et al. [5]. This experimental study was completed using

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. Karmele Urbikain).

an isothermal vertical surface and did not accounted for the framegeometry or the thermal boundary conditions of the glazing. Evenfor cases when the blind is close to the plate, the general trend ofthe convection coefficient was similar to that of an isolated verticalflat plate and decayed rapidly from the leading edge. However, aperiodic variation in the convection coefficient occurred with aspatial frequency that was equal to the slat pitch.

Zhang et al. [6] conducted a study of natural convection in adouble-glazed unit with a between-panes blind. They focused onthe closed position and modelled the blind as a vertical enclosurewith a permeable screen to the air flow in the horizontal directionduring night-time conditions. When the flow conductance C (de-fined as the ratio of the air permeability through the blind to theproduct of its height and the partition thickness) is small enough,the overall Nusselt number drops to half of the value obtainedwhen no screen is assumed. They observed a threshold value Ccfor which the impermeability of the screen begins to have an effecton the overall heat-transfer rate. The flow conductance thresholdshifted toward lower C values as the Rayleigh number increased.Fang and Ge [7] performed an experimental study with a blind lo-cated adjacent to the indoor surface of a single and a doubleglazing.

Collins et al. [8–10] analysed the problem of a set of heated lou-vers located adjacent to a vertical and isothermal wall during day-time conditions. They examined the influence of the glazingtemperature, irradiation levels, slat angle and blind-to-plate spac-ing on the convective heat transfer. The results were presented asconvective and radiative heat fluxes. Avedissian and Naylor [11]established a method to quantify the convective heat transfer rate

Nomenclature

A enclosure aspect ratio, ratio of cavity height to widthd glazing thicknessFij view factor, proportion of the radiation emitted by the

area element dAi and the radiation intercepted by thearea element dAj

G buoyancy termg gravityh heat-transfer coefficienth0 local heat-transfer coefficientH cavity heightJ radiosityk thermal conductivityp pressureu,v fluid velocityx,y Cartesian coordinatesT temperatureU U-factor, heat flux through a square meter of window

for a 1 �C temperature difference between the indoorand the outdoor air temperature.

Dimensionless variablesx⁄, y⁄ dimensionless cartesian coordinatesu⁄, v⁄ x, y dimensionless velocitiesDT⁄ dimensionless temperatureGr Grashof number, ratio of buoyancy to viscous forcesRa Rayleigh number, product of Grashof and Prandtl num-

bers

Pr Prandtl number, ratio of the momentum and thermaldiffusivities

Nu Nusselt number, a dimensionless ratio of convective toconductive heat transfer across the boundary

Greek symbolsa fluid thermal diffusivityb volumetric expansion coefficientd cavity widthdb slat widthd⁄ characteristic lengthe surface emissivityl dynamic viscositym kinematic viscosityq fluid densityh slat angler Stefan–Boltzmann constantSubscriptsb blindc convectivee externali internals surfaceT thermalw wall

M. Karmele Urbikain, J.M. Sala / International Journal of Heat and Mass Transfer 55 (2012) 1226–1235 1227

in double-glazed windows with an internal louvered blind basedon the results they obtained numerically. Experimentally, Naylorand Lai used a Mach–Zehnder interferometer [12] to obtain tem-perature-field visualisations and mean convective heat fluxes.

In the current work, a parametric study was performed to studythe convective heat-transfer rate in a vertical enclosure with aninternal louvered blind. Two broad cases were considered. First, aconvective heat-transfer problem was solved and included the con-duction along the blind slats. The Nusselt number depended on theRayleigh number, aspect ratio and slat angle. Therefore, correlationsin the form Nu = CRam � An were proposed for a double-glazed win-dow with a between-panes blind. Two blind materials were consid-ered for this study: aluminium and polyvinyl chloride (PVC).

In the second case, the overall heat fluxes were obtained, andthey accounted for the coupled effects of convection and long-wave radiation. The efficiency of the blind was defined as the ratioof the U-factor of the double-glazed unit with a between-panesvenetian blind to the U-factor of the same unit without the blind.Biquadratic equations were proposed to provide the efficiency ofthe blind in terms of the blind material, surface emissivity andblind angle. The blind efficiency revealed whether the blind con-tributed to heat loss or gain.

2. Physical model and computational method

For these problems, two conditions can be used: night-time(without solar irradiation) and daytime (with solar irradiation).The model in the present study consisted of double-glazed panelswith between–panes slats equally spaced. Adiabatic end wallswere imposed. The influence of blind thermal conductivity and slatangle on the heat transfer through a double-glazed window withinternal venetian blinds was analysed in night-time conditions,which excludes solar radiation. Two slat materials, aluminiumand PVC, were considered. The aspect ratio, defined as the ratioof cavity height to width, was varied between 42 and 74, and fiveslat angles h (open, 30�, 45�, 60�, closed) were selected.

Two different analyses were performed during night-time con-ditions. First, the convective problem was solved using an externalconvective heat-transfer coefficient of 20 W/m2 K and an internalconvective heat-transfer coefficient of 3.6 W/m2 K. From this study,the equivalent convective heat-transfer coefficients for the cavitywere calculated. Correlations for the Nusselt number were deter-mined based on these convective heat-transfer coefficients.

Second, the total heat-transfer rate, convective and long-waveradiative were evaluated to calculate the U-factor of the system.The following boundary conditions were considered: (i) an outdoortemperature of 0 �C and a global external coefficient of 23 W/m2 Kand (ii) an indoor temperature of 20 �C and a global internal coef-ficient of 8 W/m2 K. Furthermore, in the second study, the influ-ence of the slat surface emissivity was analysed. Five blindsurface emissivities e (0.1, 0.2, 0.55, 0.7 and 0.9) were considered.

Ti = 20 �C and hi = 8 W/m2 K for indoor conditions and Te = 0 �Cand he = 23 W/m2 K for outdoor conditions were considered be-cause they are the common boundary conditions in Europe forsolving heat transfer problems through windows. The EN 673[13] provides the steps for calculating the approximate glazingU-factor using 23 W/m2 K and 8 W/m2 K as the total external andinternal heat-transfer coefficients, respectively. The EN ISO 15099[14] records 20 W/m2 K and 3.6 W/m2 K as the convective heat-transfer coefficients. In the EN ISO 10077 [15,16], the boundaryconditions are an external and an internal thermal resistances of0.04 and 0.13 m2 K/W, respectively. To measure the U-factor of awindow in a hot-guarded box following the EN ISO 12567 [17],we search for a global thermal resistance of 0.17 m2 K/W.

The results obtained for a double-glazed unit with an internalblind are interesting to compare with a unit without internal lou-vers. We used these sets of conditions as the boundary conditionsbecause in Europe U-values commonly refer to these conditions.

In the overall heat-transfer problem, gray diffuse radiation be-tween the window panes and the blind was assumed. The air inthe gap between panes was assumed to be a non-participant med-ium. The radiant heat-transfer between panes was not negligible

(a) Convective heat transfer

(b) Convective and long-wave radiative heat transfer

y

x

Te

hce

Ti

hci

θ

δ

δb

y

x

Te

he

Ti

hi

θ

δ

δb

Fig. 1. Model geometry with boundary conditions.

Table 1Boundary conditions.

Convective heattransfer

Convective and long wave radiativeheat transfer

External Internal External Internal

T (�C) 0 20 0 20h (W/m2 K) 20 3.6 23 8

1228 M. Karmele Urbikain, J.M. Sala / International Journal of Heat and Mass Transfer 55 (2012) 1226–1235

because glazing has a low emissivity surface. The radiative heatexchange was solved coupled to the conductive-convective heattransfer. Radiative heat transfer depends on the surface size, sur-face to surface distance and orientation. The geometrical featuresof radiation exchange were considered by means of view factors.Absorption, emission and scattering in the medium were neglectedin the solution. Fig. 1 shows the two models adopted in the study,and Table 1 shows the boundary conditions.

The mathematical approach for solving the fluid flow and heat-transfer problem is based on the continuity, x-momentum, y-momentum and energy equations. Boundary conditions must beadded to solve the problem. The fluid is assumed to be incompress-ible. The governing equations are presented below assuming twodimensional, steady, laminar flow

@u@xþ @v@y¼ 0 ð1Þ

q u@u@xþ v @u

@y

� �¼ � @p

@xþ l @2u

@x2 þ@2u@y2

!ð2Þ

q u@v@xþ v @v

@y

� �¼ � @p

@yþ l @2v

@x2 þ@2v@y2

!þ Gy ð3Þ

u@T@xþ v @T

@y¼ a

@2T@x2 þ

@2T@y2

!ð4Þ

Conductive heat transfer through glazing panels and throughthe blind is given by Laplace’s equation.

@2T@x2 þ

@2T@y2 ¼ 0 ð5Þ

The boundary conditions are the following:

x ¼ 0 � k@T@x

����x¼0¼ heðTe � Tjx¼0Þ ð6Þ

x ¼ dþ 2d � k@T@x

����x¼dþ2d

¼ hiðTjx¼dþ2d � TiÞ ð7Þ

y ¼ 0@T@y¼ 0 ð8Þ

y ¼ H@T@y¼ 0 ð9Þ

The boundary condition at the glazing surface and the blind sur-face is given by the convective condition, which equals Fourier lawto Newton law.

�k � @T@n

����s

¼ h0 � ðT � TjsÞ ð10Þ

If the thickness dT (the distance over which the temperaturechanges from Tw to T0) << H (the height of the cavity), a thermalboundary layer is produced. In the second Navier Stokes equationthe buoyancy term is given by

Gy ¼ ðq0 � qÞg ð11Þ

then

q u@v@xþ v @v

@y

� �¼ � @p

@yþ l @2v

@x2 þ@2v@y2

!þ ðq0 � qÞg ð12Þ

where q0 is the reference density. If the equation of state q ¼ qðT;pÞis expanded as a Taylor series, the second-order terms are neglectedand the pressure correction term is neglected, the following isachieved:

u@v@xþ v @v

@y

� �¼ � 1

q0

@p@yþ m0

@2v@x2 þ

@2v@y2

!þ bgðT � T0Þ ð13Þ

where T0 is the reference temperature, and b is the volumetricexpansion coefficient,

b ¼ � 1q0

@q@T

� �P

Therefore, the Boussinesq approximation is used.Using the similarity theory, these equations can be non-dimen-

sionalised in the following manner:

@u�

@x�þ @v

�

@y�¼ 0 ð14Þ

Gr1=2 u�@u�

@x�þ v� @u�

@y�

� �¼ � @p�

@x�þ @2u�

@x�2þ @

2u�

@y�2

!ð15Þ

Gr1=2 u�@v�@x�þ v� @v

�

@y�

� �¼ � @p�

@y�þ @2v�

@x�2þ @

2v�@y�2

!

þ Gr1=2DT� ð16Þ

Gr1=2Pr u�@T�

@x�þ v� @T�

@y�

� �¼ @

2T�

@x�2þ @

2T�

@y�2ð17Þ

Conduction through glazing panels and through the blind in adimensionless form is given by the following:

@2T�

@x�2þ @

2T�

@y�2¼ 0 ð18Þ

where dimensionless variables are defined as the following:

x� ¼ xd; y� ¼ y

d; u� ¼ ud

aPrGr1=2 v� ¼ md

aPrGr1=2 p� ¼ pd2

laPrGr1=2

M. Karmele Urbikain, J.M. Sala / International Journal of Heat and Mass Transfer 55 (2012) 1226–1235 1229

DT� ¼ T � T0

DTGrd ¼

gbðT � T0Þd3

m2 Pr ¼ l � cp

kRad ¼ GrdPr

The dimensionless boundary conditions are given by the following:

x� ¼ 0@T�

@x�jx�¼0 ¼ Nux�¼0 ð19Þ

x� ¼ 1þ 2dd

@T�

@x�jx�¼1þ2d=d ¼ Nux�¼1þ2d=d ð20Þ

y� ¼ 0@T�

@y�jy�¼0 ¼ 0 ð21Þ

y� ¼ Hd

@T�

@y�jy�¼H=d ¼ 0 ð22Þ

Computational fluid dynamics (CFD) software was used to sim-ulate the system, and the pressure-velocity coupling was treatedusing the SIMPLE algorithm. A second-order upwind scheme wasused to account for the flow direction when the value of a variableat a cell side was determined.

Long-wave radiative heat transfer was calculated using theradiosity method. The radiosity of one surface can be expressedin terms of the radiosity of the other surfaces. A system of equa-tions to be solved is given below

Ji ¼ eirT4i þ ð1� eiÞ

XN

j¼1

FijJj ð23Þ

where Fij is the view factor,

Fij ¼1Ai

ZAi

ZAj

cos /i cos /j

pr2 dAidAj ð24Þ

Elemental surfaces dAi and dAj are connected by a line of lengthr, which forms the angles /i and /j, respectively, with the surfacenormals. Ti in Eq. (23) is calculated after accounting for convection.At the beginning of the problem, an arbitrary temperature distribu-tion was assumed. Eqs. )(14)–(17) were solved, and after an itera-tive process, a temperature field was produced. This was the entryfor Eq. (23). The unknowns in the equations system were the radi-osities. With the radiosities determined, the temperatures of everysurface can be calculated, and the process can be repeated again.

3. Mesh dependence

In a CFD study, the effect of the grid size on the numerical solu-tion must be analysed. The continuity, momentum and energyresiduals for the convective problem converged with an order ofmagnitude of 10�11, 10�12 and 10�17, respectively. For the globalproblem (convective–conductive plus radiative heat transfer), theresiduals were the same order of magnitude at the end of the pro-cess as for the convective problem.

When the blind was open, a block-structured grid was used. Thesystem was sub-divided into regions, each of which has a struc-tured grid. A finer mesh was used in regions where greater accu-racy is required, e.g., near the glazing panels or near the blindsurface. In the other positions, an unstructured grid was used.The mesh generation was computationally more efficient andwas concentrated where necessary.

The first simulation was performed using a 1-mm mesh to endup with a 0.2-mm mesh. The variation of the Nusselt number wasless than 1% when decreasing the dimensionless grid parameter x⁄

from 0.025 to 0.0125. Meshes that were smaller than half a milli-metre provided increased accuracy to the numerical solution andwere satisfactory from a computational point of view. Table 2 sum-marises the grid test that was carried out for an aspect ratio of 49.

The dimensionless parameter x⁄ and the Nusselt number referredto the cavity width.

4. Results

The results from the numerical simulations were validatedagainst previous experimental measurements. Garnet et al. [18]conducted experiments in a guarded-heater plate apparatus andcalculated U-values using an indoor and outdoor heat-transfercoefficient of 8 W/m2 K and 23 W/m2 K, respectively. Windowanalysis software, such as Window 6 [19], can be used to calculatethe U-value of a double-glazed unit with an internal aluminiumblind for three slat angles (open, 45� and closed). U-values for theseblind angles are given in Table 3. A difference of 2.3% was observedbetween the values obtained from this CFD study and those from[19].

Calculations were made for aspect ratios (A) between 42 and 74.Selected results for A = 59 are given in detail in Tables 4 and 5 foraluminium and PVC slats, respectively. Conclusions drawn belowalso qualitatively apply to other cavity widths. For an aspect ratioof 59, a higher heat flux was found for the double-glazed unit withthe aluminium blind in the horizontal position than for the unitwithout blind (7%). This was due to the aluminium thermal bridge,which increases the conductive heat transfer through the cavity. Asthe blind angle increased, the U-value decreased. A blind slat angleof 45� reduced the heat flux by 11% relative to the horizontal blind,and for the closed position, it decayed approximately 23%. If theblind surface emissivity was 0.1, the reduction in heat flux wouldbe 16% when the blind angle was changed from the horizontal po-sition to an angle of 45�, and the U-factor would decline 42% whenthe blind was changed from the horizontal to closed position.Therefore, a low emissivity surface on blind slats is preferred to in-crease the thermal resistance when the slats are in the closedposition.

Selected temperature distributions and velocity vectors are gi-ven in Figs. 2–4 to further demonstrate the effects of aspect ratio,slat material and slat angle. Two flow patterns exist inside doubleglazing: (1) the main flow that circulates around the perimeter ofthe cavity, travels up the hot wall and down the cold wall, and(2) the weaker secondary cells between the blind slats. Maximumvelocity was achieved in the middle of the blind-tip to panespacing.

A double-glazed unit with internal PVC slats is beneficial interms of thermal resistance, even when the slats are in the hori-zontal position. For a high surface emissivity, the heat flux is re-duced by 9% for a low thermal conductivity blind. The effect ofrotating the slats 45� reduces the U-factor by 7%, and a 10% reduc-tion is achieved as the blind moves from the horizontal to theclosed position. Therefore, the influence of slat angle is reducedwhen using a low thermal-conductivity blind.

For the case of low thermal conductivity slats, the isotherms areuniformly distributed. The temperature gradient between slats islarger than the gradient for aluminium blinds. Two flow patternsare observed: the main flow and the secondary cells between theblind’s slats. The intensity of both flows is higher than those in adouble-glazed unit with aluminium blinds. In the horizontal posi-tion, if the blind surface emissivity is reduced through a coating,little variation in temperature distribution and velocities are foundas expected. The surface emissivity has a maximum influence inthe closed position.

Figs. 2 and 3 show that when the cavity is broader, the second-ary flow between slats intensifies, and the isotherms adopt a wavyprofile. Fig. 3 illustrates that in the lower region between slats, iso-therms redistribute from the cold wall towards the hot one,whereas in the higher region of the cavity, the flux causes the

Table 2Grid test for an aspect ratio of A = 49.

Slatangle

x⁄ Number ofcells

Variation of Nu relative to the previousgrid

0 0.0417 38,454 –0 0.025 107,747 6%0 0.0125 424,000 1%

60 0.417 39,977 –60 0.025 106,127 4%60 0.0125 423,647 1%

Table 3U-values for a double-glazed unit with an internal aluminium blind and an aspectratio of 74.

U (W/m2 K) 0 30 45 60 Without

Present study 2.995 2.841 2.698 2.547 2.776Window 6 2.925 2.637 2.734

Table 4U-factors for an integrated blind with aluminium slats and an aspect ratio of 59.

e h 0 30 45 60 Closed

0.9 2.912 2.729 2.573 2.420 2.2530.7 2.902 2.708 2.536 2.357 2.1360.55 2.894 2.691 2.507 2.307 2.0360.2 2.874 2.651 2.435 2.179 1.7520.1 2.868 2.639 2.413 2.138 1.653

Table 5U-factors for an integrated blind with PVC slats and an aspect ratio of 59.

e h 0 30 45 60 Closed

0.9 2.485 2.388 2.313 2.240 2.2360.7 2.482 2.371 2.277 2.179 2.1240.55 2.480 2.359 2.253 2.134 2.0290.2 2.473 2.336 2.202 2.029 1.7550.1 2.471 2.331 2.188 1.998 1.660

1230 M. Karmele Urbikain, J.M. Sala / International Journal of Heat and Mass Transfer 55 (2012) 1226–1235

isotherms to redistribute towards the cold wall. This phenomenonis due to the secondary cells between slats and is observed regard-less of the blind thermal conductivity and the external heat-trans-fer coefficients. All figures show that in a unit with aluminiumblinds, isotherms are a set of curves that skim the slat edges andare concave towards the hot and the cold wall respectively. How-ever, isotherms for the PVC blind are wavy and uniformlydistributed.

As the blind closes from 0� to 45�, the main flow intensifies,whereas the secondary cells between slats tend to extinguish. This

Fig. 2. Temperature fields (K) and velocities (m/s)

result is best observed when comparing Figs. 3 and 4. The maxi-mum velocity occurs in the blind tip-to-plate spacing. The temper-ature distribution is comparative to the results obtained in thehorizontal position with little variation of temperatures in the re-gion between slats. When the slat emissivity is reduced in this po-sition, little variation is observed in the temperature distributionsand velocity profiles.

When the blind is closed, two channels are formed, and the slatsurface emissivity has a significant influence on the heat transfer.However, the slat material does not have a significant effect onthe heat flux. The blind in that position acts as a screen that blocksthe long-wave radiative heat flux. The decrease in the slat surfaceemissivity from 0.9 to 0.1 in a cavity with A = 59 reduces the totalheat flux by 26%. However, the difference is very small in the hor-izontal position (approximately 1.5% for aluminium blinds and0.6% for PVC blinds).

Temperature distributions along the middle plane are given inFigs. 5 and 6. A temperature oscillation appears when a blind isintroduced in a double-glazed unit. A blind in the horizontal posi-tion will favour conductive heat transfer along the slat, and a ther-mal bridge will appear between the hot and the cold side of theglazing. Therefore, a strong periodic variation of the temperaturedistribution is apparent with a period equal to the slat-to-slatspace.

The heat flux through a double-glazed unit with an aluminiuminternal blind will be larger than that of a double-glazed unit witha PVC internal blind. However, the temperature variation in themid-plane is smaller for the aluminium blind than for the PVCblind (Fig. 5). Aluminium is a material that tends to homogenisetemperatures. For instance, if a cavity with aluminium surfaces isat different temperatures, the temperature gradient in this cavity

for an aspect ratio of 74 and a slat angle of 0�.

Fig. 3. Temperature distributions (K) and velocity vectors (m/s) for an aspect ratio of 49 and a slat angle of 0�.

Fig. 4. Temperature distributions (K) and velocity vectors (m/s) for an aspect ratio of 49 and a slat angle of 45�.

M. Karmele Urbikain, J.M. Sala / International Journal of Heat and Mass Transfer 55 (2012) 1226–1235 1231

will be small. However, if the cavity is made of PVC, and the sur-faces are at different temperatures, we will have larger tempera-ture gradients compared to the aluminium case.

When the blind is rotated 45� relative to the horizontal position(Fig. 6), the oscillation is reduced because the effect of thermalbridge is diminished. The significance of the blind material dimin-ishes because the slats do not follow the principal direction of theheat transfer. When the slats are tilted, the surface emissivity be-gins to gain significance. If the slats are tilted and a low E surfaceis used, the temperature oscillation will increase. The effect oftilted low E slats provides a weak shielding that causes a temper-ature increase along the hot region of the enclosure and decreasesthe radiative heat transfer. When different double-glazed unitwidths with the same blind angle are compared, the temperatureoscillation increases with the cavity width.

5. Correlations for quantifying convective heat transfer in adouble-glazed unit with integrated blinds

Rayleigh and Nusselt numbers were calculated from the con-vective numerical solution and were used to develop a set of cor-relations. The characteristic length was defined after accountingfor three factors: (i) the cavity width where the blind is placed,(ii) the blade width and (iii) the slat angle relative to the horizontal.Thus, the characteristic length is based on these factors.

d� ¼ d� db � cos h ð25Þ

The Rayleigh number is defined as the following expression:

Rad� ¼g:b � DT � d�3

m2 Pr ð26Þ

Fig. 5. A double-glazed unit with an aspect ratio of A = 49 and a slat angle of 0� (black: aluminium, red: PVC). (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)

1232 M. Karmele Urbikain, J.M. Sala / International Journal of Heat and Mass Transfer 55 (2012) 1226–1235

The Nusselt number is given by the following expression:

Nud� ¼hcd

�

kð27Þ

According to the above equations, as the blind angle in-creases, both the Nusselt number ðNud�Þ and the Rayleigh num-ber increase. This does not contradict the fact that when theblind is rotated from the horizontal to the closed position, thetotal heat flux is reduced. When closing the blind, the increasein the Nusselt and Rayleigh numbers indicates an increase inthe ratio of the convective heat transfer rate to the pseudo-con-ductive heat transfer rate. This is also confirmed by the velocityvectors. The main flow moving up the hot wall and down thecold wall intensifies, whereas the secondary cells between theslats is reduced.

The data obtained numerically have been correlated using thefollowing expression:

Nud� ¼ C � Ramd� � A

n ð28Þ

Fig. 6. A double-glazed unit with an aspect ratio of A = 49 and a slat angle of 45� (blacklegend, the reader is referred to the web version of this article.)

The following correlations have been obtained for the inte-grated blind with aluminium slats:

Nud� ¼ 0:056755 � Ra0:1453d� � A0:3999 58:8 < A < 73:5 ð29:aÞ

Nud� ¼ 0:013092 � Ra0:1561d� � :A0:7407 53:5 < A < 58:8 ð29:bÞ

Nud� ¼ 0:047424 � Ra0:1608d� � A0:4098 49 < A < 53:5 ð29:cÞ

Nud� ¼ 0:144344 � Ra0:165d� � A0:1166 45:2 < A < 49 ð29:dÞ

0 6 h 6 60�

70 < Rad� < 5500

For the unit with PVC slats, the following correlations areobtained:

Nud� ¼ 0:003917 � Ra0:2573d� :A0:7861 58:8 < A < 73:5 ð30:aÞ

Nud� ¼ 0:078343 � Ra0:2515d� :A0:06029 53:5 < A < 58:8 ð30:bÞ

: aluminium, red: PVC). (For interpretation of the references to colour in this figure

101 102 103 104

10-0.3

10-0.2

10-0.1

Correlation 1 A=73.5 Correlation 1 A=58.8Num. data A=73.5Num. data A=58.8Correlation 2 A=58.8Correlation 2 A=53.5Num. data A=53.5Correlation 3 A=53.5Correlation 3 A=49Num. data A=49Correlation 4 A=49Correlation 4 A=45.2Num. data A=45.2

Nuδ*

Raδ*

Fig. 7. Correlations and numerical values for an integrated blind with aluminium slats.

M. Karmele Urbikain, J.M. Sala / International Journal of Heat and Mass Transfer 55 (2012) 1226–1235 1233

Nud� ¼ 0:048417 � Ra0:2315d� :A0:2133 49 < A < 53:5 ð30:cÞ

Nud� ¼ 0:127526 � Ra0:2297d� :A�0:03228 45:2 < A < 49 ð30:dÞ

0 6 h 6 60�

90 < Ra�d < 5700

The difference between the Nusselt number from the correla-tion and the Nusselt number from numerical data can be reducedby subdividing the full range into smaller intervals. In Figs. 7 and8, the correlations are compared with the numerical values for dif-ferent aspect ratios. The current correlations have been comparedwith those of Avedissian, and they predict Nusselt numbers with aminimum and maximum difference of 4% and 16%, respectively, forthe case of aluminium.

101

102

10-0.5

10-0.4

10-0.3

10-0.2

10-0.1

Nuδ*

Fig. 8. Correlations and numerical values f

6. Determination of the U-factor of double glazing withintegrated blind by means of a biquadratic equation

The thermal transmittance (U-value) is the parameter that de-scribes the thermal behaviour of a wall or a window. It is a commonway to quantify the thermal insulation of a building element. It rep-resents the heat flow through a square meter of a window for a 1 �Ctemperature difference between the indoor and outdoor air temper-atures. The heat transfer rate through a double-glazed unit with anintegrated blind is due to three modes of heat transfer: (i) convectionbetween the exterior surface and the exterior environment, betweenglazing layers and between the interior surface and interiorenvironment, (ii) long-wave radiation between the indoor and out-door surface of the glazing and its surroundings and between glazing

103

104

Raδ*

or an integrated blind with PVC slats.

Table 7Biquadratic equation coefficients for an integrated blind with polyvinyl chloride slats.

A a1 a2 2a3 2a4 2a5 a6

74 �0.2558 �0.0182 0.2008 �0.0226 �0.0170 0.959759 �0.2137 �0.0167 0.2108 �0.0756 �0.0202 0.924749 �0.2000 �0.0147 0.2050 �0.1188 �0.0224 0.9493

00.2

0.40.6

0.81

0

0.5

11.5

2

2.5

3

sin (theta)emisivity

Fig. 9. U factor curve for a double-glazed unit with an integrated blind (A = 59) interms of the surface emissivity and blade angle.

00.2

0.40.6

0.81

0

0.5

11.5

2

2.5

3

sin (theta)emisivity

Fig. 10. U factor curve for a double-glazed unit with an integrated blind (A = 49) interms of the surface emissivity and blade angle.

1234 M. Karmele Urbikain, J.M. Sala / International Journal of Heat and Mass Transfer 55 (2012) 1226–1235

layers and (iii) the conduction through the glazing and alongthe slats.

In this study, these three phenomena were analysed, and theproperties that determine the three modes were varied. To deter-mine the effect of conduction along the slats on the U-factor, twocompletely different materials were used: (i) aluminium, whichhas a high thermal conductivity, and (ii) PVC, which has a low ther-mal conductivity. The thermal conductivity of the slat has a signif-icant influence when it is in the fully open position and littleinfluence when it is in the closed position. The decrease of the sur-face emissivity is important in the case of when the slats are in aposition that is close to vertical because of the shielding effect;however, in the horizontal position, it has no significant effect.

A parametric study using mainly two parameters was per-formed. The aim of the study was the determination of the U-factorof a double-glazed window with a blind. Biquadratic curves wereobtained for several aspect ratios and two materials, i.e. aluminiumand PVC.

The efficiency for the blind has been defined as the ratio of theU-factor of a double-glazed unit with an integrated blind for a gi-ven slat angle and surface emissivity to the U-factor of the samedouble-glazed unit without a blind.

gðh; eÞ ¼ Uðh; eÞUwithout

ð31Þ

Therefore, if the U-value of the double-glazed unit is known, thethermal transmittance for the double-glazed unit with an inte-grated blind can be calculated using the blind efficiency. It isimportant to know whether the blind contributes to an increasein the heat flux through the unit or if it improves the insulating va-lue. If g > 1, then the heat flux through the double-glazed unit withan integrated blind will be higher than without a blind. If g < 1,then the heat flux will be lower, and therefore, the blind will con-tribute to thermal insulation.

With the numerical data obtained, the following matrix systemcan be solved, and the aj coefficients can be determined:

x21 y2

1 2x1y1 2x1 2y1 1x2

2 y22 2x2y2 2x2 2y2 1

. . . . . . . . . . . . . . . . . .

x2n y2

n 2xnyn 2xn 2yn 1

26664

37775

a1

a2

. . .

an

26664

37775 ¼

gðx1; y1Þgðx2; y2Þ

. . .

gðxn; ynÞ

26664

37775 ð32Þ

with xi = sinh and yi = eHence, for each of the cavity widths (aspect ratios) and for the

two blind materials (high thermal conductivity or low thermal con-ductivity), the blind efficiency can be expressed as the following:

gðh; eÞ ¼ a1 sin2 hþ a2e2 þ 2a3e sin hþ 2a4 sin hþ 2a5eþ a6 ð33Þ

The aj coefficients for different aspect ratio are given in Tables 6 and 7.Figs. 9 and 10 show U-value curves for two cavity widths in

terms of slat angle and surface emissivity. In each figure, two sur-faces are shown. The upper surface corresponds to the high con-ductivity material, and the lower surface corresponds to the lowthermal conductivity material.

The curves reveal a warped profile; the two surfaces convergewhen the blind position is close to vertical, and the surfaces moveapart when the slats are in the horizontal position. Furthermore,

Table 6Biquadratic equation coefficients for an integrated blind with aluminium slats.

A a1 a2 2a3 2a4 2a5 a6

74 �0.4205 �0.0227 0.1962 0.0150 0.0022 1.091759 �0.4022 �0.0252 0.2106 �0.0276 0.0024 1.066449 �0.3215 �0.0226 0.2062 �0.093 �0.0062 1.0431

the rise of the curves with higher emissivity is sharper when theslats are tilted and close to the vertical position compared to whenthey are open. This demonstrates the influence of the material con-ductivity when the blind is in the horizontal position and the influ-ence of the surface emissivity when it is closed.

These curves express the relation between the U-value for adouble-glazed unit with an integrated blind and the U-value for adouble-glazed unit without a blind. If the U-factor of a double-glazed unit, the blind material, the slat surface emissivity and theslat angle are known, the blind efficiency may be calculated, andthus, the thermal transmittance of a unit with the integrated blindcan be determined.

M. Karmele Urbikain, J.M. Sala / International Journal of Heat and Mass Transfer 55 (2012) 1226–1235 1235

The results were compared with the values obtained from Win-dow 6. For instance, for an aspect ratio of 59, the thermal transmit-tance using the biquadratic equation is 2.853 for a blade angle ofh = 0, 2.617 for h = 45� and 2.199 W/m2 K for h = 90�. Using Win-dow 6, U values are 2.889, 2.609 and 2.197 W/m2 K respectively.These discrepancies may be caused by the differences in the ther-mal conductivity of the aluminium and with the slat width and itscorresponding boundary conditions.

7. Conclusions

A numerical study using computational fluid dynamics wasconducted to investigate the heat transfer through a window witha between-panes venetian blind. Blinds of high and low thermalconductivity slats were analysed. A convective analysis using theEuropean boundary conditions was performed, and the resultswere presented as a set of correlations for both types of materials.The differences observed in the velocity profiles and the tempera-ture distributions have been discussed according to the materialtype. The qualitative conclusions of the present study agree withthose of Naylor and Lai [12]. Additionally, the same CFD studywas performed in night-time conditions (without solar irradiation)but included the effect of long-wave radiation. The blind efficiencywas defined by means of a biquadratic equation, which accountedfor the surface emissivity and the slat angle in a parametric study.These curves, which were obtained from numerical data for differ-ent cavity widths, allow for the prediction of the thermal transmit-tance of a double-glazed unit with an integrated blind. Theseequations have been compared with values obtained from Windows6, which enables the calculation of the U-factor for an integratedblind for three slat angles.

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