Energy Conversion and Management 46 (2005) 361–371www.elsevier.com/locate/enconman

Clear sky and all sky direct luminous efficaciesfor Florian�opolis, Brazil

Roberta G. de Souza a, Luis Robledo b,*, Alfonso Soler c,Fernando O.R. Pereira d

a Laborat�orio de Conforto Ambiental, Dpto. de Arquitetura, Universidade Federal de Santa Catarina,

Campus Universit�ario, 88.040-970 Florian�opolis, Brazil, and Dpto. TAU-EA-UFMG––R. Para�ıba, 697,Funcion�arios, Belo Horizonte, MG, Brazil

b E.P.E.S. Ciencias Ambientales, Universidad Polit�ecnica de Madrid, I.N.E.F. Avda. Mart�ın Fierro s/n,

5a Planta Ciudad Universitaria, 28040 Madrid, Spainc E.P.E.S. Ciencias Ambientales, Dpto. de F�ısica e Instalaciones Aplicadas, E.T.S. de Arquitectura,

Universidad Polit�ecnica de Madrid, Avda. Juan de Herrera, 4, 28040 Madrid, Spaind Laboratorio de Conforto Ambiental, Dpto. de Arquitetura, Universidade Federal de Santa Catarina,

Campus Universit�ario, 88.040-970 Florian�opolis, Brazil

Received 16 January 2004; accepted 21 March 2004

Available online

Abstract

This work presents a characterization of the luminous efficacy of the direct radiation for Florian�opolis,Brazil, for which several direct luminous efficacy models available in the scientific literature were evaluated.

In the first part of the work, the evaluation was restricted to clear sky conditions. The models evaluated

were used with the original coefficients given by their authors and also with local coefficients obtained for

Florian�opolis, showing their dependence on the local characteristics. None of the models presented sig-nificantly better statistics than the others, although one of them, developed by Robledo and Soler, seems to

justify better the physical behavior of the direct luminous efficacy.

In the second part of the paper, three models of direct luminous efficacy for all sky types have been

evaluated, the first two having been specifically developed for the direct luminous efficacy, whereas and the

third was developed for the global luminous efficacy and has been adapted in this paper to characterize the

direct luminous efficacy. The statistical evaluation of the three models showed that the last general model

has accuracy similar to the two other ones.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Luminous efficacy; Direct illuminance; Clear skies; All skies

* Corresponding author. Tel.: +34-91-336-4082; fax: +34-91-336-4097.

E-mail address: lrobledo@eui.upm.es (L. Robledo).

0196-8904/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2004.03.007

362 R.G. de Souza et al. / Energy Conversion and Management 46 (2005) 361–371

1. Introduction

Continuous measurements of horizontal illuminance are not as common as irradiance mea-surements. So, if a luminous efficacy model is available, together with irradiance measurements, itis possible to obtain the corresponding illuminance values in places where they are not measured.

The luminous efficacy of solar radiation is defined as the ratio of the illuminance to the irra-diance. This way, if Eb is the direct illuminance, in lux, and Ib is the direct irradiance, in W/m2, theluminous efficacy of the direct solar radiation, Kb, will be given by the following expression:

Kb ¼Eb

Ibðlm=WÞ ð1Þ

The scientific literature presents luminous efficacy models corresponding to the three mainfluxes of solar radiation: direct, diffuse and global, developed for different sky types, but if wefocus our attention on the direct radiation, we can find models developed almost exclusively forclear sky conditions. This is the case of Littlefair [1], Aydinli and Krockman [2], Ullah [3] andChung [4], who report polynomial models of different degrees for Kb, with the solar altitude, a, asthe only independent variable. Robledo and Soler [5] present a more sophisticated model as afunction of a, deduced from empirical models of direct illuminance and direct irradiance by usingEq. (1).

Now, if our attention is drawn to the developed models for skies with clouds, we will notice thatthe scientific literature is less generous. According to Molineaux et al. [6] the luminous efficacymodels for cloudy skies lacks interest. We agree with that opinion if instantaneous values areconsidered but not if mean hourly or half hourly measurements are considered. In this case, it isnecessary to model the luminous efficacy of the direct solar radiation for all sky types, once thereis a possibility that the solar disk is not shaded by clouds during a considerable part of the givenperiod, as indicated by Robledo and Soler [7] when developing two models for direct luminousefficacy models for all sky conditions––obviously excepting completely overcast skies for whichthere is no direct radiation. Both models use the solar altitude, a, and the brightness index, D, asdefined by Perez et al. [8], as independent variables. Precisely in the last cited paper, there is adirect luminous efficacy model that is valid for all sky types, although it is too complicated to usebecause it requires the calculation of thirty two coefficients, four for each of the eight sky cate-gories that the author distinguishes. In this model, there are also three independent variables, thezenith angle, Z, the brightness index, D, and the atmospheric precipitable water content, W , aquantity that is not measured in all stations.

The present work presents, in a first part, an evaluation of the clear skies models mentionedbefore. The models were tested using data obtained from the station of Florian�opolis, Brazil, withthe original coefficients given by the authors and with coefficients calculated with local data,excepting the Chung model that has the same form as Littlefair’s, as both are second degreepolynomials of a. The second part of this work is dedicated to the evaluation of all sky conditionswith two models proposed by Robledo and Soler in Ref. [7] and also a direct luminous efficacymodel that is presented in this work for the first time, based on a general luminous efficacy modelthat is reported in Robledo et al. [9]. This last paper, published after Ref. [5], provides a generalluminous efficacy model that was developed to estimate the luminous efficacy of global radiation,Kg. We consider that this form can be valid to estimate the other luminous efficacies, diffuse and

R.G. de Souza et al. / Energy Conversion and Management 46 (2005) 361–371 363

direct, just by disregarding some of the factors present in the model equation, according to thegiven flux. This model has the following mathematical expression:

Kg ¼ Aðsin aÞBe�CaDD ð2Þ

Here, the exponential factor is related to the amount of direct irradiance present in the global

radiation that will not be considered when only diffuse radiation is concerned. This model hasbeen developed in the present paper to characterize the luminous efficacy of the direct radiation.The statistical results obtained with this new model were compared to those obtained from thedirect luminous efficacy models established in Ref. [7], when all those models were used to esti-mate direct illuminance.

2. Experimental data and sky characterization

The data used in this work has been measured at the flat roof of the Laboratory of SolarRadiation of the Federal University of Santa Catarina, LabSolar––UFSC, in Florian�opolis(27�330 S, 48�440 W). They are mean half hourly values of horizontal diffuse and global illumi-nance and irradiance, measured from January 2002 to May 2003. The data from January 2002 toDecember 2002 was used to determine local model coefficients, and the data from January to May2003 was used for statistical assessment of the models. The LI-COR sensors, LI 210 SZ used forthe illuminance measurements are calibrated every year using a LI-1800 calibration device by LI-COR. The pyranometer sensors, Kipp & Zonen CM10, used for the irradiance measurements arealso calibrated every year by the personnel of LabSolar. The direct horizontal illuminance andirradiance values are obtained by subtraction of the diffuse from global values, and the solaraltitude was considered to be the middle value of each half hour considered.

The accuracy of the models was determined by using, as statistical indicators, the coefficient ofcorrelation, r, the mean bias deviation, (MBD ¼

Pðyi � xiÞ=N ) and the root mean square devi-

ation, (RMSD ¼ ½P

ðyi � xiÞ2=N �1=2), where yi is the estimated value, xi is the measured value andN is the number of values. The MBD indicates the model tendency to underestimate or overes-timate. The RMSD offers a deviation measure from the predicted values in relation to themeasured values. The MBD and RMSD values are given as percentages of the mean measuredvalues of the direct illuminance.

To characterize the sky conditions, the clearness index, e0, and the brightness index, D, asproposed by Perez et al. [8], have been used. They are defined as

e0 ¼ ½ðId þ InÞ=Id þ kZ3�=ð1þ kZ3Þ ð3Þ

D ¼ Edm=Io ð4Þ

where Id is the diffuse horizontal irradiance, In is the direct normal irradiance, Z is the solar zenithangle in radians at noon, k is a constant equal to 1.041, m is the relative optical air mass [10] and Iois the extraterrestrial irradiance.

For e0 and D, two sky categories have been considered: when e0 > 5:0 and D < 0:12, we haveclear skies conditions and without this restriction, all conditions. Skies with an e0 < 1:2 were nottaken into consideration.

364 R.G. de Souza et al. / Energy Conversion and Management 46 (2005) 361–371

3. Clear sky––direct luminous efficacy models

For simplicity, the models presented in this section are named by the first author of the papers.

3.1. Littlefair model

Littlefair [1] expresses the direct luminous efficacy for clear skies, Kbcl, as a second degreepolynomial of a, as follows:

Kbcl ¼ 51:8þ 1:646a� 0:01513a2 ðlm=WÞ ð5Þ

3.2. Chung model

Chung [4] presents the same equation as Littlefair, but with local coefficients for Hong Kong asfollows:

Kbcl ¼ 48:5þ 1:67a� 0:0098a2 ðlm=WÞ ð6Þ

3.3. Aydinli model

Aydinli and Krochman [2] present a fifth degree polynomial model of a:

Kbcl ¼ 17:2þ 4:4585a� 8:7563 � 10�2a2 þ 7:3948 � 10�4a3 � 2:167 � 106a4

� 8:4132 � 10�10a5 ðlm=WÞ ð7Þ

3.4. Ullah model

Following the three previous authors, Ullah [3] obtains the best fit of experimental points in afourth degree polynomial of a:

Kbcl ¼ 5:36139þ 7:55407a� 0:199068a2 þ 0:0227639 � 10�2a3 � 0:958058 � 10� 5a4 ðlm=WÞð8Þ

3.5. Robledo model

In Ref. [5], Robledo and Soler developed a non-polynomial model that was obtained fromdirect illuminance and irradiance models that were previously empirically developed. Both modelswould respond to the same mathematical expression:

Xbcl ¼ Aðsin aÞBe�Ca ð9Þ

where X could be either the illuminance or the irradiance. When applying Eq. (1) to obtain thedirect luminous efficacy, an equation of the same form as Eq. (9) will also be obtained. Thisequation, when used to calculate illuminances from luminous efficacies, gives better results thanwhen used simply to calculate illuminances. The explanation given in Ref. [5] for this fact consists

R.G. de Souza et al. / Energy Conversion and Management 46 (2005) 361–371 365

in assuming the existence of a factor that was not taken into account in Eq. (9) that could affectboth the illuminance and irradiance in the same way and that would disappear when the twocorrespondent equations are divided in order to obtain the luminous efficacy.

The model for the direct luminous efficacy obtained in Ref. [5] following the above-mentionedmethod, was

Kbcl ¼ 134:27ðsin aÞ0:269e�0:0045a ðlm=WÞ ð10Þ

4. Statistical assessment of clear skies direct luminous efficacy models

The graph in Fig. 1 shows the experimental values of the direct luminous efficacy obtained asthe ratio of experimental values of direct illuminance to direct irradiance against the solar altitudefor Florian�opolis clear sky data. The cloud of points presents the same shape as presented forother latitudes where similar studies have taken place. The direct luminous efficacy increases withsolar altitude until approximately 30�. From this value on, Kbcl seems to stabilize until a solaraltitude of 60�, and after this value it presents a smooth tendency to decrease. This last effect ofdecreasing for high solar altitudes has already been observed by authors from different locationswhere the sun reaches those altitudes such as Ullah [3] in Singapore and Robledo and Soler [5] inMadrid. The Littlefair model, although having been obtained for a place where solar altitudes arenot higher than 60�, predicted this decrease of Kbcl for higher solar altitudes.

In effect, Eq. (10) seems more adequate from a mathematical point of view to explain thebehavior described in the previous paragraph. Actually, in this equation, we find the product oftwo functions with opposite tendencies. One of the functions, the one that contains sin a, presentsan appreciable increase with solar altitude, while the other, the negative exponential, presents asmooth decrease that will have more importance when the a values are higher. When high solaraltitudes are considered, the function ðsin aÞB diminishes its slope and the negative exponential

60

70

80

90

100

110

120

130

0 10 20 30 40 50 60 70 80 90

Fig. 1. Clear direct luminous efficacy against solar altitude for Florian�opolis data.

366 R.G. de Souza et al. / Energy Conversion and Management 46 (2005) 361–371

factor can produce an involution in the growth of the function Kbcl, the product of those twodescribed functions.

This mathematical explanation of the behavior of the cloud of points presented in the graph ofFig. 1, is not so easy to explain when the models are expressed as polynomials of different degrees.

Table 1 shows the RMSD and MBD obtained when calculating direct illuminance with themodels described in Section 3. We can observe that all the models present RMSD values over10%, except for the Chung model that is just some few tenths lower than this value. All the modelsalso present strong underestimation, excepting again the Chung model with )1.5% MBD. Thisresult, though, is somewhat misleading and we can clearly notice that, if we look at the graphspresented in Fig. 2 where the curves corresponding to the models were superimposed on the cloudof experimental points of the direct luminous efficacy, we can see that the low MBD value ob-tained with this model is due to compensation between the underestimation presented below 45�solar altitude and the overestimation above 60�. Even more, this is the model that qualitativelyexplains the behavior of the cloud of points worse. The results presented in Table 1, along with the

Table 1

Statistical assessment of the direct luminous efficacy models for clear sky conditions with original coefficients: RMSD

and MBD values when estimating direct illuminance

Model RMSD (%) MBD (%)

Littlefair 15.5 )14.7Chung 9.4 )1.5Ullah 10.6 )10.1Aydinli 10.6 )10.2Robledo 10.7 )9.8

Mean values Kbcl ¼ 108:53 lm/W

Ebcl ¼ 68:26 klux

Original coefficients60

70

80

90

100

110

120

130

0 10 40 50 90

Robledo

Chung

Littlefair

Ullah

Aydinli

20 30 60 70 80

Fig. 2. Clear direct luminous efficacy against solar altitude: curves given by the original models of Littlefair, Chung,

Aydinli, Ullah and Robledo superimposed on the experimental points.

Table 2

Models of direct luminous efficacy for clear sky with local coefficients: RMSD and MBD values when estimating direct

illuminance

Models with local coefficients r RMSD (%) MBD (%)

Second degree

polynomial

Kbcl ¼ 85:32� 0:826aþ 0:0073a2 0.564 3.4 )1.9

Fourth degree

polynomial

Kbcl ¼ 54:066þ 4:0762a� 0:118a2

þ0:001517a3 � 7:222 � 10�6a40.617 3.4 )1.9

Fifth degree

polynomial

Kbcl ¼ 47:51þ 5:133a� 0:1769a2 þ 0:002988a3

�0:2409 � 10�4a4 þ 7:2172 � 10�8a50.618 3.3 )1.9

Robledo Kbcl ¼ 131:92 sin a0:221e�0:00279a 0.630 3.3 )2.0

Mean values Kbcl ¼ 108:53 lm/W Ebcl ¼ 68:26 klux

R.G. de Souza et al. / Energy Conversion and Management 46 (2005) 361–371 367

graphs presented in Fig. 2, indicate that the coefficients of each model seem to present a localcharacter. Because of this fact, all the functions corresponding to these models have been opti-mized for Florian�opolis data, in order to obtain the same models, but now with local coefficients.We can see in Table 2 the resulting equations for the different models with the new local coeffi-cients, along with their corresponding correlation coefficients, r, and the RMSD and MBD valuesobtained when those models are used to calculate the direct illuminance with experimental data. Itshould be noticed that the fourth degree polynomial model presented in Table 2 corresponds tothe best polynomial fit for the Florian�opolis data when a stepwise regression is done.

All the models present practically the same statistics with a RMSD something lower than 3.5%in all cases and a MBD of about )2.0%, results that we can consider as being quite good,especially when referring to the RMSD value. We have to notice here something that was alreadysaid in Robledo and Soler [5], and that is that for all polynomial models with a degree higher thantwo, the statistics are practically the same when they are used to estimate the direct illuminance.An explanation was given of this fact in this same paper [5], and it suggests that the polynomialmodels present a certain ambiguity, an ambiguity that does not appear in the Robledo model that,although presenting the same statistics, allows us to work with a function perfectly defined thatcan be obtained from analogue models of illuminance and irradiance. This way, this model showsa mathematical coherence that the polynomial models, obtained as a simple fit to experimentaldata values, lack.

5. Performance assessment of all sky direct luminous efficacy models

5.1. Robledo models

When referring to all sky conditions, Robledo and Soler [7] proposed to obtain the directilluminance, Eb, and the direct irradiance, Ib, by multiplying the previously given expressions ofEbcl and Ibcl by an attenuation factor exclusively dependent on the brightness index, D, following amethod used by Littlefair [1] to obtain the global luminous efficacy for overcast skies. Thus thefollowing expressions for Eb and Ib result:

368 R.G. de Souza et al. / Energy Conversion and Management 46 (2005) 361–371

Eb ¼ EbcluEðDÞ ðkluxÞ ð11Þ

and

Ib ¼ IbcluIðDÞ ðW=m2Þ ð12Þ

where uEðDÞ and uIðDÞ are the attenuation factors for the direct illuminance, Eb, and the directirradiance, Ib, respectively that is to say, the respective relative illuminance and irradiance to clearskies. If we use Eqs. (1), (11) and (12), we will obtain the following expression for the directluminous efficacy:

Kb ¼ KbcluKðDÞ ðlm=WÞ ð13Þ

where Kbcl is the direct luminous efficacy for clear skies that can be obtained from the modelsestablished in Section 3.4 of this work.

The relative direct luminous efficacy, uKðDÞ, was modeled by Robledo and Soler [7] using twodifferent functions. One is a linear function obtained by regression analysis of the points uK vs. D,and given by the equation:

uKðDÞ ¼ aþ bD ðlm=WÞ ð14Þ

The other is an exponential function obtained by the ratio of previously developed models of

uEðDÞ and uIðDÞ. The result was

uKðDÞ ¼ AeþBD2 ðlm=WÞ ð15Þ

where a, b, A and B were local empirical constants.

5.2. Florian�opolis local coefficients and results

If the models given by Eqs. (14) and (15) are optimized with Florian�opolis data, the equationsobtained are the following:

linear model : uKðDÞ ¼ 0:989þ 0:157D ðlm=WÞ; r ¼ 0:182 ð16Þ

exponential model : uKðDÞ ¼ 1:002eþ0:299D2 ðlm=WÞ; r ¼ 0:144 ð17Þ

Eq. (16) is obtained by the linear regression of the points in the graph of Fig. 3 in which the uK

values are represented against the brightness index, D.On the other hand, in a paper published afterwards paper Ref. [7], Robledo and Soler [9]

developed a general model for the global luminous efficacy that has the form presented in Eq. (2).In this model, the exponential factor is related to the direct radiation. When there is only diffuseradiation, Eq. (2) can be reduced to the equation of diffuse luminous efficacy reported in a pre-vious paper [11] in which the exponential factor would not be present. The factor that contains D,is related to a higher or lower attenuation by clouds and aerosols, and it can be eliminated whenmodeling the luminous efficacy for clear skies without incurring any considerable error. When thisis done, the general Eq. (2) is reduced to Eq. (9).

As a way to validate the use of a general model, such as the one given by Eq. (2), to estimate thedirect luminous efficacy, this equation was optimized with clear and intermediate sky data forFlorian�opolis, giving:

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Fig. 3. Relative direct luminous efficacy plotted against brightness index.

Table 3

Models of direct luminous efficacy for all skies with local coefficients: RMSD and MBD values when estimating direct

illuminance

Model Sky type

All Clear Intermediate

RMSD (%) MBD (%) RMSD (%) MBD (%) RMSD (%) MBD (%)

Kball ¼ Kbclð0:989þ 0:157DÞ 5.6 )3.2 3.2 )1.8 10.1 )6.0Kball ¼ Kbclð1:002e0:299D2Þ 5.7 )3.4 3.2 )1.8 10.5 )6.6Kball ¼ 143:14 sin a0:241

:e�0:0031aD0:0235.7 )3.3 3.3 )1.9 10.4 )6.2

Mean values

Kb (lm/W) 114.13 108.53 118.44

Lb (klux) 44.79 68.26 26.71

R.G. de Souza et al. / Energy Conversion and Management 46 (2005) 361–371 369

Kb ¼ 143:14ðsin aÞ0:241e�0:0031aD0:023 ðlm=WÞ; r ¼ 0:615 ð18Þ

Table 3 shows the RMSD and MBD values obtained when calculating the direct illuminance

with the following three models:

Model 1: Eq. (13) with uKðDÞ given in Eq. (16).Model 2: Eq. (13) with uKðDÞ given in Eq. (17).

In both cases, the Kbcl values are obtained from the Robledo model with local coefficients forFlorian�opolis, expressed in Table 2.

Model 3: Eq. (18).

370 R.G. de Souza et al. / Energy Conversion and Management 46 (2005) 361–371

As it can be seen in Table 3, the three evaluated models present very similar statistics for clearand intermediate skies. One may even notice that when referring to clear skies, the results arecoincident with the ones obtained when using models specially developed for this sky type (seeTable 2). The statistical indicators for intermediate skies are worse than those for clear skies,presenting values of RMSD that are appreciably higher and also a greater underestimation.Nevertheless, the more important aspect of this evaluation is to prove that the general luminousefficacy model reported in Ref. [9] by Robledo and Soler––optimized to local data––results aresatisfactory when applied to direct radiation. In the present work, as far as we know, this is beingproved for the first time, consolidating the general character of the model.

6. Conclusions

In the present paper, we have made a study of the luminous efficacy of the direct solar radiationfor Florian�opolis, Brazil. In the first part of the paper, the study was limited to clear sky con-ditions, and it consisted in evaluating five models available in the literature. In all of them, theonly independent variable is the solar altitude. Four of the models consisted of different degreepolynomials, which is very usual in the characterization of direct luminous efficacy for clear skies.The fifth is given by a somewhat more sophisticated function, not obtained by fitting of experi-mental data as is done in the case of the polynomial models but from the division of empiricalilluminance and irradiance models previously obtained.

The statistical results have shown that all the models present the same accuracy when they areused to estimate direct illuminance. Nevertheless, as it was explained in Section 4 of this paper, themodel that best interprets mathematically the experimental cloud of points, Kbcl vs. a, is the non-polynomial model given by Eq. (9), while the polynomial models do not offer good justification ofthis cloud of points. They simply fit the data, all of them offering, independently from their degree,the same accuracy when they are used to calculate direct illuminance, which gives these types ofmodels a certain ambiguity. The ambiguity disappears when Eq. (9) is taken into account becauseonce it offers a clearly defined mathematical form that can be extrapolated to any location just bycalculating local coefficients for it.

In the second part of this work, two different direct luminous efficacy models for all sky types[7] have been evaluated. Both models are based on considering an attenuation factor exclusivelydependent on the brightness index, D. They are obtained when this attenuation factor is multipliedby the luminous direct efficacy for clear sky given by Eq. (10), reported in the first part of thispaper. In Ref. [7], Robledo and Soler reported two functions of D for the attenuation factor: oneconsisting of a linear relation and the other an exponential function. These functions have beenoptimized for Florian�opolis and are reported in the present paper in Eqs. (16) and (17), respec-tively. The evaluation of the two direct luminous efficacy models derived from these equationshave shown that both present very similar statistical results which are better than the resultsobtained when this evaluation was performed with the original coefficients, reinforcing the localcharacter of the coefficients. It even presented better results than for Madrid, the location forwhich the original models were developed.

The second part of the present work also presents a general luminous efficacy model developedin Ref. [9] to characterize the luminous efficacy of global solar radiation for all sky types. A model

R.G. de Souza et al. / Energy Conversion and Management 46 (2005) 361–371 371

with the same mathematical form has been used here as a direct luminous efficacy model. We havesuggested that a model given by the form of Eq. (2) could be valid as a model for both diffuse anddirect luminous efficacy, without more than adjusting the model coefficients in order to give majoror minor weight to the diffuse or direct component. In the case of the diffuse luminous efficacy, thevalidity of this assumption has already been proved in previous works [11,12]. Indeed, here it hasbeen proved that the luminous direct efficacy behaves in accordance with the considered generalefficacy model, which given with local coefficients for Florian�opolis by Eq. (18). The statisticalevaluation of this model has shown that its prediction accuracy is practically identical to that ofthe models that Robledo and Soler developed in Ref. [7], specifically for the direct luminousefficacy for all skies. So, it has been verified that there is a type of luminous efficacy model that isvalid for any of the fluxes of solar radiation, which is the model given by Eq. (2), and thatdepending on the radiation type, each of the factors that compose the model will have a differentspecific weight, determined by the model coefficient values.

Acknowledgements

The authors wish to thank the Coordenac�~ao de Aperfeic�oamento de Pessoal de N�ıvel Superior(CAPES) and the Conselho Nacional de Desenvolvimento Cient�ıfico e Tecnol�ogico (CNPq) fortheir financial support to this work. We would also like to thank the Dpto. de F�ısica e Insta-laciones Aplicadas de la Escuela Superior de Arquitectura de Madrid, U.P.M. for propitiating themaking of this work.

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