- Email: [email protected]
A pseudo-Fresnel approach for predicting directional optical properties of coated glazing
Thin Solid Films 392 Ž2001. 282᎐288
A pseudo-Fresnel approach for predicting directional optical properties of coated glazing Peter A. van NijnattenU TNO-TPD (Institute of Applied Physics), P.O. Box 595, 5600 AN Eindho¨ en, The Netherlands
Abstract For a realistic computation of the visual light and solar energy properties of coated glazing, we need to know the angular behaviour of its optical properties. In a previous paper, the author has proposed a new type of algorithm for the prediction of this angular behaviour. The algorithm is based on a pseudo-Fresnel model in which the complex Fresnel equations and multi-layer calculations are replaced by more simple expressions. In this algorithm it is assumed that the angular dependency of the optical properties of a coated glass is mostly determined by the directional reflection at the air᎐glass and air᎐coating interfaces. The only data required by the algorithm are reflectance and transmittance values obtained at Žnear-. normal incidence. The present work has been focused on the application of the algorithm for double-glazing and improving the approach by incorporating multiple reflections between the interfaces. In this paper, different variants of the algorithm are discussed. The results obtained by these algorithms are compared with accurately measured directional optical properties. The results show that in order to obtain sufficient agreement it is necessary to divide the various types of glazing into different groups and use different algorithms for each group. 䊚 2001 Elsevier Science B.V. All rights reserved. Keywords: Glazing; Optical coatings; Optical properties
1. Introduction Recently, a three-year programme of work called the ADOPT project w1x was completed. In this project, a group of six European laboratories, one of which was the Optical Measurements & Testing group of TNOTPD, investigated the angular dependency of the optical properties of coated glazing. One of the objectives of the project was to develop an algorithm for the prediction of the angular dependency of optical properties using the optical properties measured at Žnear-. normal incidence, which are available for most commercial glazing products, as input. The algorithmŽs. are intended for improving the precision of the building energy balance simulation tools that use solar optical properties as input.
U
Corresponding author. Fax: q31-40-2650850. E-mail address: [email protected] ŽP.A. van Nijnatten..
In recent years, the author and other researchers have investigated different algorithms for the prediction of directional optical properties of glazing w2᎐8x. These algorithms differ from the so-called ‘exact’ Fresnel model w9x by their use of empirical relations and approximate models. In a previous paper by the author, a new type of algorithm was proposed for the prediction of the angular dependency of total integrated optical properties of glazing, using the Žnear-. normal Žintegrated. optical properties as input. The advantage of this approach is that calculations require little data Žsingle glazing factors instead of complete spectra., which makes it simple and fast. The preliminary results showed that the new algorithm gives an accurate description of the angular behaviour of the visible light transmittance of single coated glazing w6x. The current paper will focus on the angular dependency of the visible light transmittance, the direct solar transmittance and the solar factor g Žtotal solar energy transmittance . for the case of double glazing with and
0040-6090r01r$ - see front matter 䊚 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 0 - 6 0 9 0 Ž 0 1 . 0 1 0 4 4 - 6
P.A. ¨ an Nijnatten r Thin Solid Films 392 (2001) 282᎐288
without coating on position 3 Žoutward facing surface of the indoor pane.. The earlier proposed algorithm and two modified variants are investigated and discussed. Accurate measurement data of the directional optical properties obtained previously in the ADOPT project for a set of coated samples are used for validation of the algorithms. 2. Theory 2.1. Pseudo-Fresnel equations The specular directional reflection at an interface between two media is given by the so-called Fresnel equations w10x. In the case of a thin film system, the total reflectance of the coating is calculated using a formula that takes into account multiple reflections and interference effects w9x between the interfaces. In the present algorithms, both calculations are replaced by a more simplified approach, namely the use of equations that, by approximation, show the same behaviour as Fresnel’s equations for reflectance. We consider two types of these pseudo-Fresnel equations, namely
R P Ž . s
RS Ž . s
ž
'N 2 y sin2 Ž . y N 2 cos Ž . 'N 2 y sin2 Ž . y N 2 cos Ž .
ž
'N 2 y sin2 Ž . y cos Ž . 'N 2 y sin2 Ž . y cos Ž .
2
/
Ž 1a .
2
/
Ž 1b .
and R P Ž . s
RS Ž . s
w 1 y Ncos Ž .x 2 q K 2 cos 2 Ž . w 1 q Ncos Ž .x 2 q K 2 cos 2 Ž . w N y cos Ž .x 2 q K 2 cos 2 Ž . w N q cos Ž .x 2 q K 2 cos 2 Ž .
Ž 2a .
Ž 2b .
where N and K are the refraction and absorption coefficient, respectively Žnot to be mistaken for the real and imaginary parts of the complex refractive index.. The subscripts P and S refer to p- and s-polarisation, respectively. Eqs. Ž1a. and Ž1b. with N equal to the refractive index, is Fresnel’s equation for the power reflectance of a specular interface of an ideal dielectric material Žzero absorption. with vacuum. In the present work, Eqs. Ž1a. and Ž1b. are used to calculate the air᎐glass reflectance using for N a value of the refractive index of window glass corresponding to a total integrated Žsingle surface. reflectance at normal incidence. Different values
283
of this reflectance were used for visible daylight and solar energy properties Ž R s 0.0423 and 0.0412, respectively.. These values were obtained from the spectral single surface reflectance of a clear float glass by applying the procedure for calculating the visible daylight reflectance and direct solar energy reflectance as prescribed in EN410 w11x. Eqs. Ž2a. and Ž2b. are a somewhat crude approximation of Fresnel’s equations in the case of a strong absorbing material. In the present investigation a constant value of N is used and the absorption coefficient K is determined by the total integrated value of the spectral surface reflection of the coating at Žnear-. normal incidence. In order to make Eqs. Ž2a. and Ž2b. applicable for all types of coating, including coatings that have zero reflection Žanti-reflection coatings., the refraction coefficient was chosen to be N s 1. Both equations Eqs. Ž1a. and Ž1b. and Eqs. Ž2a. and Ž2b. are used for calculating the directional reflectance of the coating. Obviously, Eqs. Ž1a. and Ž1b. are expected to give the better approximation in the case of an all-dielectric thin film system, whereas Eqs. Ž2a. and Ž2b. should work better for a coating involving a conducting layer. The different types of angular behaviour given by these equations are demonstrated for various normal reflectance values by Fig. 1. 2.2. The in¨ estigated algorithms Three algorithms are considered, in which calculations are performed separately for p- and s-polarisation, and the final result is obtained by averaging. Algorithm 1 is the simplest and is based on the following empirical relation for directional transmittance: n
T Ž . s 1 y R g Ž . w 1 y R c Ž .x
m
Ž3.
in which R g represents the air᎐glass reflectance, R c the air᎐coating reflectance, n is the number of glass᎐air interfaces, and m the number of coated surfaces. In the preliminary investigation w6x, Eq. Ž3. was applied to the visible light transmittance of single coated glazing Ž n s m s 1.. In the present investigation, n s 3 and m s 1. The refraction or absorption coefficient necessary for the pseudo-Fresnel equation which describes the air᎐coating reflectance R c Ž ., is calculated from its value at normal incidence given by R c Ž0. s 1 y T Ž0. 1 y R g Ž0.
y3
Ž4.
where T Ž0. is the visible daylight or solar energy transmittance at normal incidence of the double glazing unit as determined by the measurements. Algorithm 2 is a slightly modified version of algo-
P.A. ¨ an Nijnatten r Thin Solid Films 392 (2001) 282᎐288
284
Fig. 1. Angular dependency of the single surface reflection for 5 different values of the reflectance at normal incidence wNs 1 in Eqs. Ž2a. and Ž2b., N s 1x.
rithm 1, obtained by introducing a term representing the internal Žbulk. transmittance: n
m
T Ž . s 1 y R g Ž . w 1 y R c Ž .x Ž . .
Ž5.
In this case R c Ž0. is the visible light or solar energy reflectance of the coated glass sample Žcoating side reflectance . and in the present investigation, n s 3 and m s 1. The internal transmittance is determined from the total transmittance of the glazing at normal incidence and is given by Ž . s 1r
1y w sin Ž .rN g x 2
'
.
coated sample are determined by the standard measurement procedure. With R c Ž0. calculated by Eq. Ž9., the internal transmittance is determined in the same manner as for algorithm 2. The total reflectance R1 of pane 1 is determined by R1Ž . s R g Ž .w 1 q T1Ž . 1Ž .x .
Ž 10 .
The total reflectance R 2 of pane 2 is determined by either Eqs. Ž1a. and Ž1b. or Eqs. Ž2a. and Ž2b.. N or K is determined from the measured Žnear-. normal value of the coated glass reflectance R 2 .
Ž6. 2.3. Validation
Algorithm 3 is based on the calculation of double glazing properties from the properties of the individual panes: T Ž . s
T1Ž . T2 Ž . 1 y R1Ž . R 2 Ž .
Ž7.
in which T1 and T2 are the transmittances of panes 1 and 2, respectively, given by Ti Ž . s
1 y R g Ž . w 1 y R x Ž .x i Ž . 1 y R g Ž . R x Ž . i2 Ž .
,
i s 1,2.
Ž8.
In our case of T1 , R x s R g , and the internal transmittance 1 is given by Eq. Ž6. with determined from the total transmittance of pane 1 at normal incidence. In the case of the second Žcoated. pane, R x s R c , which is determined by either Eqs. Ž1a. and Ž1b. or Eqs. Ž2a. and Ž2b., and N or K is determined from its value for normal incidence given by R c Ž0. s R 2 Ž0. y R g Ž0.
T2 Ž 0 . 1 y R g Ž0.
2
.
Ž9.
The reflectance R 2 Ž0. and transmittance T2 Ž0. of the
The algorithms have been tested using the known directional optical properties of 1 double glazing unit consisting of 2 clear glass panes and 8 coated double glazing units, each consisting of 1 clear glass outdoor pane and 1 coated glass indoor pane with the coating facing outwards Ža typical glazing in northern European countries.. The optical properties considered are the visible light transmittance, the direct solar energy transmittance and the solar factor g, which are the principle glazing factors. The different coatings and properties of these glazing for normal incidence are listed in Table 1 below. The coated glass samples were specially made for the ADOPT project w1x. Their composition closely resembles those of actual glazing products, but since they were custom-made, more detailed information on the layer composition was available than would have been the case with real products. Their directional reflectance and transmittance was measured in the wavelength range 250᎐2500 nm at near-normal incidence and at 30, 45, 60 and 75⬚ using state-of-the-art equipment w12,13x. The glazing factors were calculated from these data according to EN410 w11x. The thickness of the coating layers reported in Table 1 where determined by means of RBS.
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Table 1 Investigated double glazing ŽDG. units; the listed values of the glazing factors were obtained for Žnear-. normal incidence DG
Tvis
Tsol
g
Coating design
0 1 2 3 4 5 6 7 8
0.823 0.391 0.578 0.766 0.345 0.453 0.534 0.776 0.779
0.737 0.245 0.379 0.522 0.439 0.333 0.425 0.639 0.648
0.753 0.315 0.459 0.600 0.546 0.518 0.574 0.671 0.690
Without coating 32 nm SnO2 , 20 nm Ag, 1 nm NiCr and 59 nm SnO2 22 nm SnO2 , 15 nm Ag, 1 nm NiCr and 52 nm SnO2 29 nm SnO2 , 8 nm Ag, 1 nm NiCr and 29 nm SnO2 5 nm Si 10 nm TiN 5 nm TiO2 , 8 nm TiN and 10 nm TiO2 150 nm SnO2 114 nm ITO
q0.232T Ž 60⬚. q 0.134T Ž 75⬚. .
3. Results and discussion
Ž 12. By substituting T in Eqs. Ž3., Ž5. and Ž7. for the visible daylight transmittance Tvis , the direct solar energy transmittance Tsol or the solar factor g; the algorithms determine the angular dependency of these glazing factors. The errors obtained by the various combinations of algorithms with Eqs. Ž1a., Ž1b., Ž2a. and Ž2b. for the coating reflectance are summarised in Tables 2᎐4. These errors represent a weighted average according to ␦T s 0.3⌬T Ž 30⬚. q 0.27⌬T Ž 45⬚. q 0.23⌬T Ž 60⬚. q 0.14⌬T Ž 75⬚.
Ž 11.
with ⌬T the absolute difference between the measured and calculated data. Eq. Ž11. is based on the assumption that not all angles are equally important and that one must be able to calculate the hemispherical value of the glazing factor most accurately from the directional values by T hem s
r2
H0
T Ž . sin Ž 2 . d f 0.134T Ž 15⬚.
q0.232T Ž 30⬚. q 0.268T Ž 45⬚.
In Eq. Ž11. it is assumed that ⌬T Ž15⬚. s 0.5 ⌬T Ž30⬚. wsince ⌬T Ž0⬚. s 0 and T Ž15⬚. was not measuredx. On average, Algorithm 3 in combination with Eqs. Ž1a. and Ž1b. gives the best performance, but leads to large errors in some cases. Different algorithms perform better for different glazing and there seems to be a correlation with the type of coating. The different types of glazing appear to be divided into three groups. One group is formed by the silver-based coatings, DG1᎐DG3, and another by the silicon and titanium coatings, DG4᎐DG6. The third group is formed by glazing DG0 without coating, and the two tin oxide type coatings DG7 and DG 8. Their data seem to agree well with Algorithm 3 in combination with Eqs. Ž1a. and Ž1b. for all three glazing factors Žsee Fig. 2.. In the case of the silver based coatings, the solar energy properties seem to be best predicted with Algorithm 3 in combination with Eqs. Ž2a. and Ž2b.. However, the visible light transmittance can be predicted with better accuracy using a more simpler algorithm like Algorithm 2 in combination with Eqs. Ž1a. and Ž1b. Žsee Fig. 3., although for DG1 a better result is obtained with Algorithm 1. The best results for glazing units DG4᎐DG6 are obtained with Algorithm 1 in combina-
Table 2 Errors obtained in the prediction of the directional visible light transmittance Glazing unit
Algorithm 1 Eqs. Ž1a.Ž1b.
Eqs. Ž2a.Ž2b.
Algorithm 2 Eqs. Ž1a.Ž1b.
Eqs. Ž2a.Ž2b.
Algorithm 3 Eqs. Ž1a.Ž1b.
Eqs. Ž2a.Ž2b.
DG0 DG1 DG2 DG3 DG4 DG5 DG6 DG7 DG8 Average
0.019 0.035 0.027 0.015 0.016 0.033 0.018 0.026 0.011 0.022
0.052 0.006 0.019 0.021 0.017 0.008 0.024 0.061 0.037 0.027
0.015 0.022 0.014 0.005 0.029 0.028 0.027 0.023 0.011 0.019
0.050 0.021 0.036 0.031 0.053 0.050 0.055 0.060 0.037 0.044
0.006 0.024 0.018 0.011 0.029 0.026 0.023 0.015 0.008 0.018
0.037 0.017 0.027 0.021 0.050 0.045 0.048 0.047 0.023 0.035
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286
Table 3 Errors obtained in the prediction of the directional solar energy transmittance Glazing unit
Algorithm 1 Eqs. Ž1a.Ž1b.
Eqs. Ž2a.Ž2b.
Eqs. Ž1a.Ž1b.
Algorithm 2 Eqs. Ž2a.Ž2b.
Eqs. Ž1a.Ž1b.
Algorithm 3 Eqs. Ž2a.Ž2b.
DG0 DG1 DG2 DG3 DG4 DG5 DG6 DG7 DG8 Average
0.008 0.044 0.053 0.052 0.033 0.038 0.035 0.017 0.029 0.034
0.030 0.021 0.018 0.012 0.010 0.007 0.006 0.026 0.018 0.016
0.013 0.020 0.020 0.014 0.015 0.026 0.027 0.009 0.005 0.017
0.030 0.019 0.028 0.032 0.038 0.040 0.042 0.031 0.031 0.032
0.005 0.019 0.021 0.017 0.014 0.025 0.025 0.004 0.002 0.015
0.032 0.005 0.006 0.010 0.037 0.041 0.044 0.032 0.024 0.026
Table 4 Errors obtained in the prediction of the directional solar factor g Glazing unit
Algorithm 1 Eqs. Ž1a.Ž1b.
Eqs. Ž2a.Ž2b.
Eqs. Ž1a.Ž1b.
Algorithm 2 Eqs. Ž2a.Ž2b.
Eqs. Ž1a.Ž1b.
Eqs. Ž2a.Ž2b.
DG0 DG1 DG2 DG3 DG4 DG5 DG6 DG7 DG8 Average
0.007 0.039 0.042 0.035 0.011 0.016 0.010 0.005 0.014 0.020
0.034 0.009 0.008 0.012 0.031 0.027 0.033 0.036 0.028 0.024
0.014 0.028 0.025 0.015 0.015 0.025 0.026 0.012 0.008 0.019
0.029 0.019 0.027 0.030 0.036 0.042 0.040 0.032 0.031 0.032
0.006 0.028 0.027 0.021 0.010 0.021 0.020 0.006 0.002 0.016
0.033 0.006 0.004 0.011 0.041 0.047 0.047 0.036 0.029 0.028
tion with Eqs. Ž2a. and Ž2b. for the visible light and direct solar energy transmittance, and Eqs. Ž1a. and Ž1b. for the solar factor Žsee also Fig. 4..
Algorithm 3
The results obtained for the silver based coated glazing, the glazing coated with tin dioxide, and the non-coated glazing seem quite satisfactory. However, as
Fig. 2. Calculations Žsolid line. vs. measurements Ždotted line with markers. for the three silver-based coated glazing units.
P.A. ¨ an Nijnatten r Thin Solid Films 392 (2001) 282᎐288
287
Fig. 3. Calculations Žsolid line. vs. measurements Ždotted line with markers. for the silicon and titanium nitride coated glazing units.
Fig. 3 clearly illustrates, a poor agreement is obtained for the remaining group DG4᎐DG6. Generally, the agreement is better for the solar properties than for the visible light transmittance. 4. Conclusion Three different algorithms for the prediction of the angular dependency of glazing factors have been investigated. These algorithms are based on a pseudo-Fresnel model in which the complex Fresnel equations and
multi-layer calculations are replaced by more simple expressions. The only data required by the algorithms are the reflectance and transmittance values obtained at Žnear-. normal incidence. The algorithms are intended to be used in combination with building simulation tools. The results obtained by these algorithms have been compared with accurately measured directional optical properties of 9 different double glazing systems. The results show that in order to obtain sufficient agreement, it is necessary to divide the various types of
Fig. 4. Calculations Žsolid line. vs. measurements Ždotted line with markers. for the uncoated and tin dioxide coated glazing units.
288
P.A. ¨ an Nijnatten r Thin Solid Films 392 (2001) 282᎐288
glazing into different groups and use different algorithms for each group. A good agreement between measurements and algorithms has been obtained for most glazing. The agreement is better for solar transmittance and its factor than for visible light transmittance. The agreement was poor in the case of the silicon and titanium nitride coated glazing. A different algorithm or a modified version of one of the present algorithms should be developed for this group of materials. References w1x P.A. van Nijnatten. Proceedings of the Conference on Industrial Technology, Toulouse, October 1997. w2x P. Pfrommer, Energy Build. 21 Ž1994. 101᎐110.
w3x P. Polato, M. Montecchi, La rivista della staz Sper. del Vetro 5 Ž1995. 165᎐175. w4x A. Roos, J. Non-Cryst. Sol. 3651 Ž1997.. w5x M. Montecchi, P. Polato, La Rivista della Staz Sper. del Vetro 2 Ž1998. 55᎐62. w6x P.A. van Nijnatten, Thin Solid Films 351 Ž1999. 295᎐300. w7x M. Rubin, R. Powles, K. von Rottkay, Sol. Energy 66 Ž1999. 267᎐276. w8x J. Karlsson, A. Roos, Sol. Energy 69 Ž2000. 321᎐329. w9x P.A. van Nijnatten, SPIE Series 2255 Ž1994. 753᎐762. w10x M. Born, E. Wolf, Principles of Optics, 6th ed, Pergamon, Oxford, 1980. w11x EN 410, European Committee for Standardization, doc. CENrTC 129 N 123 E, 1992. w12x P.A. van Nijnatten. Proceedings of the World Renewable Energy Conference, Part I, 2000, pp. 300᎐306. w13x M.G. Hutchins, A.J. Topping, Proc. World Renewable Energy Congress VI, Ed. A. Sayigh, 2000, pp. 265᎐270.
A pseudo-Fresnel approach for predicting directional optical properties of coated glazing Peter A. van NijnattenU TNO-TPD (Institute of Applied Physics), P.O. Box 595, 5600 AN Eindho¨ en, The Netherlands
Abstract For a realistic computation of the visual light and solar energy properties of coated glazing, we need to know the angular behaviour of its optical properties. In a previous paper, the author has proposed a new type of algorithm for the prediction of this angular behaviour. The algorithm is based on a pseudo-Fresnel model in which the complex Fresnel equations and multi-layer calculations are replaced by more simple expressions. In this algorithm it is assumed that the angular dependency of the optical properties of a coated glass is mostly determined by the directional reflection at the air᎐glass and air᎐coating interfaces. The only data required by the algorithm are reflectance and transmittance values obtained at Žnear-. normal incidence. The present work has been focused on the application of the algorithm for double-glazing and improving the approach by incorporating multiple reflections between the interfaces. In this paper, different variants of the algorithm are discussed. The results obtained by these algorithms are compared with accurately measured directional optical properties. The results show that in order to obtain sufficient agreement it is necessary to divide the various types of glazing into different groups and use different algorithms for each group. 䊚 2001 Elsevier Science B.V. All rights reserved. Keywords: Glazing; Optical coatings; Optical properties
1. Introduction Recently, a three-year programme of work called the ADOPT project w1x was completed. In this project, a group of six European laboratories, one of which was the Optical Measurements & Testing group of TNOTPD, investigated the angular dependency of the optical properties of coated glazing. One of the objectives of the project was to develop an algorithm for the prediction of the angular dependency of optical properties using the optical properties measured at Žnear-. normal incidence, which are available for most commercial glazing products, as input. The algorithmŽs. are intended for improving the precision of the building energy balance simulation tools that use solar optical properties as input.
U
Corresponding author. Fax: q31-40-2650850. E-mail address: [email protected] ŽP.A. van Nijnatten..
In recent years, the author and other researchers have investigated different algorithms for the prediction of directional optical properties of glazing w2᎐8x. These algorithms differ from the so-called ‘exact’ Fresnel model w9x by their use of empirical relations and approximate models. In a previous paper by the author, a new type of algorithm was proposed for the prediction of the angular dependency of total integrated optical properties of glazing, using the Žnear-. normal Žintegrated. optical properties as input. The advantage of this approach is that calculations require little data Žsingle glazing factors instead of complete spectra., which makes it simple and fast. The preliminary results showed that the new algorithm gives an accurate description of the angular behaviour of the visible light transmittance of single coated glazing w6x. The current paper will focus on the angular dependency of the visible light transmittance, the direct solar transmittance and the solar factor g Žtotal solar energy transmittance . for the case of double glazing with and
0040-6090r01r$ - see front matter 䊚 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 0 - 6 0 9 0 Ž 0 1 . 0 1 0 4 4 - 6
P.A. ¨ an Nijnatten r Thin Solid Films 392 (2001) 282᎐288
without coating on position 3 Žoutward facing surface of the indoor pane.. The earlier proposed algorithm and two modified variants are investigated and discussed. Accurate measurement data of the directional optical properties obtained previously in the ADOPT project for a set of coated samples are used for validation of the algorithms. 2. Theory 2.1. Pseudo-Fresnel equations The specular directional reflection at an interface between two media is given by the so-called Fresnel equations w10x. In the case of a thin film system, the total reflectance of the coating is calculated using a formula that takes into account multiple reflections and interference effects w9x between the interfaces. In the present algorithms, both calculations are replaced by a more simplified approach, namely the use of equations that, by approximation, show the same behaviour as Fresnel’s equations for reflectance. We consider two types of these pseudo-Fresnel equations, namely
R P Ž . s
RS Ž . s
ž
'N 2 y sin2 Ž . y N 2 cos Ž . 'N 2 y sin2 Ž . y N 2 cos Ž .
ž
'N 2 y sin2 Ž . y cos Ž . 'N 2 y sin2 Ž . y cos Ž .
2
/
Ž 1a .
2
/
Ž 1b .
and R P Ž . s
RS Ž . s
w 1 y Ncos Ž .x 2 q K 2 cos 2 Ž . w 1 q Ncos Ž .x 2 q K 2 cos 2 Ž . w N y cos Ž .x 2 q K 2 cos 2 Ž . w N q cos Ž .x 2 q K 2 cos 2 Ž .
Ž 2a .
Ž 2b .
where N and K are the refraction and absorption coefficient, respectively Žnot to be mistaken for the real and imaginary parts of the complex refractive index.. The subscripts P and S refer to p- and s-polarisation, respectively. Eqs. Ž1a. and Ž1b. with N equal to the refractive index, is Fresnel’s equation for the power reflectance of a specular interface of an ideal dielectric material Žzero absorption. with vacuum. In the present work, Eqs. Ž1a. and Ž1b. are used to calculate the air᎐glass reflectance using for N a value of the refractive index of window glass corresponding to a total integrated Žsingle surface. reflectance at normal incidence. Different values
283
of this reflectance were used for visible daylight and solar energy properties Ž R s 0.0423 and 0.0412, respectively.. These values were obtained from the spectral single surface reflectance of a clear float glass by applying the procedure for calculating the visible daylight reflectance and direct solar energy reflectance as prescribed in EN410 w11x. Eqs. Ž2a. and Ž2b. are a somewhat crude approximation of Fresnel’s equations in the case of a strong absorbing material. In the present investigation a constant value of N is used and the absorption coefficient K is determined by the total integrated value of the spectral surface reflection of the coating at Žnear-. normal incidence. In order to make Eqs. Ž2a. and Ž2b. applicable for all types of coating, including coatings that have zero reflection Žanti-reflection coatings., the refraction coefficient was chosen to be N s 1. Both equations Eqs. Ž1a. and Ž1b. and Eqs. Ž2a. and Ž2b. are used for calculating the directional reflectance of the coating. Obviously, Eqs. Ž1a. and Ž1b. are expected to give the better approximation in the case of an all-dielectric thin film system, whereas Eqs. Ž2a. and Ž2b. should work better for a coating involving a conducting layer. The different types of angular behaviour given by these equations are demonstrated for various normal reflectance values by Fig. 1. 2.2. The in¨ estigated algorithms Three algorithms are considered, in which calculations are performed separately for p- and s-polarisation, and the final result is obtained by averaging. Algorithm 1 is the simplest and is based on the following empirical relation for directional transmittance: n
T Ž . s 1 y R g Ž . w 1 y R c Ž .x
m
Ž3.
in which R g represents the air᎐glass reflectance, R c the air᎐coating reflectance, n is the number of glass᎐air interfaces, and m the number of coated surfaces. In the preliminary investigation w6x, Eq. Ž3. was applied to the visible light transmittance of single coated glazing Ž n s m s 1.. In the present investigation, n s 3 and m s 1. The refraction or absorption coefficient necessary for the pseudo-Fresnel equation which describes the air᎐coating reflectance R c Ž ., is calculated from its value at normal incidence given by R c Ž0. s 1 y T Ž0. 1 y R g Ž0.
y3
Ž4.
where T Ž0. is the visible daylight or solar energy transmittance at normal incidence of the double glazing unit as determined by the measurements. Algorithm 2 is a slightly modified version of algo-
P.A. ¨ an Nijnatten r Thin Solid Films 392 (2001) 282᎐288
284
Fig. 1. Angular dependency of the single surface reflection for 5 different values of the reflectance at normal incidence wNs 1 in Eqs. Ž2a. and Ž2b., N s 1x.
rithm 1, obtained by introducing a term representing the internal Žbulk. transmittance: n
m
T Ž . s 1 y R g Ž . w 1 y R c Ž .x Ž . .
Ž5.
In this case R c Ž0. is the visible light or solar energy reflectance of the coated glass sample Žcoating side reflectance . and in the present investigation, n s 3 and m s 1. The internal transmittance is determined from the total transmittance of the glazing at normal incidence and is given by Ž . s 1r
1y w sin Ž .rN g x 2
'
.
coated sample are determined by the standard measurement procedure. With R c Ž0. calculated by Eq. Ž9., the internal transmittance is determined in the same manner as for algorithm 2. The total reflectance R1 of pane 1 is determined by R1Ž . s R g Ž .w 1 q T1Ž . 1Ž .x .
Ž 10 .
The total reflectance R 2 of pane 2 is determined by either Eqs. Ž1a. and Ž1b. or Eqs. Ž2a. and Ž2b.. N or K is determined from the measured Žnear-. normal value of the coated glass reflectance R 2 .
Ž6. 2.3. Validation
Algorithm 3 is based on the calculation of double glazing properties from the properties of the individual panes: T Ž . s
T1Ž . T2 Ž . 1 y R1Ž . R 2 Ž .
Ž7.
in which T1 and T2 are the transmittances of panes 1 and 2, respectively, given by Ti Ž . s
1 y R g Ž . w 1 y R x Ž .x i Ž . 1 y R g Ž . R x Ž . i2 Ž .
,
i s 1,2.
Ž8.
In our case of T1 , R x s R g , and the internal transmittance 1 is given by Eq. Ž6. with determined from the total transmittance of pane 1 at normal incidence. In the case of the second Žcoated. pane, R x s R c , which is determined by either Eqs. Ž1a. and Ž1b. or Eqs. Ž2a. and Ž2b., and N or K is determined from its value for normal incidence given by R c Ž0. s R 2 Ž0. y R g Ž0.
T2 Ž 0 . 1 y R g Ž0.
2
.
Ž9.
The reflectance R 2 Ž0. and transmittance T2 Ž0. of the
The algorithms have been tested using the known directional optical properties of 1 double glazing unit consisting of 2 clear glass panes and 8 coated double glazing units, each consisting of 1 clear glass outdoor pane and 1 coated glass indoor pane with the coating facing outwards Ža typical glazing in northern European countries.. The optical properties considered are the visible light transmittance, the direct solar energy transmittance and the solar factor g, which are the principle glazing factors. The different coatings and properties of these glazing for normal incidence are listed in Table 1 below. The coated glass samples were specially made for the ADOPT project w1x. Their composition closely resembles those of actual glazing products, but since they were custom-made, more detailed information on the layer composition was available than would have been the case with real products. Their directional reflectance and transmittance was measured in the wavelength range 250᎐2500 nm at near-normal incidence and at 30, 45, 60 and 75⬚ using state-of-the-art equipment w12,13x. The glazing factors were calculated from these data according to EN410 w11x. The thickness of the coating layers reported in Table 1 where determined by means of RBS.
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Table 1 Investigated double glazing ŽDG. units; the listed values of the glazing factors were obtained for Žnear-. normal incidence DG
Tvis
Tsol
g
Coating design
0 1 2 3 4 5 6 7 8
0.823 0.391 0.578 0.766 0.345 0.453 0.534 0.776 0.779
0.737 0.245 0.379 0.522 0.439 0.333 0.425 0.639 0.648
0.753 0.315 0.459 0.600 0.546 0.518 0.574 0.671 0.690
Without coating 32 nm SnO2 , 20 nm Ag, 1 nm NiCr and 59 nm SnO2 22 nm SnO2 , 15 nm Ag, 1 nm NiCr and 52 nm SnO2 29 nm SnO2 , 8 nm Ag, 1 nm NiCr and 29 nm SnO2 5 nm Si 10 nm TiN 5 nm TiO2 , 8 nm TiN and 10 nm TiO2 150 nm SnO2 114 nm ITO
q0.232T Ž 60⬚. q 0.134T Ž 75⬚. .
3. Results and discussion
Ž 12. By substituting T in Eqs. Ž3., Ž5. and Ž7. for the visible daylight transmittance Tvis , the direct solar energy transmittance Tsol or the solar factor g; the algorithms determine the angular dependency of these glazing factors. The errors obtained by the various combinations of algorithms with Eqs. Ž1a., Ž1b., Ž2a. and Ž2b. for the coating reflectance are summarised in Tables 2᎐4. These errors represent a weighted average according to ␦T s 0.3⌬T Ž 30⬚. q 0.27⌬T Ž 45⬚. q 0.23⌬T Ž 60⬚. q 0.14⌬T Ž 75⬚.
Ž 11.
with ⌬T the absolute difference between the measured and calculated data. Eq. Ž11. is based on the assumption that not all angles are equally important and that one must be able to calculate the hemispherical value of the glazing factor most accurately from the directional values by T hem s
r2
H0
T Ž . sin Ž 2 . d f 0.134T Ž 15⬚.
q0.232T Ž 30⬚. q 0.268T Ž 45⬚.
In Eq. Ž11. it is assumed that ⌬T Ž15⬚. s 0.5 ⌬T Ž30⬚. wsince ⌬T Ž0⬚. s 0 and T Ž15⬚. was not measuredx. On average, Algorithm 3 in combination with Eqs. Ž1a. and Ž1b. gives the best performance, but leads to large errors in some cases. Different algorithms perform better for different glazing and there seems to be a correlation with the type of coating. The different types of glazing appear to be divided into three groups. One group is formed by the silver-based coatings, DG1᎐DG3, and another by the silicon and titanium coatings, DG4᎐DG6. The third group is formed by glazing DG0 without coating, and the two tin oxide type coatings DG7 and DG 8. Their data seem to agree well with Algorithm 3 in combination with Eqs. Ž1a. and Ž1b. for all three glazing factors Žsee Fig. 2.. In the case of the silver based coatings, the solar energy properties seem to be best predicted with Algorithm 3 in combination with Eqs. Ž2a. and Ž2b.. However, the visible light transmittance can be predicted with better accuracy using a more simpler algorithm like Algorithm 2 in combination with Eqs. Ž1a. and Ž1b. Žsee Fig. 3., although for DG1 a better result is obtained with Algorithm 1. The best results for glazing units DG4᎐DG6 are obtained with Algorithm 1 in combina-
Table 2 Errors obtained in the prediction of the directional visible light transmittance Glazing unit
Algorithm 1 Eqs. Ž1a.Ž1b.
Eqs. Ž2a.Ž2b.
Algorithm 2 Eqs. Ž1a.Ž1b.
Eqs. Ž2a.Ž2b.
Algorithm 3 Eqs. Ž1a.Ž1b.
Eqs. Ž2a.Ž2b.
DG0 DG1 DG2 DG3 DG4 DG5 DG6 DG7 DG8 Average
0.019 0.035 0.027 0.015 0.016 0.033 0.018 0.026 0.011 0.022
0.052 0.006 0.019 0.021 0.017 0.008 0.024 0.061 0.037 0.027
0.015 0.022 0.014 0.005 0.029 0.028 0.027 0.023 0.011 0.019
0.050 0.021 0.036 0.031 0.053 0.050 0.055 0.060 0.037 0.044
0.006 0.024 0.018 0.011 0.029 0.026 0.023 0.015 0.008 0.018
0.037 0.017 0.027 0.021 0.050 0.045 0.048 0.047 0.023 0.035
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Table 3 Errors obtained in the prediction of the directional solar energy transmittance Glazing unit
Algorithm 1 Eqs. Ž1a.Ž1b.
Eqs. Ž2a.Ž2b.
Eqs. Ž1a.Ž1b.
Algorithm 2 Eqs. Ž2a.Ž2b.
Eqs. Ž1a.Ž1b.
Algorithm 3 Eqs. Ž2a.Ž2b.
DG0 DG1 DG2 DG3 DG4 DG5 DG6 DG7 DG8 Average
0.008 0.044 0.053 0.052 0.033 0.038 0.035 0.017 0.029 0.034
0.030 0.021 0.018 0.012 0.010 0.007 0.006 0.026 0.018 0.016
0.013 0.020 0.020 0.014 0.015 0.026 0.027 0.009 0.005 0.017
0.030 0.019 0.028 0.032 0.038 0.040 0.042 0.031 0.031 0.032
0.005 0.019 0.021 0.017 0.014 0.025 0.025 0.004 0.002 0.015
0.032 0.005 0.006 0.010 0.037 0.041 0.044 0.032 0.024 0.026
Table 4 Errors obtained in the prediction of the directional solar factor g Glazing unit
Algorithm 1 Eqs. Ž1a.Ž1b.
Eqs. Ž2a.Ž2b.
Eqs. Ž1a.Ž1b.
Algorithm 2 Eqs. Ž2a.Ž2b.
Eqs. Ž1a.Ž1b.
Eqs. Ž2a.Ž2b.
DG0 DG1 DG2 DG3 DG4 DG5 DG6 DG7 DG8 Average
0.007 0.039 0.042 0.035 0.011 0.016 0.010 0.005 0.014 0.020
0.034 0.009 0.008 0.012 0.031 0.027 0.033 0.036 0.028 0.024
0.014 0.028 0.025 0.015 0.015 0.025 0.026 0.012 0.008 0.019
0.029 0.019 0.027 0.030 0.036 0.042 0.040 0.032 0.031 0.032
0.006 0.028 0.027 0.021 0.010 0.021 0.020 0.006 0.002 0.016
0.033 0.006 0.004 0.011 0.041 0.047 0.047 0.036 0.029 0.028
tion with Eqs. Ž2a. and Ž2b. for the visible light and direct solar energy transmittance, and Eqs. Ž1a. and Ž1b. for the solar factor Žsee also Fig. 4..
Algorithm 3
The results obtained for the silver based coated glazing, the glazing coated with tin dioxide, and the non-coated glazing seem quite satisfactory. However, as
Fig. 2. Calculations Žsolid line. vs. measurements Ždotted line with markers. for the three silver-based coated glazing units.
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Fig. 3. Calculations Žsolid line. vs. measurements Ždotted line with markers. for the silicon and titanium nitride coated glazing units.
Fig. 3 clearly illustrates, a poor agreement is obtained for the remaining group DG4᎐DG6. Generally, the agreement is better for the solar properties than for the visible light transmittance. 4. Conclusion Three different algorithms for the prediction of the angular dependency of glazing factors have been investigated. These algorithms are based on a pseudo-Fresnel model in which the complex Fresnel equations and
multi-layer calculations are replaced by more simple expressions. The only data required by the algorithms are the reflectance and transmittance values obtained at Žnear-. normal incidence. The algorithms are intended to be used in combination with building simulation tools. The results obtained by these algorithms have been compared with accurately measured directional optical properties of 9 different double glazing systems. The results show that in order to obtain sufficient agreement, it is necessary to divide the various types of
Fig. 4. Calculations Žsolid line. vs. measurements Ždotted line with markers. for the uncoated and tin dioxide coated glazing units.
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glazing into different groups and use different algorithms for each group. A good agreement between measurements and algorithms has been obtained for most glazing. The agreement is better for solar transmittance and its factor than for visible light transmittance. The agreement was poor in the case of the silicon and titanium nitride coated glazing. A different algorithm or a modified version of one of the present algorithms should be developed for this group of materials. References w1x P.A. van Nijnatten. Proceedings of the Conference on Industrial Technology, Toulouse, October 1997. w2x P. Pfrommer, Energy Build. 21 Ž1994. 101᎐110.
w3x P. Polato, M. Montecchi, La rivista della staz Sper. del Vetro 5 Ž1995. 165᎐175. w4x A. Roos, J. Non-Cryst. Sol. 3651 Ž1997.. w5x M. Montecchi, P. Polato, La Rivista della Staz Sper. del Vetro 2 Ž1998. 55᎐62. w6x P.A. van Nijnatten, Thin Solid Films 351 Ž1999. 295᎐300. w7x M. Rubin, R. Powles, K. von Rottkay, Sol. Energy 66 Ž1999. 267᎐276. w8x J. Karlsson, A. Roos, Sol. Energy 69 Ž2000. 321᎐329. w9x P.A. van Nijnatten, SPIE Series 2255 Ž1994. 753᎐762. w10x M. Born, E. Wolf, Principles of Optics, 6th ed, Pergamon, Oxford, 1980. w11x EN 410, European Committee for Standardization, doc. CENrTC 129 N 123 E, 1992. w12x P.A. van Nijnatten. Proceedings of the World Renewable Energy Conference, Part I, 2000, pp. 300᎐306. w13x M.G. Hutchins, A.J. Topping, Proc. World Renewable Energy Congress VI, Ed. A. Sayigh, 2000, pp. 265᎐270.