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A note on the extinction coefficient and absorptivity of glass
Available online at www.sciencedirect.com
ScienceDirect Solar Energy 114 (2015) 196–197 www.elsevier.com/locate/solener
Brief Note
A note on the extinction coefficient and absorptivity of glass Robert E. Parkin Department of Mechanical Engineering, University of Massachusetts, Lowell, United States Received 6 February 2014; received in revised form 6 November 2014; accepted 6 January 2015 Available online 23 February 2015 Communicated by: Associate Editor Yanjun Dai
Abstract The standard technique for describing the attenuation through glass is to use the extinction coefficient in Beer’s law. What is demonstrated here is that a simpler model that uses a single number for absorptivity of the glass to normal incident radiation is fully satisfactory. Ó 2015 Elsevier Ltd. All rights reserved.
Keywords: Extinction; Coefficient; Beer’s law
The extinction coefficient of a medium is the measure of the rate of diminution of transmitted light by scattering and absorption in that medium (Derr, 1980). Assuming that the intensity of light is proportional to the distance x traveled in the medium, and proportional to the intensity I of light, then dI ¼ kI dx ð1Þ where k is known as the extinction coefficient. Integrating over distance ‘ produces Z ‘ I‘ ¼ kI dx ¼ I o ek‘ ð2Þ 0
where I 0 is the intensity of light entering the medium. Eq. (2) is known as Bouguer’s law Duffie and Beckman (2013), or the Lambert–Beer law, or simply Beer’s law. For glass of thickness L and light at angle h2 to the normal, this law becomes I L ¼ I 0 ekL= cos h2
ð3Þ
Further, only absorption occurs in glass, at least glass suitable for solar collection purposes, and scattering can be ignored, so the energy absorbed is a ¼ I 0 I L ¼ I 0 ð1 ekL= cos h2 Þ . Normalizing, so I 0 ¼ 1, http://dx.doi.org/10.1016/j.solener.2015.01.004 0038-092X/Ó 2015 Elsevier Ltd. All rights reserved.
aðk; h2 Þ ¼ 1 ekL= cos h2
ð4Þ
Physicists discuss absorptivity through a partially transparent medium in terms of the extinction coefficient and complex (as in complex numbers) refractive indices. The mediums they consider include window glass. However, glass manufacturers consider simpler models, and typically will quote a single number for absorptivity. Assuming the refractive index of air is g1 ¼ 1, for glass is g2 ¼ 1:515, and the angle of incidence on the glass is h1 , then the angle h2 in the glass as given by Snell’s law Hecht (2001) is 1 g1 h2 ¼ sin sin h1 ð5Þ g2 The reflectivity at the air/glass interface as given by Fresnel’s equation is 1 sin2 ðh2 h1 Þ cos2 ðh2 þ h1 Þ q¼ 1 þ ð6Þ 2 sin2 ðh2 þ h1 Þ cos2 ðh2 h1 Þ which takes the average of two polarizations of light, parallel and perpendicular (Duffie and Beckman, 2013). For normal incident radiation the cosine terms are unity and so
R.E. Parkin / Solar Energy 114 (2015) 196–197
absorptivity
The deviations of the simple model from the extinction curve of the points for h1 ¼ 45 and h1 ¼ 90 are minimal. Further the responses of aðk; 45Þ and aðk; 90Þ are virtually straight lines. Therefore, for clear window glass with realistic absorptivity, it is safe to assume that absorptivity can be modeled by the simple expression a ¼ cosa0h2 , where a0 is the absorptivity to normal incident radiation. If we had applied Taylor series to Eq. (4) we would have
0.12 0.1
90
0.08
45 0
0.06 0.04
1 ekL= cosh2 ¼
0.02 0
5
10
15
20
25
197
30
Fig. 1. Absorptivity at three incidence angles.
h1 !0
ð7Þ
The extinction coefficient k for window glass is in the range from 4 m 1 to 32 m1. For flat glass of thickness 3 mm and the three incident angles 0 , 45 and 90 , the absorbed energy defined by Eq. (4) is plotted as shown in Fig. 1: at the incidence value of h1 ¼ 90 ; h2 ¼ 41:3049 , the critical angle, so any light at that angle inside the sheet of glass with flat and parallel surfaces will bounce back and forth off the surfaces at that critical angle until all is absorbed. However, no light can enter the glass when the incidence angle is 90 , and for angles close to 90 the reflectance is almost 100%, as will be considered after discussion of the simple absorption model. Also shown in this figure with the symbol ‘o’ are the values of 0:0382 , for the same cos h2 three angles 0 , 45 and 90 , plotted at k ¼ 13, and 0:0915 cos h2 for the same three angles plotted at k ¼ 32. að13; 45Þ ¼ 0:0431;að13;90Þ ¼ 0:0509;að32;45Þ ¼ 0:1029 and að32;90Þ ¼ 0:1200. Compare these numbers to 0:0382 0:0382 0:0915 ¼ 0:0432; ¼ 0:0509; cos 27:823 cos 41:305 cos 27:823 0:0915 ¼ 0:1035 and ¼ 0:1218 cos 41:305
ð8Þ
but a0 – kL. With kL ¼ 13 0:003 ¼ 0:039 we have a0 ¼ 0:0382, and with kL ¼ 32 0:003 ¼ 0:096 we have a0 ¼ 0:0915. The small deviation of the simple model which occurs for high incidence angles and high absorptivity coefficients is even less significant when one looks at the total reflectivity
k
qnormal ¼ lim sin2 ðh2 h1 Þ sin2 ðh2 þ h1 Þ ¼ 0:0419
2 3 kL 1 kL 1 kL kL þ . .. ’ cos h2 2! cos h2 3! cos h2 cos h2
2
R¼qþq
ð1 qÞ ð1 aÞ
2
1 q2 ð1 aÞ2
ð9Þ
and the total transmissivity 2
T ¼
ð1 qÞ ð1 aÞ 1 q2 ð1 aÞ
2
ð10Þ
With glass having a0 ¼ 0:05 and h1 ¼ 89 ; R ¼ 0:9296 and T ¼ 0:0296 . Thus, the simple model is fully satisfactory when considering the absorptivity of glass covers for solar collectors. However, when considering attenuation of solar radiation through the atmosphere, the simple model is unsatisfactory, and the extinction coefficient model reigns (Derr, 1980). References Derr, V.E., 1980. Estimation of the extinction coefficient of clouds from multiwavelength lidar backscatter measurements. Appl. Opt. 19 (14), 2310–2314. Duffie, J.A., Beckman, W.A., 2013. Solar Engineering of Thermal Processes, Fourth ed. John Wiley, Hoboken, New Jersey. Hecht, E., 2001. Optics, Fourth ed. Addison Wesley, Boston, Mass.
ScienceDirect Solar Energy 114 (2015) 196–197 www.elsevier.com/locate/solener
Brief Note
A note on the extinction coefficient and absorptivity of glass Robert E. Parkin Department of Mechanical Engineering, University of Massachusetts, Lowell, United States Received 6 February 2014; received in revised form 6 November 2014; accepted 6 January 2015 Available online 23 February 2015 Communicated by: Associate Editor Yanjun Dai
Abstract The standard technique for describing the attenuation through glass is to use the extinction coefficient in Beer’s law. What is demonstrated here is that a simpler model that uses a single number for absorptivity of the glass to normal incident radiation is fully satisfactory. Ó 2015 Elsevier Ltd. All rights reserved.
Keywords: Extinction; Coefficient; Beer’s law
The extinction coefficient of a medium is the measure of the rate of diminution of transmitted light by scattering and absorption in that medium (Derr, 1980). Assuming that the intensity of light is proportional to the distance x traveled in the medium, and proportional to the intensity I of light, then dI ¼ kI dx ð1Þ where k is known as the extinction coefficient. Integrating over distance ‘ produces Z ‘ I‘ ¼ kI dx ¼ I o ek‘ ð2Þ 0
where I 0 is the intensity of light entering the medium. Eq. (2) is known as Bouguer’s law Duffie and Beckman (2013), or the Lambert–Beer law, or simply Beer’s law. For glass of thickness L and light at angle h2 to the normal, this law becomes I L ¼ I 0 ekL= cos h2
ð3Þ
Further, only absorption occurs in glass, at least glass suitable for solar collection purposes, and scattering can be ignored, so the energy absorbed is a ¼ I 0 I L ¼ I 0 ð1 ekL= cos h2 Þ . Normalizing, so I 0 ¼ 1, http://dx.doi.org/10.1016/j.solener.2015.01.004 0038-092X/Ó 2015 Elsevier Ltd. All rights reserved.
aðk; h2 Þ ¼ 1 ekL= cos h2
ð4Þ
Physicists discuss absorptivity through a partially transparent medium in terms of the extinction coefficient and complex (as in complex numbers) refractive indices. The mediums they consider include window glass. However, glass manufacturers consider simpler models, and typically will quote a single number for absorptivity. Assuming the refractive index of air is g1 ¼ 1, for glass is g2 ¼ 1:515, and the angle of incidence on the glass is h1 , then the angle h2 in the glass as given by Snell’s law Hecht (2001) is 1 g1 h2 ¼ sin sin h1 ð5Þ g2 The reflectivity at the air/glass interface as given by Fresnel’s equation is 1 sin2 ðh2 h1 Þ cos2 ðh2 þ h1 Þ q¼ 1 þ ð6Þ 2 sin2 ðh2 þ h1 Þ cos2 ðh2 h1 Þ which takes the average of two polarizations of light, parallel and perpendicular (Duffie and Beckman, 2013). For normal incident radiation the cosine terms are unity and so
R.E. Parkin / Solar Energy 114 (2015) 196–197
absorptivity
The deviations of the simple model from the extinction curve of the points for h1 ¼ 45 and h1 ¼ 90 are minimal. Further the responses of aðk; 45Þ and aðk; 90Þ are virtually straight lines. Therefore, for clear window glass with realistic absorptivity, it is safe to assume that absorptivity can be modeled by the simple expression a ¼ cosa0h2 , where a0 is the absorptivity to normal incident radiation. If we had applied Taylor series to Eq. (4) we would have
0.12 0.1
90
0.08
45 0
0.06 0.04
1 ekL= cosh2 ¼
0.02 0
5
10
15
20
25
197
30
Fig. 1. Absorptivity at three incidence angles.
h1 !0
ð7Þ
The extinction coefficient k for window glass is in the range from 4 m 1 to 32 m1. For flat glass of thickness 3 mm and the three incident angles 0 , 45 and 90 , the absorbed energy defined by Eq. (4) is plotted as shown in Fig. 1: at the incidence value of h1 ¼ 90 ; h2 ¼ 41:3049 , the critical angle, so any light at that angle inside the sheet of glass with flat and parallel surfaces will bounce back and forth off the surfaces at that critical angle until all is absorbed. However, no light can enter the glass when the incidence angle is 90 , and for angles close to 90 the reflectance is almost 100%, as will be considered after discussion of the simple absorption model. Also shown in this figure with the symbol ‘o’ are the values of 0:0382 , for the same cos h2 three angles 0 , 45 and 90 , plotted at k ¼ 13, and 0:0915 cos h2 for the same three angles plotted at k ¼ 32. að13; 45Þ ¼ 0:0431;að13;90Þ ¼ 0:0509;að32;45Þ ¼ 0:1029 and að32;90Þ ¼ 0:1200. Compare these numbers to 0:0382 0:0382 0:0915 ¼ 0:0432; ¼ 0:0509; cos 27:823 cos 41:305 cos 27:823 0:0915 ¼ 0:1035 and ¼ 0:1218 cos 41:305
ð8Þ
but a0 – kL. With kL ¼ 13 0:003 ¼ 0:039 we have a0 ¼ 0:0382, and with kL ¼ 32 0:003 ¼ 0:096 we have a0 ¼ 0:0915. The small deviation of the simple model which occurs for high incidence angles and high absorptivity coefficients is even less significant when one looks at the total reflectivity
k
qnormal ¼ lim sin2 ðh2 h1 Þ sin2 ðh2 þ h1 Þ ¼ 0:0419
2 3 kL 1 kL 1 kL kL þ . .. ’ cos h2 2! cos h2 3! cos h2 cos h2
2
R¼qþq
ð1 qÞ ð1 aÞ
2
1 q2 ð1 aÞ2
ð9Þ
and the total transmissivity 2
T ¼
ð1 qÞ ð1 aÞ 1 q2 ð1 aÞ
2
ð10Þ
With glass having a0 ¼ 0:05 and h1 ¼ 89 ; R ¼ 0:9296 and T ¼ 0:0296 . Thus, the simple model is fully satisfactory when considering the absorptivity of glass covers for solar collectors. However, when considering attenuation of solar radiation through the atmosphere, the simple model is unsatisfactory, and the extinction coefficient model reigns (Derr, 1980). References Derr, V.E., 1980. Estimation of the extinction coefficient of clouds from multiwavelength lidar backscatter measurements. Appl. Opt. 19 (14), 2310–2314. Duffie, J.A., Beckman, W.A., 2013. Solar Engineering of Thermal Processes, Fourth ed. John Wiley, Hoboken, New Jersey. Hecht, E., 2001. Optics, Fourth ed. Addison Wesley, Boston, Mass.