- Email: [email protected]
Simple analytic formula for the light-tube optical efficiency under overcast sky conditions
Solar Energy 194 (2019) 47–50
Contents lists available at ScienceDirect
Solar Energy journal homepage: www.elsevier.com/locate/solener
Simple analytic formula for the light-tube optical efficiency under overcast sky conditions
T
Jaromír Petržala ICA, Slovak Academy of Sciences, Dúbravská cesta 9, 845 03 Bratislava, Slovakia
A R T I C LE I N FO
A B S T R A C T
Keywords: Straight light pipe Optical efficiency Analytic formula Overcast sky
In a previous study we developed an analytic tool for express optical efficiency analysis of cylindrical light-tubes under various standardized sky conditions. In case of the clear sky luminance pattern dominated by direct sunlight, which was of main interest, the tool generally provides better efficiency predictions than some other analytic formulae used for this purpose. However, this trend is not so clearly fulfilled in the case of the standard overcast sky, because some approximations made for diffuse light were not accurate enough due to the effort to apply them on all the sky types. The case of overcast sky is often regarded by lighting engineers as the reference one. Therefore, this contribution offers a new improved and moreover simplified analytic formula for the pipe’s optical efficiency calculation, the results of which are in good coincidence with precise ray-tracing numerical computations and also with some experiments. In general, they appear to be better than those calculated according to the transmission efficiency formula proposed by CIE.
1. Introduction Tubular daylight guidance systems are now being widely installed in various types of buildings to deliver solar light into spaces with insufficient conventional glazing. This is related to the need of a simple calculation tool which would allow practicing designers to predict the performance of these systems, and thus to assess their possibilities within the interior lighting design. Some mathematical models or design tools have been proposed for this purpose (e.g., Swift and Smith (1995), Zhang and Muneer (2000), Jenkins and Muneer (2003), Jenkins et al. (2005)). The paper Petržala et al. (2018) discussed an approximate analytical model enabling fast estimation of light transmission efficiency of straight cylindrical mirror pipes in dependence on their technical parameters and luminance characteristics of the sky. Although attention was paid primarily to the clear sky conditions, which yield the highest illuminances, there have also been derived the relations for predictions of the pipe’s optical efficiency under cloudy skies defined in terms of the CIE standard types (ISO, 2004). Particularly for the overcast sky (CIE type 1) the authors stated that the developed method provides slightly worse results than the obviously used standard formula (CIE, 2006). It is just the overcast sky which lighting engineers concerning in interior lighting design usually refer to, because it represents the worst scenario. Therefore the aim of this brief note is to present an improvement of the previous calculation tool under the overcast conditions and to offer a new analytic formula for
the pipe’s optical efficiency which provides better results than the obviously used standard CIE formula. An accuracy of the new formula has been evaluated against numerical simulations as well as several measurements. The proposed expression is still relatively simple and so easy to apply in routine calculations. Therefore it could find application in elementary evaluation of interior lighting designs using light guides. Also within the standard CIE method. 2. Improved analytic formula The luminance of the standardized overcast sky is according to ISO (2004) described by the following function of zenith angle ϑ φ (ϑ)
L (ϑ) = Lz φ (0)
φ (ϑ) = 1 + 4exp
( ), −0.7 cosϑ
(1)
where Lz is the zenith luminance. Let us consider the vertical cylindrical light-tube with the height H, radius R, and internal reflectance ρ . If we take into account the sky luminance given by Eq. (1), in accordance with Eqs. (2) and (3) in Petržala et al. (2018) the luminous flux through the upper aperture of the tube will be
E-mail address: [email protected]. https://doi.org/10.1016/j.solener.2019.10.032 Received 15 March 2019; Received in revised form 11 October 2019; Accepted 15 October 2019 0038-092X/ © 2019 International Solar Energy Society. Published by Elsevier Ltd. All rights reserved.
Solar Energy 194 (2019) 47–50
J. Petržala
Nomenclature luminous flux through the upper aperture (lm) luminous flux through the lower aperture (lm) random function (unitless) fixed parameters (unitless) luminance of the sky element (cd/m2) zenith luminance (cd/m2) height of the vertical tube (m)
F1 F2 fR k1,k2 L Lz H
L
z F1 = 2π 2R2 φ (0) κ
κ = ∫0
π /2
η≈
φ (ϑ)cosϑsinϑd ϑ.
Lz φ (0)
∫0
π /2
(2)
φ (ϑ)cosϑsinϑ
{∫ 0
R
∫0
r0 ⎡ ⎣
2π
ξ=
H tanϑ R
ηCIE =
∫0 r0 [∫0 ρ N (r0,ϑ, ϕ) dϕ] dr0 ≈ πR2ρ(1 + ξ )/2 (1 + plnρξ + qln2 ρξ 2) p = 0.124573 q = 0.025514.
L
(4)
π /2
φ (ϑ)cosϑsinϑG (ϑ) d ϑ
G (ϑ) = e νtanϑ (1 + 2pν tanϑ + 4qν 2tan2 ϑ).
+ k2
ν2 (λ − ν )2
)
(8)
e νtanΘ , [1 − ν tanΘ]1/2
(9)
where Θ is the angular radius of the zenith surroundings the luminances of which are taken into account. It is recommended to take the value Θ = π/6 for the formula gives realistic efficiencies (CIE, 2006). The optical efficiencies were predicted for the three different internal relectances ρ = 0.94, 0.96, 0.98, and for the aspect ratio changing from 2 to 10. The results are shown in Fig. 1 which depicts the graph of the efficiency against an absolute value of the parameter ν . It is evident that the efficiencies determined by Eq. (8) are in very good agreement with the numerical simulations results, while the predictions made by Eq. (9) are overestimated against them. The deviations of the CIE values grow mainly for the higher aspect ratios and lower reflectances. In Fig. 2 one can see the percentage error of the values obtained by Eq. (8) and by Eq. (9) against the numerical simulations. The mean absolute percentage error of the values gained by Eq.
The coefficients p and q represent numerical values of some integrals occurring in the calculation. For the sake of brevity, let’s introduce a dimensionless parameter ν = (H / D)lnρ , where D is the tube’s diameter. The ratio H / D is commonly known as aspect ratio of a light-tube. Thus, using Eq. (4) we can write z F2 ≈ 2π 2R2 φ (0) ρ ∫0
ν 1λ−ν
An accuracy of the obtained formula for the optical efficiency was primarily validated by precise backward ray-tracing numerical simulations. Simultaneously, the same was done for the CIE formula, which can be in our notation written as
The number of beam reflections N is approximated by Eq. (6) in Petržala et al. (2018). The integrals according to r0 and ϕ can be after some algebra expressed as follows 2π
(1 + k
3. Formula validation and benchmark
}
ρ N (r0,ϑ, ϕ) dϕ⎤ dr0 d ϑ. ⎦ (3)
R
ρ γ κ (λ − ν )2
ν = (H / D)lnρ κ = 1.16424 γ = 4.603 λ = 1.988 k1 = 0.498292 k2 = 0.612324.
The integral in Eq. (2) can be computed numerically and it yields κ = 1.16424. Analogically, using Eqs. (2), (4) and (5) in Petržala et al. (2018) we get the following expression for the luminous flux through the lower aperture
F2 = 2π
fixed parameters (unitless) tube radius (m) azimuth angle (rad) fixed parameters (unitless) optical efficiency of the light-tube (unitless) zenith angle (rad) parameter characterizing the tube (unitless) internal surface reflectance (unitless) luminance gradation function (unitless)
p ,q R α γ ,κ ,λ η ϑ ν ρ φ
(5)
It is desirable to transform the last integral according to ϑ into the integral according to the variable χ = tanϑ . This requires to express the function φ (ϑ)cos3 ϑsinϑ as some function of χ . It is impossible to make such transformation exactly, but it turns out that the mentioned function can be fitted quite well by the function γ tanϑexp(−λ ϑ) , where the coefficients γ = 4.603 and λ = 1.988 are determined by minimizing the corresponding residual norm. So we get
F2 ≈ 2π 2R2
Lz ργ φ (0)
∫0
∞
χe−(λ − ν) χ (1 + 2pνχ + 4qν 2χ 2 ) dχ .
(6)
The integral in Eq. (6) is solvable analytically and it yields
F2 ≈ 2π 2R2
Lz γ ν2 ⎞ ⎛1 + k1 ν + k2 , ρ 2 (λ − ν ) ⎝ λ−ν (λ − ν )2 ⎠ φ (0) ⎜
⎟
(7)
where k1 = 4p and k2 = 24q . Finally, the efficiency η of the light transmission through the tube is given by the ratio F2/ F1, so it can be approximately determined from the following formula
Fig. 1. The optical efficiency of a vertical cylindrical light-pipe under the standard overcast sky resulting from ray-tracing numerical simulations and computed analytically by Eq. (8) and by the CIE formula (9). 48
Solar Energy 194 (2019) 47–50
J. Petržala
corrected for the transmission losses of these components. According to Carter (2002), these were specified by the transmittances 0.88 and 0.6. The mentioned comparison is illustrated in Fig. 3. The numerical values of the measured and calculated total efficiences togehter with the corresponding percentage errors are introduced in Table 1. Although the data inferred from Carter (2002) are just approximate, one can see that in most of the cases Eq. (8) offers better estimations than the CIE formula. It holds mainly at the higher aspect ratios, where the CIE formula gives overestimated efficiencies in regard to the measured ones. Some more significant deviations from the measured values may be caused by the fact, that the Carter’s data are obtained under the real sky. And the real overcast sky may, of course, differ from the idealized model (even the azimuthal symmetry may not be fulfilled, see, e.g. Riechelmann et al. (2013)). The situation is worse for light pipes with very high reflectances (over 0.99). As some experiments made by Lo Verso et al. (2011) in a sky simulator indicate, there are relatively significant discrepancies between measured and theoretically predicted efficiencies. For example, they measured the efficiency 0.84 for the pipe with the aspect ratio 3, while the numerical computations as well as Eq. (8) provide the value approximately 0.96. This is close to the value 0.97 resulting from Eq. (9). But also close to the value 0.95 calculated through simulations in Lo Verso et al. (2011). In effort to find out if this discrepancy could be, at least partially, caused by small deviations of the simulated luminance distribution from the model one, a few numerical simulations were performed. The exact luminance L (ϑ) given by Eq. (1) was loaded by random errors providing the perturbed luminance distribution
Fig. 2. Percentage errors of the efficiencies calculated by Eq. (8) and by the CIE formula (9) against the results of the numerical simulations.
Lperturb (α, ϑ) = L (ϑ) + εfR (α, ϑ) Lz ,
where α is an azimuth angle of the sky element, ε is an “error measure”, and fR (α, ϑ) is a function generating random values in the range [-1,1]. Two such perturbed luminance distributions had been generated for ε = 0.1 and 0.2. Consequently the optical efficiency of the pipe with the aspect ratio 3 and the reflectance 0.99 was computed numerically using the ray tracing method. In case of the exact model luminance, the efficiency was 0.9576. In case of the perturbed luminance, we obtained the values 0.9571 for ε = 0.1 and 0.9567 for ε = 0.2. It is evident, that small random inaccuracies of the luminance distribution have negligible influence on the pipe’s optical efficiency. So the origin of the abovementioned discrepancy is probably to be found in the optical characteristics of the pipe (e.g. in a directional dependence of the surface reflectance).
Fig. 3. Total light guide efficiency - measured and calculated. Table 1 Measured (Carter (2002), Love and Dratnal (1995)) and calculated total efficiences of light-guids (including the dome and diffuser with transmittances 0.88 and 0.6). Total efficiency (–)
Percentage error (%)
Aspect ratio
Measured
CIE form.
Eq. (8)
CIE form.
Eq. (8)
1.15 1.85 2.3 3.7 4.6 5.56 7.4 8.4 1.9 3.7 5.56
0.547 0.452 0.5 0.4 0.35 0.364 0.323 0.326 0.47 0.415 0.363
0.502 0.487 0.477 0.449 0.432 0.415 0.384 0.368 0.486 0.449 0.415
0.479 0.459 0.447 0.412 0.392 0.373 0.34 0.324 0.458 0.412 0.373
8.2 7.7 4.6 12.3 23.4 14.0 18.9 12.9 3.4 8.3 14.2
12.4 1.5 10.6 3.0 12.0 2.5 5.3 0.6 2.6 0.6 2.6
(10)
4. Conclusions The new improved analytic formula to predict the light transmission efficiency of straight mirror tubes under the overcast sky was derived. At least for not too high inner reflectances it seems to provide results which are in good agreement with the precise numerical simulations as well as measurements. In the former case, the efficiencies predicted by Eq. (8) are resolutely more accurate than the ones calculated according to the standard CIE approach. The similar statement can be concluded also from the comparison with several experimental data. With regard to these facts, Eq. (8) seems to be more appropriate to apply in “handcalculation” methods used by practicing lighting designers. However, the coincidence of the theoretical and experimental values is disrupted for the pipes with very high reflectances (over 0.99). It implies that a new theoretical model considering more accurate optical properties of the pipe’s material should be developed.
(8) is 0.5% , while for the Eq. (9) it is 9.3%. To compare the theoretical predictions with measurements, we took a few available experimental values of the tubular light-guide efficiency presented in Fig. 5 in Carter (2002). There are also involved three pipe efficiencies originating from Love and Dratnal (1995). Since Carter’s measurements were done by light-guides including a collecting dome and diffuser, the efficiencies calculated by Eqs. (8) and (9) have to be
Declaration of Competing Interest The authors declared that there is no conflict of interest. 49
Solar Energy 194 (2019) 47–50
J. Petržala
Acknowledgement
Jenkins, D., Muneer, T., Kubie, J., 2005. A design tool for predicting the performances of light pipes. Energy Build. 37, 485–492. Love, J.A., Dratnal, P., 1995. Photometric comparison of mirror light pipes. Unpublished report, University of Calgary. Lo Verso, V.R.M., Pellegrino, A., Serra, V., 2011. Light transmission efficiency of daylight guidance systems: An assessment approach based on simulations and measurements in a sun/sky simulator. Sol. Energy 85, 2789–2801. Petržala, J., Kocifaj, M., Kómar, L., 2018. Accurate tool for express optical efficiency analysis of cylindrical light-tubes with arbitrary aspect ratios. Sol. Energy 169, 264–269. Riechelmann, S., Schrempf, M., Seckmeyer, G., 2013. Simultaneous measurement of spectral sky radiance by a non-scanning multidimensional spectroradiometer (MUDIS). Meas. Sci. Technol. 24, 125501. Swift, P.D., Smith, G.B., 1995. Cylindrical mirror light pipes. Sol. Energy Mater. Sol. Cells 36, 159–168. Zhang, X., Muneer, T., 2000. Mathematical model for the performance of light pipes. Lighting Res. Technol. 32 (3), 141–146.
This work was supported by the slovak National Grant Agency VEGA No. 2/0016/16. References Carter, D.J., 2002. The measured and predicted performance of passive solar light pipe systems. Lighting Res. Technol. 34, 39–52. Commission International de l’Eclairage (CIE), 2006. Tubular daylight guidance systems. CIE Publication 173:2006, Bureau Central CIE, Vienna, Austria. International Standards Organization (ISO), 2004. Spatial distribution of daylight - CIE Standard General Sky, ISO Standard 15469, Geneva, Switzerland. Jenkins, D., Muneer, T., 2003. Modelling light-pipe performances – a natural daylighting solution. Build. Environ. 38, 965–972.
50
Contents lists available at ScienceDirect
Solar Energy journal homepage: www.elsevier.com/locate/solener
Simple analytic formula for the light-tube optical efficiency under overcast sky conditions
T
Jaromír Petržala ICA, Slovak Academy of Sciences, Dúbravská cesta 9, 845 03 Bratislava, Slovakia
A R T I C LE I N FO
A B S T R A C T
Keywords: Straight light pipe Optical efficiency Analytic formula Overcast sky
In a previous study we developed an analytic tool for express optical efficiency analysis of cylindrical light-tubes under various standardized sky conditions. In case of the clear sky luminance pattern dominated by direct sunlight, which was of main interest, the tool generally provides better efficiency predictions than some other analytic formulae used for this purpose. However, this trend is not so clearly fulfilled in the case of the standard overcast sky, because some approximations made for diffuse light were not accurate enough due to the effort to apply them on all the sky types. The case of overcast sky is often regarded by lighting engineers as the reference one. Therefore, this contribution offers a new improved and moreover simplified analytic formula for the pipe’s optical efficiency calculation, the results of which are in good coincidence with precise ray-tracing numerical computations and also with some experiments. In general, they appear to be better than those calculated according to the transmission efficiency formula proposed by CIE.
1. Introduction Tubular daylight guidance systems are now being widely installed in various types of buildings to deliver solar light into spaces with insufficient conventional glazing. This is related to the need of a simple calculation tool which would allow practicing designers to predict the performance of these systems, and thus to assess their possibilities within the interior lighting design. Some mathematical models or design tools have been proposed for this purpose (e.g., Swift and Smith (1995), Zhang and Muneer (2000), Jenkins and Muneer (2003), Jenkins et al. (2005)). The paper Petržala et al. (2018) discussed an approximate analytical model enabling fast estimation of light transmission efficiency of straight cylindrical mirror pipes in dependence on their technical parameters and luminance characteristics of the sky. Although attention was paid primarily to the clear sky conditions, which yield the highest illuminances, there have also been derived the relations for predictions of the pipe’s optical efficiency under cloudy skies defined in terms of the CIE standard types (ISO, 2004). Particularly for the overcast sky (CIE type 1) the authors stated that the developed method provides slightly worse results than the obviously used standard formula (CIE, 2006). It is just the overcast sky which lighting engineers concerning in interior lighting design usually refer to, because it represents the worst scenario. Therefore the aim of this brief note is to present an improvement of the previous calculation tool under the overcast conditions and to offer a new analytic formula for
the pipe’s optical efficiency which provides better results than the obviously used standard CIE formula. An accuracy of the new formula has been evaluated against numerical simulations as well as several measurements. The proposed expression is still relatively simple and so easy to apply in routine calculations. Therefore it could find application in elementary evaluation of interior lighting designs using light guides. Also within the standard CIE method. 2. Improved analytic formula The luminance of the standardized overcast sky is according to ISO (2004) described by the following function of zenith angle ϑ φ (ϑ)
L (ϑ) = Lz φ (0)
φ (ϑ) = 1 + 4exp
( ), −0.7 cosϑ
(1)
where Lz is the zenith luminance. Let us consider the vertical cylindrical light-tube with the height H, radius R, and internal reflectance ρ . If we take into account the sky luminance given by Eq. (1), in accordance with Eqs. (2) and (3) in Petržala et al. (2018) the luminous flux through the upper aperture of the tube will be
E-mail address: [email protected]. https://doi.org/10.1016/j.solener.2019.10.032 Received 15 March 2019; Received in revised form 11 October 2019; Accepted 15 October 2019 0038-092X/ © 2019 International Solar Energy Society. Published by Elsevier Ltd. All rights reserved.
Solar Energy 194 (2019) 47–50
J. Petržala
Nomenclature luminous flux through the upper aperture (lm) luminous flux through the lower aperture (lm) random function (unitless) fixed parameters (unitless) luminance of the sky element (cd/m2) zenith luminance (cd/m2) height of the vertical tube (m)
F1 F2 fR k1,k2 L Lz H
L
z F1 = 2π 2R2 φ (0) κ
κ = ∫0
π /2
η≈
φ (ϑ)cosϑsinϑd ϑ.
Lz φ (0)
∫0
π /2
(2)
φ (ϑ)cosϑsinϑ
{∫ 0
R
∫0
r0 ⎡ ⎣
2π
ξ=
H tanϑ R
ηCIE =
∫0 r0 [∫0 ρ N (r0,ϑ, ϕ) dϕ] dr0 ≈ πR2ρ(1 + ξ )/2 (1 + plnρξ + qln2 ρξ 2) p = 0.124573 q = 0.025514.
L
(4)
π /2
φ (ϑ)cosϑsinϑG (ϑ) d ϑ
G (ϑ) = e νtanϑ (1 + 2pν tanϑ + 4qν 2tan2 ϑ).
+ k2
ν2 (λ − ν )2
)
(8)
e νtanΘ , [1 − ν tanΘ]1/2
(9)
where Θ is the angular radius of the zenith surroundings the luminances of which are taken into account. It is recommended to take the value Θ = π/6 for the formula gives realistic efficiencies (CIE, 2006). The optical efficiencies were predicted for the three different internal relectances ρ = 0.94, 0.96, 0.98, and for the aspect ratio changing from 2 to 10. The results are shown in Fig. 1 which depicts the graph of the efficiency against an absolute value of the parameter ν . It is evident that the efficiencies determined by Eq. (8) are in very good agreement with the numerical simulations results, while the predictions made by Eq. (9) are overestimated against them. The deviations of the CIE values grow mainly for the higher aspect ratios and lower reflectances. In Fig. 2 one can see the percentage error of the values obtained by Eq. (8) and by Eq. (9) against the numerical simulations. The mean absolute percentage error of the values gained by Eq.
The coefficients p and q represent numerical values of some integrals occurring in the calculation. For the sake of brevity, let’s introduce a dimensionless parameter ν = (H / D)lnρ , where D is the tube’s diameter. The ratio H / D is commonly known as aspect ratio of a light-tube. Thus, using Eq. (4) we can write z F2 ≈ 2π 2R2 φ (0) ρ ∫0
ν 1λ−ν
An accuracy of the obtained formula for the optical efficiency was primarily validated by precise backward ray-tracing numerical simulations. Simultaneously, the same was done for the CIE formula, which can be in our notation written as
The number of beam reflections N is approximated by Eq. (6) in Petržala et al. (2018). The integrals according to r0 and ϕ can be after some algebra expressed as follows 2π
(1 + k
3. Formula validation and benchmark
}
ρ N (r0,ϑ, ϕ) dϕ⎤ dr0 d ϑ. ⎦ (3)
R
ρ γ κ (λ − ν )2
ν = (H / D)lnρ κ = 1.16424 γ = 4.603 λ = 1.988 k1 = 0.498292 k2 = 0.612324.
The integral in Eq. (2) can be computed numerically and it yields κ = 1.16424. Analogically, using Eqs. (2), (4) and (5) in Petržala et al. (2018) we get the following expression for the luminous flux through the lower aperture
F2 = 2π
fixed parameters (unitless) tube radius (m) azimuth angle (rad) fixed parameters (unitless) optical efficiency of the light-tube (unitless) zenith angle (rad) parameter characterizing the tube (unitless) internal surface reflectance (unitless) luminance gradation function (unitless)
p ,q R α γ ,κ ,λ η ϑ ν ρ φ
(5)
It is desirable to transform the last integral according to ϑ into the integral according to the variable χ = tanϑ . This requires to express the function φ (ϑ)cos3 ϑsinϑ as some function of χ . It is impossible to make such transformation exactly, but it turns out that the mentioned function can be fitted quite well by the function γ tanϑexp(−λ ϑ) , where the coefficients γ = 4.603 and λ = 1.988 are determined by minimizing the corresponding residual norm. So we get
F2 ≈ 2π 2R2
Lz ργ φ (0)
∫0
∞
χe−(λ − ν) χ (1 + 2pνχ + 4qν 2χ 2 ) dχ .
(6)
The integral in Eq. (6) is solvable analytically and it yields
F2 ≈ 2π 2R2
Lz γ ν2 ⎞ ⎛1 + k1 ν + k2 , ρ 2 (λ − ν ) ⎝ λ−ν (λ − ν )2 ⎠ φ (0) ⎜
⎟
(7)
where k1 = 4p and k2 = 24q . Finally, the efficiency η of the light transmission through the tube is given by the ratio F2/ F1, so it can be approximately determined from the following formula
Fig. 1. The optical efficiency of a vertical cylindrical light-pipe under the standard overcast sky resulting from ray-tracing numerical simulations and computed analytically by Eq. (8) and by the CIE formula (9). 48
Solar Energy 194 (2019) 47–50
J. Petržala
corrected for the transmission losses of these components. According to Carter (2002), these were specified by the transmittances 0.88 and 0.6. The mentioned comparison is illustrated in Fig. 3. The numerical values of the measured and calculated total efficiences togehter with the corresponding percentage errors are introduced in Table 1. Although the data inferred from Carter (2002) are just approximate, one can see that in most of the cases Eq. (8) offers better estimations than the CIE formula. It holds mainly at the higher aspect ratios, where the CIE formula gives overestimated efficiencies in regard to the measured ones. Some more significant deviations from the measured values may be caused by the fact, that the Carter’s data are obtained under the real sky. And the real overcast sky may, of course, differ from the idealized model (even the azimuthal symmetry may not be fulfilled, see, e.g. Riechelmann et al. (2013)). The situation is worse for light pipes with very high reflectances (over 0.99). As some experiments made by Lo Verso et al. (2011) in a sky simulator indicate, there are relatively significant discrepancies between measured and theoretically predicted efficiencies. For example, they measured the efficiency 0.84 for the pipe with the aspect ratio 3, while the numerical computations as well as Eq. (8) provide the value approximately 0.96. This is close to the value 0.97 resulting from Eq. (9). But also close to the value 0.95 calculated through simulations in Lo Verso et al. (2011). In effort to find out if this discrepancy could be, at least partially, caused by small deviations of the simulated luminance distribution from the model one, a few numerical simulations were performed. The exact luminance L (ϑ) given by Eq. (1) was loaded by random errors providing the perturbed luminance distribution
Fig. 2. Percentage errors of the efficiencies calculated by Eq. (8) and by the CIE formula (9) against the results of the numerical simulations.
Lperturb (α, ϑ) = L (ϑ) + εfR (α, ϑ) Lz ,
where α is an azimuth angle of the sky element, ε is an “error measure”, and fR (α, ϑ) is a function generating random values in the range [-1,1]. Two such perturbed luminance distributions had been generated for ε = 0.1 and 0.2. Consequently the optical efficiency of the pipe with the aspect ratio 3 and the reflectance 0.99 was computed numerically using the ray tracing method. In case of the exact model luminance, the efficiency was 0.9576. In case of the perturbed luminance, we obtained the values 0.9571 for ε = 0.1 and 0.9567 for ε = 0.2. It is evident, that small random inaccuracies of the luminance distribution have negligible influence on the pipe’s optical efficiency. So the origin of the abovementioned discrepancy is probably to be found in the optical characteristics of the pipe (e.g. in a directional dependence of the surface reflectance).
Fig. 3. Total light guide efficiency - measured and calculated. Table 1 Measured (Carter (2002), Love and Dratnal (1995)) and calculated total efficiences of light-guids (including the dome and diffuser with transmittances 0.88 and 0.6). Total efficiency (–)
Percentage error (%)
Aspect ratio
Measured
CIE form.
Eq. (8)
CIE form.
Eq. (8)
1.15 1.85 2.3 3.7 4.6 5.56 7.4 8.4 1.9 3.7 5.56
0.547 0.452 0.5 0.4 0.35 0.364 0.323 0.326 0.47 0.415 0.363
0.502 0.487 0.477 0.449 0.432 0.415 0.384 0.368 0.486 0.449 0.415
0.479 0.459 0.447 0.412 0.392 0.373 0.34 0.324 0.458 0.412 0.373
8.2 7.7 4.6 12.3 23.4 14.0 18.9 12.9 3.4 8.3 14.2
12.4 1.5 10.6 3.0 12.0 2.5 5.3 0.6 2.6 0.6 2.6
(10)
4. Conclusions The new improved analytic formula to predict the light transmission efficiency of straight mirror tubes under the overcast sky was derived. At least for not too high inner reflectances it seems to provide results which are in good agreement with the precise numerical simulations as well as measurements. In the former case, the efficiencies predicted by Eq. (8) are resolutely more accurate than the ones calculated according to the standard CIE approach. The similar statement can be concluded also from the comparison with several experimental data. With regard to these facts, Eq. (8) seems to be more appropriate to apply in “handcalculation” methods used by practicing lighting designers. However, the coincidence of the theoretical and experimental values is disrupted for the pipes with very high reflectances (over 0.99). It implies that a new theoretical model considering more accurate optical properties of the pipe’s material should be developed.
(8) is 0.5% , while for the Eq. (9) it is 9.3%. To compare the theoretical predictions with measurements, we took a few available experimental values of the tubular light-guide efficiency presented in Fig. 5 in Carter (2002). There are also involved three pipe efficiencies originating from Love and Dratnal (1995). Since Carter’s measurements were done by light-guides including a collecting dome and diffuser, the efficiencies calculated by Eqs. (8) and (9) have to be
Declaration of Competing Interest The authors declared that there is no conflict of interest. 49
Solar Energy 194 (2019) 47–50
J. Petržala
Acknowledgement
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This work was supported by the slovak National Grant Agency VEGA No. 2/0016/16. References Carter, D.J., 2002. The measured and predicted performance of passive solar light pipe systems. Lighting Res. Technol. 34, 39–52. Commission International de l’Eclairage (CIE), 2006. Tubular daylight guidance systems. CIE Publication 173:2006, Bureau Central CIE, Vienna, Austria. International Standards Organization (ISO), 2004. Spatial distribution of daylight - CIE Standard General Sky, ISO Standard 15469, Geneva, Switzerland. Jenkins, D., Muneer, T., 2003. Modelling light-pipe performances – a natural daylighting solution. Build. Environ. 38, 965–972.
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