Solar light transmission of polymer optical fibers

Available online at www.sciencedirect.com

Solar Energy 83 (2009) 2039–2049 www.elsevier.com/locate/solener

Solar light transmission of polymer optical fibers Murat Tekelioglu a,*, Byard D. Wood b b

a Mechanical Engineering Department, University of Nevada, Reno, NV 89557, USA Mechanical and Aerospace Engineering Department, Utah State University, Logan, UT 84322, USA

Received 29 October 2008; received in revised form 16 July 2009; accepted 6 August 2009 Available online 2 September 2009 Communicated by: Associate Editor J.-L. Scartezzini

Abstract Light transfer (10 m) has been shown in recent experiments that used large-core optical fibers. Theoretical models are not extensive, however, and a further correlation between the theory and experiments has not been given. In this paper, straight and bent fiber subsystem models are introduced with skew and meridional rays to predict the light transmission of POFs (plastic optical fibers). Such fibers have been realized, for example, in HSL (hybrid solar lighting) systems. The purpose of this paper is to combine the straight and bent fiber subsystems to estimate the light transmission of HSL systems. It is shown that meridional rays, for which the optical-loss parameters were estimated, better represent the experimental results compared to skew rays (5:3% vs 24:7% of %-difference). Model predictions were compared with the results of a commercial software. Sensitivity analysis on the subsystems indicated the most-to-least significant parameters in light transmission. Ó 2009 Elsevier Ltd. All rights reserved. Keywords: Hybrid lighting; Optical fibers; Light transmission; Solar lighting

1. Introduction POFs have been preferred in solar lighting applications because of their low-cost and high-flexibility. HSL systems with POFs have been demonstrated in transfer of the visible part of the optical spectrum into luminaries for home/ office space lighting (ORNL, 2004). Optical and geometrical properties of the fiber, input condition (arrangement of the rays, incident angle, power profile), type of the rays (skew and meridional), and number and configuration of the straight and bent sections affect the light transmission of an HSL system. A hybrid day-lighting system with halide and fluorescent back-up lighting has been shown (Tsangrassoulis et al., 2005). Although substantial energy savings were reported * Corresponding author. Address: Mechanical Engineering Department, University of Nevada, 1664 N Virginia St., Reno, NV 89557, USA. Tel.: +1 775 376 3178. E-mail address: [email protected] (M. Tekelioglu).

0038-092X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2009.08.002

(55.4% for January and 58.8% for June), the system cost ($/lm) was stated high. This was attributed to the low collection efficiency of the large-core liquid optical fiber. Experimental light transmission with a point light source was given for five different fibers (Feuermann et al., 2002). They showed that the light transmission substantially decreased for rays entering the fibers beyond their nominal NA, for example, 16% vs 75% for L=d ¼ 3000   and NA ¼ 0:66 between h ¼ 30 and 48 . A test room was evaluated experimentally with a roof-mount tubular tube having 0:25 m of diameter and 1 m of length (Paroncini et al., 2007). Average illuminance levels were measured for select months through six sensors installed inside the room. Light pipes, also called air-clad POFs, can collect the near-graze incident rays without refraction. A solarcanopy system included light pipes to transfer the light into the building interior (Rosemann et al., 2008). Direct window-collected and transmitted solar light illuminance values were compared. An analytical flux confinement diagram (FCD), which was based on the critical angle

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M. Tekelioglu, B.D. Wood / Solar Energy 83 (2009) 2039–2049

Nomenclature Letters BRDF CCW CW c D d HSL HSLS k k L l l m MPD N NA n n POF R R RPP r T V X Z

Bidirectional reflection distribution function (sr1) Counter clock-wise Clock-wise Constants Defects loss coefficient () Diameter (mm) Hybrid solar lighting Hybrid solar lighting system Wave vector (rad/nm) Radial ray () Length (m) Angular ray () Length (mm) Ratio of bend radii () Modal power distribution () Number of rays () Numerical aperture () Refractive index () Number of reflection points () Plastic optical fiber Radius, bend (mm) Reflectivity () Radiant power profile () Radius (mm) Transmitted-refracted light () V-number () Optical or geometrical fiber parameter Sensitivity coefficient ()

Greek a D k

Loss coefficient (mm1) Finite changes Wavelength (nm)

information of a light pipe was defined on rectangular light pipes (Derlofske and Hough, 2004). A flux transmission difference of 5% was observed between their FCD model and computer modeling for a light pipe having a bend angle of 45°, thickness of 6.35 mm, and width of 25.4 mm. In their model, they only considered the loss due to refraction at the core–air interface of the light pipe. In the other paper, raytracing with Harvey scatter model was shown to take an order of magnitude less time compared to the re-built model of ten thousand point cloud (Kaminski and Koshel, 2003). Harvey model calculations, however, was stated to require about 5% more time compared to the nominal (smooth-surface) reflection design calculations. Planar or slab light pipes were studied, with the sharp, moderate, thinning, and broadening geometry of bends, in an attempt to parameterize the light transmission (Koshel and Gupta, 2005). Filament and arc-based

U u u r s h C

Radiant power () Bend angle (degree) Angle, internal (degree) Rms roughness height (nm) Transmitted light () Angle (degree) Fraction of power ()

Subscripts a Absorption a Aperture a Incident air Air b Bent cl Clad co Core cr Critical design Design f Fresnel i Reflection point inc Incident max Maximum min Minimum ? Perpendicular r Refracted s Scattering s Straight user User-defined 0 Maximum Superscripts 0 Unit piece of fiber m Parameter

light sources were modeled with direct-measured and rayassigned radiance profiles (Kaminski et al., 2002). The authors showed, for a light source located at the focus of a parabolic reflector, the difference in intensity between 0 and 639 hrs of operation from a commercial ray-trace software. Effect of change of the fiber external refractive index on the bent fiber light transmission was studied for a fiber: rco ¼ 0:5 mm, nco ¼ 1:492, and ncl ¼ 1:417 (Arrue et al., 1998). They compared their results with experiments after stripping off the clad layer and exposing the fiber to a distilled water (n ¼ 1:333) and to a solution containing 85% glucose (n ¼ 1:44). Light transmission models on the straight and bent fibers were included as design guidelines (Saraiji et al., 1996). The authors of that paper introduced a coefficient for the overall optical-loss of the fiber which, however, did not account for the specular and spectral dependencies of the bulk and interface losses of the fiber.

M. Tekelioglu, B.D. Wood / Solar Energy 83 (2009) 2039–2049

The straight fiber specular (u)-spectral (k) light loss coefficients were given for the single skew and meridional rays (Tekelioglu and Wood, 2005). Their model predicted the experimental data available for rays that make a small  angle of u (u < 20 ) with the optical fiber axis. Although there have been separate studies of individual straight and bent fibers, the combination of both received little attention. In this paper, straight and bent fiber subsystem models are introduced and evaluated to predict the light transmission of POFs. Then, the two model results are combined to predict the light transmission of HSL systems. This is a parameterized study of the straight and bent fibers which are combined together. The fiber interface loss due to roughness and defects was incorporated from the present model. The results are compared with both the experiments and commercial software. Parametric sensitivity analysis indicates the most-to-least significant light transmission parameters.

2041

Solar light

Solar Collector Straight 1 Bend 1

Input POF

Straight 2

Luminaire

AC Power + Photo-sensor

Bend 2 Straight 3 Fig. 1. Layout of an HSL distribution system.

2. Methodology

2.1. Subsystems

In fiber optic systems, the approximate criterion nco rco ffi 2  rco ð¼ d co Þ >> k is used for the geometrical optics condition, or more specifically V-number is defined (Cancellieri and Ravaioli, 1984)

A typical HSL system is shown in Fig. 1. Light transmission was predicted via the straight and bent fiber subsystems (See Fig. 2). The optical and geometrical parameters for the subsystems were described in Table 1. In the present model, r and D on the bent fiber subsystems were not defined as the refracted amount of light over bends surpassed the light loss due to roughness at the core–clad interface (e.g., 63.0% vs 1.3% for nco ¼ 1:498, ncl ¼ 1:35,  r ¼ 5 nm, k ¼ 580 nm, h ¼ 45 ).

V ¼

2p rco ðn2co  n2cl Þ1=2 k

for V >> 1

ð1Þ

where nco and ncl are the refractive indices of the fiber core and clad materials, respectively. A large-core POF with rco ¼ 6:3 mm, nco ¼ 1:498, ncl ¼ 1:35, and k ¼ 780 nm yields, e.g., V ¼ 32; 950 which implies multi-mode fiber, or solution with geometrical optics. When there are multi modes (thousands or millions), the results from EMW (electromagnetic wave) picture can be inaccurate as proved by the experiments but the geometrical optics results can easily be compared with the experiments (Pollock and Lipson, 2003). The radiant power of a ray over the visible spectrum is denoted with Uðhinc Þ: The power distribution at a straight or bent fiber subsystem inlet was given from the Snell’s law of refraction ninc sin hinc ¼ nco sin hco

ð2Þ

where ninc is the refractive index for air (ninc ¼ 1), hinc is the angle of incidence in air (degree), and hco is the angle of refraction inside the fiber core (degree). Analogous modal power distributions (MPDs) have been found previously experimentally (Loke and McMullin, 1990). Fresnel reflectivity (Rf ) loss was quantified at the inlet and exit ends of the POF (Pollock and Lipson, 2003), " # 1 tan2 ðhinc  hco Þ sin2 ðhinc  hco Þ ð3Þ Rf ¼ þ 2 tan2 ðhinc þ hco Þ sin2 ðhinc þ hco Þ

2.1.1. Straight fiber subsystem Light propagation inside an optical fiber can be described using skew and meridional rays as shown in Fig. 3 (Tekelioglu and Wood, 2005). Meridional rays cross the fiber axis after each reflection and skew rays spiral around inside the optical fiber. Unlike their counterpart, skew rays possess an angular momentum. The single skew and meridional ray light loss coefficients were given, respectively, on the straight fiber subsystem (Tekelioglu and Wood, 2005) 2

a¼

ðaa þ as Þ f1  Rf exp½ð2k ? rÞ g tan u þ 1=2 cos u 2ðr2co  d 2 Þ D tan u þ 2ðr2co  d 2 Þ1=2

ð4Þ

2

a¼

ðaa þ as Þ f1  Rf exp½ð2k ? rÞ g tan u þ cos u 2ðrco þ dÞ D tan u þ 2ðrco þ dÞ

ð5Þ

where aa is the fiber absorption loss coefficient (mm1), as is the Rayleigh scattering loss coefficient (mm1), k ? is the perpendicular component of the wave vector at the core– clad interface (rad/nm), r is the core–clad interface rms

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M. Tekelioglu, B.D. Wood / Solar Energy 83 (2009) 2039–2049

System: HSL Subsystem 1: Straight fiber

Subsystem 2: Bent fiber

(a)

Light out

Light In

Parameters: rco , rcl , nco , ncl , α a , α s , σ , D , Ls

Parameters: rco , rcl , nco , ncl , α a , α s , R , ϕb

Light In

ϕb R

Light out

(c)

(b)

Fig. 2. (a) Arrangement of an HSL system. POF characterization by optical and geometrical properties: (b) Straight fiber, (c) Bent fiber subsystem.

Table 1 Typical ranges for the subsystem parameters. Symbol

Parameter

Typical rangea

rco rcl nco ncl aa as r D Ls R=rco ub

Core radius Clad radius Core refractive index Clad refractive index Absorption loss coefficient Scattering loss coefficient Rms roughness height Interface defects loss coefficient Length (Straight fiber subsystem) Bend radius (Bent fiber subsystem) Bend angle (Bent fiber subsystem)

ð0:1  20Þ mm > rco ð> 1:0  1:6Þ < nco ð103  106 Þmm1 ð104  107 Þmm1 ð0  14Þnm ð0  105 Þ ð0  10Þm ð2  80Þ ð0  360Þ

a

Values encountered in practice.

θ θ l

l′ y

ϕ d rco x

2 co −

2 1/2

(r d ) (a)

l′

l ϕ

rco

y

d

x

roughness height (nm), D is the core–clad interface defects loss coefficient (dimensionless), Rf is the Fresnel reflectivity (Rf ¼ 1 for TIR rays), and u, rco , d, and h are described in Fig. 3. It was presumed in Eqs. (4) and (5) that some of bulk and surface scattered light being trapped inside and which can exit the fiber is ignored. More on the individual loss factors contributing to the overall loss of an optical fiber can be found (Tekelioglu and Wood, 2005). Through the available commercial ray-trace software (See TracePro, 2006), the %-loss at the core–clad interface of the fiber (at each turning point) can be quantified, however, interface parameters r and D are not direct inputs by the user. This paper defines, in the physical model, these parameters explicitly. The straight fiber subsystem light transmission was calculated from PN s PN s k¼1 l¼1 Uk;l expðaðu; d; kÞk;l  Ls Þ ð6Þ ss ¼ PN s PN s k¼1 l¼1 Uk;l where ss is the transmitted light (dimensionless), k is the radial ray coordinate number (dimensionless), l is the angular ray coordinate number (dimensionless), Ls is the length of the subsystem (m), a is given by Eqs. (4) and (5), U is the normalized ray power from (k; l) (dimensionless), and N s is the number of angular and radial rays. In Eqs. (4) and (5), for Ls P l0 all three terms and for Ls < l0 , only the bulk losses (first term) are considered in the FORTRAN code. No range limit on Ls was set.

(b)

Fig. 3. Single ray geometry showing l the path length, l0 the unit fiber length, rco the fiber core radius, and d the local radius. (a) Skew ray. (b) Meridional ray.

2.1.2. Bent fiber subsystem Amount of refracted light (T r ) at a turning point on the fiber was found from

M. Tekelioglu, B.D. Wood / Solar Energy 83 (2009) 2039–2049

½cos h þ

ðcos2

h

cos2

1=2

hcr Þ

50

1=2 2



ð7Þ

where the polarization state of the incident electric field is linearly polarized (LP) (s and p-polarized), h is the incident angle at a turning point (degree), and hcr is the critical angle, hcr ¼ a sinðncl =nco Þ (degree) of the fiber. This equation is valid for h < hcr (or as long as the Snell’s law holds), assumes that all refracted light into the clad is lost, and is the scalar or weakly guiding approximation form of the Fresnel equation, nco ffi ncl (Arrue et al., 1998), (Arrue and Zubia, 1996). Eq. (7) was implemented in the analysis of multi-mode bent fibers (Arrue et al., 1998). Thus, the bent fiber light transmission (sb ) was given as n  Nb P Nb P Q Uk;l ð1  Ca;k;l;i  Cs;k;l;i Þ  ð1  T r;k;l;i Þ k¼1 l¼1 i¼1 sb ¼ ð8Þ Nb P Nb P Uk;l k¼1 l¼1

where n is the number of reflection points, N b is the number of radial and angular rays, and Ca and Cs are, respectively, the absorbed and scattered fractions of power between the ith and ði  1Þth turning points, Ca;s ¼ expðaa;s  l0 Þ. The tunneling losses of the TIR rays (<1%) were ignored at turning points (Arrue et al., 1998), (Arrue and Zubia, 1996). Coordinates x-, y-, and z- of a turning point was found by numerical iteration after describing the core–clad interface as a torus. An initial distance of ð5%Þ  rco taken at a turning point was incremented by 1% of rco in the direction of the outgoing ray vector as calculated from the incoming ray and surface normal vectors. It was noted that available commercial CAD packages such as (TracePro, 2006) have more accurate analytical means in locating the turning point coordinates. Fiber manufacturers specify a critical value for the bend radius to avoid substantial light loss on the order of > ð3  5Þ%. For example, 3M (LF120B) Rcr =rco ¼ 16, Lumenyte (SEL 500) Rcr =rco > 24, and PolyOptics (Poly 120) Rcr =rco > 12.

1.2

(b)

1.1

45

1

40

0.9 35

0.8

30

0.7

NAdesign = 0.3 NAactual = 0.4 (θinc,user = 24º)

25 20 15

0.6 0.5 0.4 0.3

10

(a)

0.2

5

Terrestrial Solar Irradiance 2 (W/(m nm))

4 cos hðcos2 h  cos2 hcr Þ

Light Source Irradiance 2 (W/(m nm))

T r ðhÞ ¼

2043

0.1

0 0 300 350 400 450 500 550 600 650 700 750 800

λ(nm) Fig. 4. Spectral irradiance. (a) Cogent Light (CL 350) illuminator. (b) Terrestrial–solar direct–normal (AM = 1.5).

Arrangement of rays, incident angle, and angular power profile assigned to the rays are shown in Fig. 5 where RPPi is the radiant power (flux) profile shown with Ui . RPP1, RPP2, and RPP3 define, respectively, the angular uniform, angular concave, and angular convex power profiles. The angular and spatial power profiles are the same due to equal number of rays striking the optical fiber inlet. CCW (Conical-converging), collimated, and CW (Conical-diverging) arrangement of rays with these power pro-

(a)

(c)

(b) i) Meridional rays

2.2. Input condition The spectral irradiance of the fiber-optic illuminator  (Cogent Light, CL 350) with NA ¼ 0:4 (hinc;user ¼ 24 ) (NAdesign ¼ 0:3) and of sunlight with air mass (AM) = 1.5 are shown in Fig. 4. The spatial power distribution (U) at the illuminator exit port was modeled uniform (Wavien, 2006) with meridional rays. Power input at the fiber inlet, considered in the results of Figs. 7–9, was weighted over the visible spectrum. The spectral-specular case with spectral-specular input of the both light source power and optical properties of the fiber was given in Fig. 10. The shape of the spatial power profile at the fiber inlet is determined by the type of solar collector (paraboloidal, hyperboloidal, e.g.) and fiber arrangement (single or bundle).

(d)

(e) ii) Skew rays

Φ(θ inc ) 1

Φ(θ inc )

RPP1 1

RPP3 RPP2

θinc,user

(f)

θinc,user

θinc,user

(g)

θinc,user

iii) Radiant power profiles Fig. 5. Arrangement of rays (arrows) at subsystem inlet. (i) Meridional   rays: (a) Conical converging, hinc;user > 0 , (b) Collimated, hinc;user ¼ 0 , and  c) Conical diverging, hinc;user < 0 . (ii) Skew rays: (d) CCW for conicalconverging rays, (e) CW for conical-diverging rays, iii) Radiant power profiles: (f) conical-converging rays, (g) conical-diverging rays.

M. Tekelioglu, B.D. Wood / Solar Energy 83 (2009) 2039–2049 40 30 20

%-Difference

100

Skew (Present model) Skew Max (Present model) Meridional (Commercial) Meridional Max (Commercial) Meridional (Present model) Meridional Max (Present model)

96

10 0 Setup 1

Setup 2

Setup 3

Transmitted light (%)

2044

92

150.5 lm/W 149.4 lm/W

88 84 80 76 72 68

RPP1 (m=0) RPP2 (m=2) RPP2 (m=4) RPP2 (m=6) RPP3 (m=2) RPP3 (m=4) RPP3 (m=6)

Top

130.8 lm/W

Side

64

-10

146.6 lm/W

117.1 lm/W 110.7 lm/W 106.7 lm/W

60 56

-20

I

Transmitted light (%)

Transmitted light (%)

100 96 92 88 84 80 76 72 68 64 60 56 52 48 44

rco = 1.0 mm RPP1 rco = 5.0 mm RPP1 rco = 10.0 mm RPP1 rco = 1.0 mm RPP2 rco = 5.0 mm RPP2 rco = 10.0 mm RPP2

III

IV

V

VI

149.2 lm/W RPP1 (m=0) RPP2 (m=2) RPP2 (m=4) RPP2 (m=6) RPP3 (m=2) RPP3 (m=4) RPP3 (m=6)

146.3 lm/W 138.9 lm/W

Top

105.6 lm/W

Side

II

III

79.5 lm/W 69.3 lm/W 63.6 lm/W IV

V

VI

(b) Common Parameters and Values

II

III

IV

V

(a)

Transmitted light (%)

100 96 92 88 84 80 76 72 68 64 60 56 52 48 44 40 36 32

I

I

100 96 92 88 84 80 76 72 68 64 60 56 52 48 44 40

II

(a)

Fig. 6. Present model and commercial software %-differences as directly measured values and as maximum %-differences due to experimental uncertainty.

II

rcl = 4.3mm

nco = 1.6

ncl = 1.45

α a = 0.00001mm −1

α s = 0.000001mm−1

D = 0.000005

θinc,user = 23ο

Fig. 8. An HSL system light transmission with U: (a) skew rays (CCW), (b) meridional rays (conical-converging). I First bent section (R1 =rco ¼ 55;   ub;1 ¼ 135 ), II second bent section (R2 =rco ¼ 25; ub;2 ¼ 85 ), III third bent  section (R3 =rco ¼ 20;, ub;3 ¼ 95 ), IV fourth bent section (R4 =rco ¼ 15;  ub;4 ¼ 90 ), V straight section (Ls ¼ 5:0m; r ¼ 6 nm), VI HSL system (overall).

rco = 1.0 mm RPP1 rco = 5.0 mm RPP1 rco = 10.0 mm RPP1 rco = 1.0 mm RPP2 rco = 5.0 mm RPP2 rco = 10.0 mm RPP2

I

rco = 4mm

III

IV

V

(b) Common Parameters and Values nco = 1.4

ncl = 1.29

α a = 0.00001mm −1

α s = 0.000001mm−1

D = 0.000002

θinc,user = 30ο

RPP1 ( m = 0 )

RPP2 ( m = 4 )

Fig. 7. An HSL system light transmission with rco : (a) skew rays (CCW), (b) meridional rays (conical-converging). I First bent section (R1 =rco ¼ 60;   ub;1 ¼ 110 ), II second bent section (R2 =rco ¼ 45; ub;2 ¼ 95 ), III third bent  section (R3 =rco ¼ 35, ub;3 ¼ 115 ), IV straight section (Ls ¼ 5:4m; r ¼ 5 nm)), V HSL system (overall).

files were considered on the skew (meridional) rays. The RPP2 and RPP3 power profiles of Fig. 5(f) were represented by the functions Uðhinc Þ ¼ U0  ½cosððp=2=hinc;user Þ  hinc Þm

for RPP2

m

for RPP3

Uðhinc Þ ¼ U0  ½sinððp=2=hinc;user Þ  hinc Þ

and ð9Þ

where U0 is the power amplitude (U0 ¼ 1), Uðhinc Þ is the power from a specific incident angle hinc (0 6 Uðhinc Þ 6 U0 ), hinc;user is the half-cone angle for the incident rays (degree), and m is a parameter for the width  of the power profile. For collimated rays, hinc;user ¼ 0 , functions similar to those in Eq. (9) were introduced replacing the hinc dependency by the spatial coordinates y and z. The power of a ray inside the fiber core was given from Eq. (9) with hinc replaced by hco and hinc;user by hinc;co where

M. Tekelioglu, B.D. Wood / Solar Energy 83 (2009) 2039–2049 100 92

150.5 lm/W 149.4 lm/W

88 84 80 76 72 68

Transmitted light (%)

Transmitted light (%)

96

RPP1 (m=0) RPP2 (m=2) RPP2 (m=4) RPP2 (m=6) RPP3 (m=2) RPP3 (m=4) RPP3 (m=6)

64

Top

146.7 lm/W

Side

133.4 lm/W 121.6 lm/W 116.4 lm/W 113.3 lm/W

60 56

I

II

III

IV

V

VI

(a)

2045

80 -6 -1 76 □, ○: αa = 1.122·10 ·exp(0.0058·λ) mm 72 αs = 5.492·10-15·λ3.200 , D(λ) = 3.371·10-11·λ2 - 2.661·10-8·λ + 6.382·10-6 68 -5 -1 -6 -1 -6 64 ●, ■: αa = 10 mm , αs = 10 mm , D = 10 60 ■, □ Φ (θ,λ) = 1 (m=0) 56 m 52 ●, ○ Φ (θ,λ) = Φ 0[cos(θ)] ·AM1.5(λ) (m=4) 48 44 40 36 32 28 24 20 16 12 8 4 0 380 420 460 500 540 580 620 660 700 740 780

(a) 150.6 lm/W RPP1 (m=0) RPP2 (m=2) RPP2 (m=4) RPP2 (m=6) RPP3 (m=2) RPP3 (m=4) RPP3 (m=6)

149.5 lm/W 146.0 lm/W

Top 118.8 lm/W 96.7 lm/W 86.1 lm/W 79.3 lm/W

Side

I

II

Light output (lm/W)

Transmitted light (%)

λ (nm) 100 96 92 88 84 80 76 72 68 64 60 56 52 48 44 40

III

IV

V

VI

(b) Common Parameters and Values

rco = 4mm

rcl = 4.3mm

nco = 1.6

ncl = 1.45

α a = 0.00001mm −1

α s = 0.000001mm−1

D = 0.000005

θinc,user = −23ο

Φ 0·[cos(θ)]m·AM1.5(λ)

(m=4)

Luminous efficacy, IESNA

420

460

500

540

580

620

660

700

740

780

(b) Common Parameters and Values

hinc;co ¼ a sin½ðninc =nco Þ sin hinc;user  from Eq. (2). Numerical aperture (NA) (Pollock and Lipson, 2003) of the optical fiber 1=2

■, □ Φ(θ,λ) = 1 (m=0) ●, ○ Φ(θ,λ) =

λ (nm)

Fig. 9. An HSL system light transmission with U: (a) skew rays (CW), (b) meridional rays (conical-diverging). I First bent section (R1 =rco ¼ 55;   ub;1 ¼ 135 ), II second bent section (R2 =rco ¼ 25; ub;2 ¼ 85 ), III third bent  section (R3 =rco ¼ 20; ub;3 ¼ 95 ), IV fourth bent section (R4 =rco ¼ 15;  ub;4 ¼ 90 ), V straight section (Ls ¼ 5:0m; r ¼ 6 nm), VI HSL system (overall).

NA ¼ nair  sinðhinc;max Þ ¼ ðn2co  n2cl Þ

700 650 600 550 500 450 400 350 300 250 200 150 100 50 0 380

ð10Þ

where hinc;max , the angle of acceptance for the incident rays, was used as an upper limit for hinc;user  (0 < hinc;user < hinc;max ). We did not model the distributions of spatial irradiance (W =m2 ) and/or illuminance (lm=m2 ) at the fiber exit. See, e.g., (Jongewaard, 2002) for the spatial irradiance distributions at the exit end of the planar fiber. 2.2.1. Number of rays Number of rays used on the straight fiber subsystem simulations was 90  103. It was found that the maximum difference in light transmission (rco;min ¼ 0:1 mm), calculated from Eq. (6), that can be observed between simulating

rco = 7.4mm

rcl = 7.8mm

nco = 1.2

ncl = 1.09

Δλ = 5nm

θinc,user = 20

Fig. 10. Spectral results. (a) Light transmission (%), (b) light output (lm/  W). I First bent section (R1 =rco ¼ 50; ub;1 ¼ 120 ), II second bent section   (R2 =rco ¼ 20; ub;2 ¼ 75 ), III third bent section (R3 =rco ¼ 10; ub;3 ¼ 90 ), IV straight section (Ls ¼ 6:3m; r ¼ 5 nm), VI HSL system (overall).

250  103 and 90  103 rays was < þ0:06% and < þ0:11% for RPP1 and RPP2 power profiles, respectively. The bent fiber subsystem simulations used 2025 rays with n ¼ 50. The difference in light transmission between simulating 2025 and 10  103 rays (rco;min ¼ 0:1 mm) was found to be higher for R=rco < 20 (but still less than 0:45%). 2.3. HSL system Light transmission was given due to Eqs. (6) and (8). In order to determine the amount of transmitted light, transmitted power due to the source power profile was overlapped with the luminous efficacy value over the visible spectrum (380nm < k < 780nm) (IESNA, 2000). In the case

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M. Tekelioglu, B.D. Wood / Solar Energy 83 (2009) 2039–2049

of the spectral values of the properties of the optical fiber, power profile, and light transmission, the light transmission was given from 81 discrete k’s over the visible spectrum. The configuration of the subsystems in 3-D space was immaterial due to the axial symmetry of the power profile as defined in Eq. (9) and depicted in Fig. 5. The configurations can be explained by proper definitions of the MPDs at subsystem inlets. The FORTRAN simulation time for an HSL system with four bends and three straight sections was nearly 10 mins with 92,025 rays (90  103 rays on the straight and 2025 rays on the bent fiber subsystem). 3. Results and discussion 3.1. Straight and bent fiber subsystems The optical light loss coefficient (overall) of the straight fiber subsystem (a ¼ 4:1  105 mm1) was found from a cut-back experiment: Light transmission was measured before and after cutting 1-m section on 3-m of the fiber and Eq. (6) was applied. 72.7% of a was assumed aa (3  105 mm1) and 14:5%  a as as (6  106 mm1) with (as =aa ¼ 0:20), D was estimated from an earlier study (Tekelioglu and Wood, 2005) (D ¼ 2  105 ), and r ¼ 4 nm closely approximated the experimental light transmission result to within < 0:64% (%-difference) at Ls ¼ 5 m. %-Difference was defined as [(simulation  experiment)/experiment]100. The optical properties were estimate for 3M fiber, conical-converging rays, and RPP1 (m ¼ 0). For typical values of aa , as , r, and D, given for Lumenyte fiber, See (Tekelioglu and Wood, 2005). The bent fiber subsystem experiments used bend radius  R=rco of 16, 19, 21, 27, 32, 43, 54, 64 with ub ¼ 90 and their results were compared with the simulation results. A Labsphere spectrophotometer (FIMS-400P) was employed to measure the transmitted amount of light in the experiments (Labsphere, 2006). Effect of experimental uncertainty on these results was quite significant.

4 nm, D ¼ 0:2  104 . A prototype HSL system employing the same 3M fiber was operational on eight-fiber bundle design (ORNL, 2004). In the present model, RPP1 power  profile was assigned to the converging rays (hinc;user ¼ 24 ; m ¼ 0) at the inlets of the first and subsequent subsystems. Table 2 includes the subsystem dimensions and Table 3 includes (R=rco and rcl ! 1) the simulation and experimental results. It was found from Table 3 that the experimental HSL system light transmission can be estimated to within 5:3% with meridional rays and to within 24:7% with skew rays. After incorporating into the ray-trace software, our model and TracePro results (rcl ! 1) were compared to experimental results of Table 3 with meridional rays, r ¼ 4 nm, and D ¼ 0:2  104 . Our model parameters r and D of Eq. (5) were translated on TracePro into BRDF with ABg scatter model. Again, in most of the available ray-trace software, there are no physical models to define r and D explicitly. We found (rcl ! 1), respectively, 56.2%, 51.2% and 57.9% for the HSLS 1, 2, and 3. After incorporating our model on TracePro (rcl ¼ 6:75 mm), 56.2%, 54.6%, and 57.9% were found for the same HSL systems, which showed that HSLS 2 (rcl ¼ 6:75 mm) returned some of the refracted rays back into the fiber core. %(%-max)-Differences on TracePro (rcl ¼ 6:75 mm) were 10.1(16.9%), +2.2(+11.4%), and +1.4(+10.5%) for the HSLS 1, 2, and 3, respectively. TracePro and present model results were shown, respectively, to be within 10:1% vs 5:3% of the measured values (See Fig. 6). Any mixture of the skew and meridional rays was not considered in the current simulations. The spectrophotometer accuracy was ð5 8Þ% (Labsphere, 2006) and the light source accuracy due to the illuminator instability and fiber positioning was estimated 3%. These values yielded a combined accuracy of 8:2% on the measured values of Table 3 (Coleman and Steele, 1999). As a result, a 10% of prediction range for the HSL system light transmission was considered reasonable. The second and third sets of experiments with skew rays fell outside this range.

3.2. HSL system (3M fiber, experimental) 3.3. HSL system (any fiber, projected) The light transmission of the three HSL systems was validated (3M fiber): rco ¼ 6:3 mm, rcl ¼ 6:75 mm, nco ¼ 1:498, ncl ¼ 1:35, aa ¼ 3  105 mm1, as ¼ 6  106 mm1, r ¼

The three HSL system designs with user-defined optical and geometrical properties were simulated. Fig. 7(a) and

Table 2 Subsystem dimensions for the HSL systems.a Subsystemb

HSLS 1

HSLS 2

HSLS 3

s1, Ls,1 (mm) b1, R1/rco (dimensionless) s2, Ls,2 (mm) b2, R2/rco (dimensionless) s3, Ls,3 (mm) b3, R3/rco (dimensionless) s4, Ls,4 (mm) b4, R4/rco (dimensionless)

1000 26.8 (ub = 90°) 1000 16.1 (ub = 90°) 1000 56.8 (ub = 90°) 2000 0

820 26.8 (ub = 90°) 1000 16.1 (ub = 90°) 1000 56.8 (ub = 90°) 1850 16.1 (ub = 90°)

900 26.8 (ub = 90°) 2900 16.1 (ub = 90°) 900 21.4 (ub = 90°) 0 0

a b

Total length: HSLS1 = 5.99 m, HSLS2 = 5.82 m, HSLS3 = 5.34 m. s denotes the straight and b the bent fiber subsystem.

M. Tekelioglu, B.D. Wood / Solar Energy 83 (2009) 2039–2049

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Table 3 Simulation and experimental results. Subsystems and HSL system

b1 b2 b3 b4 stotal HSLSa HSLSb HSLSc Accuracy ranged (%) %-Differencee (%) %-Differencef (%)

HSLS 1 (%)

HSLS 2 (%)

HSLS 3 (%)

Skew

Meridional

Skew

Meridional

Skew

Meridional

97.5 97.7 94.0 – 75.1 67.3 – 62.5 57.3–67.7 +7.6 (+17.3) –

97.1 86.8 96.2 – 75.0 60.8 56.2 62.5 57.3–67.7 2.7 (11.0) 10.1 (16.9)

97.5 97.7 94.0 97.5 76.1 66.5 – 53.4 49.0–57.8 +24.5 (+35.7) –

97.1 86.8 96.2 86.8 76.0 53.5 54.6 53.4 49.0–57.8 +0.2 (+8.4) +2.2 (+11.4)

97.5 97.7 98.3 – 76.0 71.2 – 57.1 52.4–61.8 +24.7 (+35.9) –

97.1 86.8 93.8 – 75.9 60.1 57.9 57.1 52.4–61.8 +5.3 (+13.5) +1.4 (+10.5)

a

Present model. Commercial software. c Experimental. d Range of systematic uncertainty from the general method (Coleman and Steele, 1999). e Between present model and experimental values. f Between commercial model and experimental values, given for meridional rays. Outside-the-parentheses values are %-differences in light transmission calculated from [(simulation-experiment)/experiment]100 and inside-the-parentheses are %-max differences calculated on the lower and upper bounds of the experimental uncertainty. (+) represents an over-prediction and () represents an under-prediction of the experiments. b

(b) show the results for skew and meridional rays, respectively. In this figure, light transmission of the straight fiber subsystems was found to increase with rco due to a decrease on the number of reflection points, thereby, the core–clad interface losses. The HSL system light transmission was determined to be higher with RPP2 compared to RPP1, e.g., 81.5% vs 44.4% for meridional rays and 80.2% vs 48.8% for skew rays (rco ¼ 1 mm and rcl ¼ 1:1 mm). Skew rays with RPP1 on the bent fiber subsystems gave a higher light transmission compared to meridional rays. This was attributed to the relatively lower loss of the skew rays entering toward the edge of the fiber. The same yet subtle effect was also observed on the straight fiber subsystem (See Table 3 in this paper and Figs. 4 and 8 in (Tekelioglu and Wood, 2005)). Fig. 7 results show that the light transmission through bends and straight sections can be comparable in size. The effect of arrangement of rays is shown on the other HSL system in Figs. 8 and 9. It is derived from the figures that the light transmission with RPP1, RPP2, and RPP3 increases by changing the ray arrangement from conicalconverging to conical-diverging. The power profile parameter (m) had a subtle effect on the light transmission compared to the arrangement of rays and that for m > 6, only slight difference was observed on the light transmission of conical-converging and conical-diverging rays. It was found by comparing Fig. 8(a) to (b) and Fig. 9(a) to (b) that the light transmission of the skew rays was higher with RPP1 and RPP3. The higher light transmission (RPP3) was due to the power increase of the toward-the-edge rays where the skew rays lost less power. The k-specific light transmission (%) and light output (lm/W) are shown in Fig. 10 on the other user-defined

HSL system design where meridional rays were used. In this figure, the filled symbols are with k dependency of the only ar term and the empty symbols with k dependencies of the aa , as , ar , and D terms. An exponential regression (aa / expðc1 kÞ) was applied on aa and a power-law regression (as / kc2 ) on as both starting from aa ¼ 105 mm1 and as ¼ 106 mm1 at k ¼ 380 nm. Power profile was specified by the specular (u) and spectral (k) components of the terrestrial spectral solar light (See Fig. 4). Effect of the light source u and k-dependencies was identified by comparing the light transmission of the square and circle symbols. In the results of Figs. 7–10 and Table 3, the optical properties of the collection hardware, i.e., primary and secondary mirrors along with the coupling losses between the solar collector and optical fiber were not included. Visible spectrum reflection and transmission characteristics of the collection hardware can lower these results by as much as 10%. 4. Sensitivity analysis A sensitivity analysis was included to determine which of the optical, geometrical, or external parameters was more significant. First, we found the sensitivity coefficients on the subsystems: The changes on the parameters required to create a Ds ¼ 2% change on the light transmission were calculated and the ratios of fractional changes were determined (as fractional change of light transmission to fractional change of a parameter) which was represented by Z ¼ ððDs=sÞ=ðDX =X ÞÞ, where X is one of the parameters listed in Fig. 2(b) and (c). The smallest ratio of the fractional changes gave the least significant parameter while the largest ratio of the fractional changes gave the most sig-

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M. Tekelioglu, B.D. Wood / Solar Energy 83 (2009) 2039–2049

nificant parameter. That is, the largest of the ratios meant a significant parameter that could create a 2% change on the light transmission with a little fractional change of the parameter, compared to the parameter of the smallest ratio which could create the same change with a comparatively large fractional change. Following values were used:  as ¼ 6  106 mm1, hinc ¼ 25 , aa ¼ 3  105 mm1,  hinc;user ¼ 30 , k ¼ 550 nm, r ¼ 4 nm, rco ¼ 6:3 mm, D ¼ 2  105 , nco ¼ 1:498, Ls ¼ 5 m. The local radius of the fiber (d) was related to the fiber core radius (rco ) via d ¼ rco  ðtan hinc = tan hinc;user Þ for conical-converging rays; ss ¼ expða  Ls Þ was used for the straight fiber and sb ¼ ð1  T r Þ for the bent fiber subsystem. The results are shown in Tables 4 and 5: Most-to-least significant parameters were determined: Straight fiber subsystem (Skew ray) ðDs ¼ 2%Þrco > hinc > k > r > Ls > nco > aa > as > D ðDs ¼ þ2%Þhinc > rco > r > k > Ls > nco > aa > as > D Straight fiber subsystem (meridional ray) ðDs ¼ 2%Þhinc > Ls > k > r > aa > nco > rco > as > D ðDs ¼ þ2%ÞLs > hinc > r > aa > k > nco > as > rco > D Bent fiber subsystem (skew and meridional rays) ðDs ¼ 2%Þnco > ncl > h

Table 5 Parametric sensitivity analysis results (bent fiber subsystem). Z Parameter

h nco ncl

Ds = 2%

Ds = +2%

Skew and meridional

Skew and meridional

12.467 31.552 +31.373

+17.000 +37.697 37.875

straight fiber subsystem was quantified with parameters r and D . The system-level light transmission was predicted to within 5:3% on the measured values and to less than 13:5% maximum with accuracy ranges on the same values. The sensitivity analysis used 3M fiber, select operational points (geometrical and optical parameters of the fiber), and RPP1. Thus, the analysis needs to be further elaborated to represent the fiber overall light transmission. This paper was intended to take a step toward developing parametric models, e.g., with unit subsystems, MPDs, or more elaborate sensitivity analyses. These models, in turn, will help predict the light transmission of HSL systems of any configuration and optical and geometrical property. Experimental verification will be necessary.

ðDs ¼ þ2%Þncl > nco > h It was found (Ds ¼ þ2%) that the most significant parameters were hinc , rco , r, and k for a skew ray on the straight fiber subsystem and Ls , hinc , r, and aa for a meridional ray on the same subsystem, and ncl and nco for the both ray types on the bent fiber subsystem. Note that hinc and k were external parameters not pertaining to the fiber. 5. Conclusions The straight and bent fiber subsystem models were evaluated with skew and meridional rays. The model results were combined to estimate the light transmission of the HSL systems. The loss at the core–clad interface of the Table 4 Parametric sensitivity analysis results (straight fiber subsystem). Z Parameter

aa as r D hinc k nco Ls rco

Ds = 2%

Ds = +2%

Skew

Meridional

Skew

Meridional

+0.154 +0.031 +0.596 +0.001 +0.885 0.628 0.371 +0.476 0.964

+0.154 +0.031 +0.202 +0.000 +0.313 0.229 0.152 +0.280 0.067

0.158 0.032 0.582 0.001 0.833 +0.550 +0.308 0.492 +0.743

0.158 0.032 0.181 0.000 0.270 +0.154 +0.092 0.288 +0.039

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