Approximation of within-stack absorptances for diffuse irradiation

Solar Energy Vol. 37, No. 2, pp. 175-178, 1986

0038-092X/86 $3.00 + .(~1 © 1986 Pergamon Journals Ltd.

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TECHNICAL NOTE Approximation of within-stack absorptances for diffuse irradiation PENELOPE H. SMITH and THEODORE F. SMITH Department of Mechanical Engineering, College of Engineering, The University of Iowa, Iowa City, Iowa 52242 (Received 5 August 1985; Accepted 5 March 1986) INTRODUCTION

parent plates covering an absorbing background. The embedding method examines the effect of adding one plate, called the Nth plate, to a stack of N - 1 plates, with plate N = 1 being the absorber at the bottom of the stack. The within-stack absorptances are dependent on the polar angle of incidence and wavelength. The hemispherical stack properties for diffuse irradiation were obtained by integrating the directional stack properties over the hemisphere.

Windows and solar collectors typically consist of one or more semitransparent plates and an absorbing background, either a room interior or an absorber plate. When evaluating the heat transfer through these systems it is desirable to know the fraction of incident solar energy absorbed by each plate and by the absorbing background. For beam irradiation, directional property calculations are made for the direction of incidence of the beam. For diffuse irradiation, hemispherical properties are obtained by integrating the directional properties over the hemisphere. The latter calculations would be simplified were there one incident angle that yielded approximate, yet accurate, hemispherical properties when inserted into the directional property expressions. For surfaces with directional specular reflectance as given by the Fresnel relations[l], Look[2] found the directional reflectance at a polar angle of incidence of 60° to be a good approximation of the hemispherical reflectance. Brandemuehl and Beckman[3] present polar angles of incidence that yield transmittances for diffuse irradiation when inserted into expressions for beam transmittance for a tilted flat plate. These results do not address the within-stack absorptance[4] as needed to evaluate the heat loss through a multiplate system. The purpose of this note is to determine whether there is one angle that can be used to evaluate the plate absorptances when irradiation is diffuse. The savings in computational time and effort might be significant when many calculations are made, such as for the heat transfer analysis of several window configurations or for computation of the thermal response of a building.

RESULTS AND DISCUSSION

ANALYSIS Analyses were done for three cases: surface reflection, overall properties for a single plate, and a stack of semitransparent plates. Expressions for the directional specular reflectance, derived from electromagnetic theory, for incident radiation polarized parallel to and perpendicular to the plane of incidence were obtained from Siegal and Howell[l]. A "mixed" component of the reflectance is defined as the average of the parallel and perpendicular components. Integral expressions for the parallel, perpendicular, and mixed components of the hemispherical reflectance of a diffusely irradiated, specularly reflecting surface were also obtained from [1]. The overall properties for a semitransparent plate were determined with the net-radiation method of Siegal and Howell[l]. The overall hemispherical properties for diffuse irradiation were obtained by integrating the overall directional properties over the hemisphere. Calculations were made for the parallel, perpendicular, and mixed components. Edwards' embedding technique[4] was used to obtain the within-stack absorptances for a stack of semitrans-

Within-stack absorptances were studied for three absorbing backgrounds, namely, black, directional specular dielectric, and directional specular aluminum. The dielectric and aluminum backgrounds were chosen as representative of window shades. A refractive index of 2.0 was assigned to the dielectric background as typical of a painted surface with a high emittance. The refractive and absorptive indices for aluminum were selected at wavelengths of 0.95 and 2.07 p,m and were obtained from Ordal. et al.[5]. Each background was embedded with one, two, or three glass plates, each having a refractive index of 1.5, an absorptive coefficient of 0.299 c m - J, and a thickness of 0.317 cm. All results presented are for the mixed values of the within-stack absorptance. Directional within-stack absorptances are presented first in order to assist in selection of a particular value of the polar angle of incidence to approximate the hemispherical within-stack absorptance. Within-stack absorptances for two glass plates and a black background as a function of the polar angle of incidence are shown in Fig. l(a). Results for one and three glass plates are similar. Within-stack absorptances of the glass plates increase slightly until about polar angle of incidence 0 = 60°, then the absorptance of the inner plate, A2,3, decreases steadily to zero at 0 = 90°. The absorptance of the outer plate, A3.3, continues to increase, peaking at 0 = 80°, then drops sharply to zero at 0 = 90°. The glass plate absorptances are low, not rising above 0.14, and A3,3 > A2.3. The absorptance of the black background, At.3, is much higher than the plate absorptances, being greatest at 0 = 0° where its value is 0.708. It exhibits a steady decrease until about 0 = 60°, whereupon it drops more sharply until reaching zero at 0 = 90 °. The hemispherical absorptances for diffuse irradiation are plotted as horizontal lines in this and the other plots to be discussed. It is these values that are to be approximated by the directional values. Within-stack absorptances for a directional specular dielectric background embedded by two glass plates are presented in Fig. l(b). The behavior of these results is similar to that in Fig. l(a). The absorptance for the dielectric background is less than that for the black background, having a maximum value of 0.638. The glass plate absorptances, however, are slightly greater when the background is a dielectric. This is due to the plates absorbing energy reflected from the dielectric background.

175

Technical Note

176

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(b) Fig. 1. Within-stack absorptance. (a) Two glass plates and a black background. (b) Two glass plates and a directional specular dielectric background.

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Fig. 2. Within-stack absorptance for two glass plates and a directional specular aluminum background.

Technical Note

177

Table 1. Percentage error between hemispherical within-stack absorptances and directional withinstack absorptances

% Error Background

i

Ahemi

@ = 0°

Black N = 2

2 1

0.1015 0.7616

Ii.0 -10.2

2.] -8.0

-6.1 -0.58

-7.0 I.i

Black N = 3

3 2 1

0.1106 0.0795 0.6164

13.5 4.2 -14.8

4.5 -3.6 -Ii.i

-5.8 -6.9 0.02

-7.2 -6.4 2.6

Black N = 4

4 3 2 1

0.1152 0.0868 0.0642 0.5087

13.6 6.7

-1.3

-17.7

-7.3 -12.9

-5.4 -6.7 -7.3 0.08

-7.0 -6.9 -6.1 3.3

Dielectric N = 2

2 1

0.1125 0.6778

12.2 -10.9

3.3 -8.7

-5.9 -1.2

-7.0 0.74

Dielectric N = 3

3 2 1

0.1165 0.0886 0.5587

12.6 5.4 -14.1

4.0 -2.5 -10.9

-5.2 -6.7 -1.4

-6.5 -6.5 1.0

Dielectric N=4

4 3 2 1

0.1188 0.0920 0.0714 0.4656

12.5 6.0 0.28 -16.4

4.2

-4.7 -6.0 -6.9 -2.0

-6.1 -6.1 -5.6 0.92

Aluminum (~ = 0.95Bm) N=2

2 1

0.1799 0.0868

8.7 9.3

0.22 7.6

-6.4 -3.2

-6.9 -5.6

Aluminum (l = 0.95~m) N = 3

3 2 1

0.1634 0.1508 0.0741

7.6 3.4 5.9

-0.12

-5.8 -7.0 -5.1

-6.1 -6.7 -7.4

Aluminum (l = 0.95~m) N = 4

4 3 2 1

0. 1520 0.1350 0.1260 0.0631

7.5 1.8 -1.5 3.0

-4.7 -7.5 2.9

-5.1 -6.1 -6.5 -6.8

-5.5 -5.4 -5.2 -8.9

Aluminum (X = 2.07~m) N=2

2 i

0.1875 0.0208

9.2 15.9

0.75 13.9

-6.5 ].4

-7.i -1.4

Aluminum (k = 2.07~m) N = 3

3 2 1

0.1689 0.1582 0.0177

7.8 4.2 12.4

0.12 -3.1 10.7

-5.9 -6.9 -1.1

-6.3 -6.7 -4.0

Aluminum 4 (i = 2.07Bin) 3 N = 4 2 1

0. 1560 0.1402 0.1326 0.0150

7.5 2.1 -0.68 8.7

0.32 -4.3 -6.6 8.7

-5.3 -6.1 -6.3 -3.3

-5.8 -5.6 -5.3 -6.0

No.

@ = 40 °

in Ai,N(8 ) 8 = 58 °

8 = 60 °

of Plates

-0.47

4.9

-1.5 -6.6 -12.4

-3.9 5.0

0.26

Technical Note

178

Within-stack absorptances for two glass plates embedded on directional specular aluminum backgrounds for wavelengths of 0.95 i~m and 2.07 I~m are presented in Figure 2. Because the aluminum is highly reflecting, the within-stack absorptance of the background is less than that of the glass plates. The surface reflectance of aluminum is higher at a wavelength of 2.07 ~tm than at 0.95 ~m; therefore, AI.3 for h = 2.07 Ixm is less than A~.3 for h = 0.95 ~m. Also, rather than steadily decreasing as it does for the black and dielectric backgrounds, A ~.3 for the aluminum background increases, reaching its peak value at a polar angle of incidence between 75° and 85°, then decreases to zero at 0 = 90 °. Although A2,3 and A3,3 have similar distributions as the corresponding plate absorptances for the dielectric and black backgrounds, their magnitudes are higher because of the highly reflecting aluminum background, Shown in Table 1 is the percentage error between the directional within-stack absorptance at polar angles of incidence of 0°, 40 °, 58°, and 60° and the hemispherical within-stack absorptance for twelve stacks of plates. The hemispherical within-stack absorptance for each plate is also given. The directional within-stack absorptances for the particular value of 0 can be obtained by inserting information from the table into the following equation: r ai u(O) = ahemi [ 1 •

t

Error] I~ J

(1)

A polar angle of incidence of 0° was chosen for convenience, since the sine and cosine terms in the Fresnel expressions are easily evaluated. The error in using the directional within-stack absorptance at 0 = 0 deg to approximate the hemispherical within-stack absorptance, however, can be as high as 18%. Using 0 = 40 °, the error for plate 1 is less than 14%, and the error for the other plates is less than 8%. The errors in using 0 = 58° and 60°

are similar to each other. Both angles yield errors of less than 10% for all plates. If greater accuracy is required, the hemispherical within-stack absorptances can be represented by the directional within-stack absorptances calculated at the polar angles of reflection given by the following menu: for plate 1 use 0 = 58° for N = 2, 3, or 4 for p l a t e 2 u s e 0 = 40 ° f o r N = 2 o r 3 and0 =

0° f o r N = 4

for plates 3 and 4 use 0 = 40° for N = 3 and 4 As seen in Table 1, where these values are in bold print, the error is less than 5% except for plate 1 for aluminum at h = 0.95 ~tm with N = 3 and 4. In summary, it is possible to select one incident angle that yields accurate hemispherical within-stack plate absorptances when inserted into the directional expressions.

REFERENCES

1. R. Siegal and J. R. Howell, Thermal Radiation Heat Transfer, 2rid ed. McGraw-Hill, New York (1981). 2. D. C. Look, Approximate hemispherical radiative properties. AIAA Journal 17, 443 (1979). 3. M.J. Brandemuehl and W. A. Beckman, Transmission of diffuse radiation through CPC and flat plate collector glazings. Solar Energy 24, 511 (1980)• 4. D. K. Edwards, Solar absorption by each element in an absorber-coverglass array. Solar Energy 19, 401 (1977). 5. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander, Jr. and C. A. Ward, Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared. Applied Optics 22, 1099 (1983).