Performance characteristics of rectangular honeycomb solar-thermal converters

Solar Energy, VoL 13,pp. 193-221.

Pergamon Press, 1971.

Printed in Great Britain

P E R F O R M A N C E C H A R A C T E R I S T I C S OF R E C T A N G U L A R HONEYCOMB SOLAR-THERMAL CONVERTERS H. B U C H B E R G , O. A. L A L U D E and D. K. EDWARDS*

(Received 26 November ! 969) Abatraet-An array of rectangular cells, properly shaped with highly reflecting specular walls absorbing to infrared radiation, is shown to be an effective device to limit heat losses when placed over a solar absorber. Theoretical relationships are presented which permit the prediction of the performance of closed-cell and air transpiration systems. The theory rests on analytical and experimental studies of radiant transmission through cells with speeularly reflecting, polarizing walls coated with an absorbing film and studies of the suppression of natural convection in cells heated from below. Good agreement was obtained between theory and experiments for the air transpiration system. Three experimental honeycombs of difft/¢~t geometries were tested over a l-sq ft blackened fiberglass porous absorber, assembled in a well-insulated and glazed housing mounted on a tilting platform to permit any desired orientation. Experiments over a l-yr period, and theoretical predictions, indicated that the thermal efficiency could be given as a linear function of the ratio of temperature difference between the average temperature of the working fluid and ambient air to the incident solar power, virtually independent of flow, angle of tilt and time of collection. For the best honeycomb tested, the thermal efficiency ranged from 82 to 63 per cent for values of (Toy-- To)/G, equal to 0-10 and 0.30°F/ (B.t.u./hr fts), respectively. The honeycomb-porous bed solar-air heater is shown to perform as well as the best previously reported solar-air heater at moderate temperatures. At higher temperatures, of the order of 200°F, it is greatly superior. R/~am~--On montre qu'une rangte de piles rectangulaires, bien formtes avec des parois sptculaires haute rtflexion absorbant ies radiations infra-rouges, est un dispositif efficace pour iimiter ies pertes de chaleur si on les place au-dessus d'un absorbeur solaire. Des relations thtoriques sont pr~sentg, es qui permettcat de pr~lire la performance de systtmes it ceilule enfern~e et/~ filtration d'air. La thtorie repose sur les ~tudes aaatytiques et exlakr/mantales de ia transmission des radiations/L travers its piles avec purois miroitantes de polarisation endures d'une peilioule abeorbante, et des 6tudes de la supln~sioa de ia convection natureile darts les piles c ~ par le has. Pour le syst~me iL filtration d'air, raccord entre la thtorie et rexptrience ~ait satisfaisant. Trois gtom~tries difftrentes de nids d'abeilles eXl~'rimentaux ont 6t~ raises /~ I'essai sur un absorbeur poreux en fibre de verre noircie de 900 cmz de surface, mont6 darts un coffrage vitrifi~ et parfaitement isoi6 piac~ sur une piate-forme inclinable afin de permettre de prendre rorientation dtsirte. Les exl~riences effectutes sur une ptriode d'un an eties prtdictions thtoriques ont montr6 que le rendement thermique pouvait/~tre donn6 en rant que fonction lintaire du rapport de ia difference de temptrature entre la temptrature moyenne du fluide de travail et de rair ambiant et de ia puissance solalre incidente, virtuellement indtpendante de l'tcoulement, de rangle d'incfinaison et du temps de recueil. Pour le meilleur hid d'abeilles soumis h l'essai, le rendement thermique variait de 82 ~ 63 pour cent, pour des valeurs respectires de ( T , , . - To)/G, 6galent h 0,10 et 0,30°F/(B.t.u. hr ftz). On montre que le radiateur air-solaire/l touche poreuse hid d'abeilles peut donner des performances aussi bonnes que le meilleur radiateur air-solaire jusqu'ici mentionnt, h des temptratures modtrtes. A des temptratures plus 61evtes de rordre de 200°F, il est nettement suptrieur. R ~ u m e n - S e demuestra que un conjunto de pilas rectangulares, debidamente configuradas y con paredes espectaculares altamente reflectantes y absorbentes a ia radiacitn infrarroja, constituye un medio eficaz para la iimitacitn de perdidas col6ricas cuando va montado encima de un absorbedor de hiz solar. Se presentan relaciones te6ricas que permiten pronosticar ia actuaci6n de los sistemas de pilas cerradas y de transpiracitn del aire. La teoria se funda en estudios anallticos y experimentales sobre la transmissitn radiante a travts de pilas con paredes reflectantes y polarizadoras revestidas de une pelicas absorbente, asf como estudios sobre la supresi6n de ia conveccitn natural en pilas dotadas de calefaccitn inferior. Se ha establecido buena concordancia entre teorla y experimentaci6n con respecto al sistema de transpiraci6n del aire. Tres panales experimentales de geometriias diversas fueron ensayados sobre un absorbedor poroso de fibra de vidrio ennegrecida, el cual media I pie cuadrado e iba alojado en une envuelta bien aislada y encristalada, montada sobre une plataforma basculante en orden a lograr cualquier orientaci6n deseada. Los experimentos efectuados durante espacio de un afio, asi como las predicciones tetricas, indicaron que el rendimiento t~rmico podria venir expresado como funcitn lineal de la raztn de la diferencia de temperaturas entre la promedia del fluido operante y aire ambiente y la de la energia solar incidente, y ello en forma pr~cticamente independiente de los factores de flujo, ~ngulo de inclinacitn y tiempo de captaci6n. En el mejor panal de los ensayados, el rendimiento ttrmico oscil6 entre el 82 y 63 por ciento, con valores de ( T , v - Ta)/G, iguales a 0,10 y 0,30°F * School of Engineering and Applied Science, University of California, Los Angeles, U.S.A. 193

194

H. BUCHBERG. O. A. L A L U D E and D. K. EDWARDS

(B.t.u./hr ft2), respectivamente. Se demuestra que el calentador de aire por energia solar de tipo panal con lecho poroso funciona tan bien, a temperaturas moderadas, como el mejor calentador de aire por energia solar hasta ahora comunicado. A temperaturas mayores, del orden de 200*F, su rendimiento es muy superior. INTRODUCTION

A CELLULARstructure placed over a solar collector has considerable appeal. The open structure permits the transmission of incident radiation, and the multiple cells may inhibit the initiation of free convection above the heated absorber surface and may significantly reduce the net long wavelength radiation loss. As long ago as 1929, Veinberg and Veinberg[1] experimented with cell structures formed by stacked wire meshes or vertical sheets of lacquered, corrugated paper laid over absorbing media. Later, Francia[2] demonstrated the effectiveness of cell structures (transparent glass cylinders) laid over medium- and high-temperature solar absorbers. Following Francia's investigations, Perrot e t al.[3] have experimented with transparent and opaque cell structures laid over low- and medium-temperature absorbers. Hollands[4] has presented a simplified analysis of honeycomb solar collectors. None of the past studies has taken account of spectral, directional and polarizing effects of cell walls on the radiant transmittance of rectangular cross-section cells, nor do they present a theory sufficiently detailed to permit parametric design studies. In the present paper, application of fundamental studies of radiation transfer and free convection is made to show the effect of significant design parameters on the performance of rectangular honeycomb solar-thermal converters. Theoretical relationships are presented for the performance of both nonflow and air transpiration converters. The validity of the analyses is demonstrated by comparisons with measurements. The analyses rest upon the analytical methods developed for radiant transmission through long passages with specularly reflecting, polarizing walls [5]. These methods are extended to account for walls coated with thick or thin films of absorbing materials and rectangular cells for both solar and infrared wavelength distributions. The nontranspiration theory utilizes, in addition, analytical and experimental studies which have resulted in knowledge of the Rayleigh number at which free convection initiates in closed vertical rectangular cells heated from below [6], and the cellular convection heat transfer versus Rayleigh number after initiation [7]. The experimental results used to validate analyses were obtained with a l-sq ft instrumented test module, assembled in a glazed and well-insulated housing mounted on a tilting platform to permit any desired orientation. The honeycomb structures investigated (see Fig. 1) consist of an arrangement of multiple rectangular cells with highly specularly reflecting walls coated with a dielectric film. The film is transparent to solar radiation and absorbing to long-wave radiation. For the transpired air tests, the cells were located over a blackened fiberglass porous bed absorber. For the nontranspiration system, a black flat-plate absorber was used. The authors chose to investigate selectively-reflecting cellular structures rather than selectively-transmitting ones, because they felt that the former could be manufactured at a much lower cost per unit area than the latter. The same methods of analysis may be applied to cellular structures with transmitting walls. THEORY

A description of the thermal behavior of a honeycomb solar-thermal converter involves consideration of the three modes of heat transfer: conduction, convection and

Performance characteristics of rectangular honeycomb solar-thermal converters

195

SOLAR INPUT

•~

ti/

,

d /

0LA 'N

<

_--"" - ; " . --.'% s -"~. J' /~I A ~ I ~Ill ~ " . "..":-,~5 " . >'-_>.~<...: T~".~..~

~ ~

I

1

"

(AW.,.,ZEO. CLEAR-RESIN OVERCO,TE0 PAPER)

COLD AIR IN THROUGH SPACE BETWEEN GLAZING AND HONEYCOMB

HOT AIR OUT Fig. 1. Rectangular honeycomb solar-thermal converter.

radiation between the solar absorber and glazing, separated by the rectangular multiple cell structure, shown in Fig. 1, in addition to the heat exchanges with the surroundings. Conduction between the glazing and solar absorber occurs in the cell walls and through the intervening fluid. Natural convection may be initiated in the cells for a specified temperature difference depending upon the cell dimensions and simultaneous occurrence of the other heat transfer modes. Radiation entering the cells obliquely, from above or below, suffers reflections from a composite surface consisting of an absorbing film over an absorbing substrate which comprises the cell walls. Analysis is conveniently carried out by the formulation of a cell transmittance property which may be a function of wavelength, direction and state of polarization, as well as the dimensions of the cell. Theoretical relationships are presented first to describe the thermal performance of the rectangular honeycomb system operated without transpiration, i.e. with no forced fluid flow through the cells, and second operated as a transpiration system where air is drawn through the cells and an absorbing porous medium, as shown in Fig. 1.

Closed-cell system A model may be developed on the basis of uncoupled heat transfer modes; namely, radiation exchange plus conduction and natural convection between glazing and a black absorber, each considered to be in an equilibrium state. It is assumed that the glazing emits as a blackbody. For radiation exchange only, within the cell bounded on top by the glazing (transparent to solar radiation) and on the bottom by the absorber, the change in net radiant flux streaming toward the top through any cross-section per unit change in the coordinate x must be zero; i.e. the net radiant flux absorbed by any wall element is zero. Thus, the governing integrodifferential equation becomes

~'o.(-;.~--'~x)+ [(rTc'--~rrw'(x)] -+

fo: d(x_x,)2

+ [o'T.'-o'rw'(x)]

,

de(L--x)] d(L--x) J

[°'Tw'(x')-°'r'~4(x)] dx'+ fL de(x'-x) "~Z'~ [crTw'(x')--o'T,,'(x)]dx' (1)

196

H. B U C H B E R G , O. A. L A L U D E and D. K. E D W A R D S

where the derivatives are related to ~- factors [5] as follows: -~

d r = :T factor for radiant exchange between the cell entrance plane and a wall element of width dr, confined to the solar wavelength band;

dz _ u dx = ~- factor for radiant exchange between the glazing and a wall element of dr width dr;

at(L--x) d(L-x)

d r = ~ factor for radiant exchange between the absorber and a wall element of width dx;

d2~'(x-x') d r ' d x = ~- factor for radiant exchange between any wall element of width d(x-x) 2 dx' (x' < x) and a wall element of width dx; d2r(x ' - x ) d r ' d r = 8~',fa~or for radiant exchange between any wall element of width d(x' ~ X ) 2

dx (x' > x) and a wall element of width dx.

The net radiation qr leaving the cell may be determined in the manner of [5] after the following steps: quadrature by parts, integration over x, collocation at x =: L/2 and x = L (assuming wall emission varies linearly with x) and solving for ,r[T,,'(L)

- T~,'(0)]

-

aE --

tr(TA'-- TG') [ 1 + ~'(L) -- 27"(L/2) ] -- G : ~ , [ 1 + r, (L) - 2~'0(L/2) ] 1 + a-(L) -- 2r (L/2) -- 2F (L)/L + 4F(L/2)/L

(2)

and orTw'(0) = {'rg,G,[ 1 - r, (L/2) ] + o'1"o[ 1 - ~'(L/2) ] + trT,~ [r (L/2) - (L) ] -AE[r(L/2)-~'(L)

+F(L)/L-2F(L/2)/L]}[I +~'(L)]-k

(3)

Then

o'Tw'(L ) = hE + crTw'(O )

(4)

trTw'(L/2) = crETw4(0) + Tw'(L)]/2.

(5)

and

Finally

q,. = rg, G,[1 --r,(L/2) ] + 7"(L/2)[crT,44--O'TG 4] +AE[2F(L/2)/L--~'(L/2)].

(6)

To account for conduction and natural convection in the intervening space between the absorber and glazing, Nu k/L(Ta--TG) was added to the right-hand member of Eq. (6). Sun and Edwards [7] have investigated the problem of natural convection in closed rectangular cells heated from below, and present a method for calculating the

Performance characteristics of rec'tangular honeycombsolar-thermal converters

197

Nusselt number in terms of a Rayleigh number based on cell height and an adjusted wavenumber.

Transpiration system The governing equations that describe the thermal behavior of a rectangular honeycomb-porous bed transpiration solar convector may be obtained by expressing the energy exchange resulting from all modes of transfer at the glazing surfaces, within a single honeycomb cell, and within the porous medium. As shown in Fig. 1, the transport fluid, considered to be air in this study, enters between the glazing and honeycomb structure and flows down through the cells and porous bed absorber. For the purposes of this analysis, it is postulated that (1) steady-state pertains; (2) all properties are constant; (3) the condition in one cell is repeated in all; (4) the cell walls are conducting, reflect specularly, and emit only in the long-wave region; (5) the transport fluid filling all cells is conducting and is transparent to all radiation; (6) the porous bed is semi-infinite and uniform, black, emitting only in the long-wave region, and is perfectly conducting; (7) the glazing is black with respect to all radiation it sees from within the collector; (8) the convective conductance at the cell walls is infinite. The governing energy equations in nondimensional form may be given as follows: (The distance X is zero at the honeycomb entrance plane and increases toward the absorber.) -

Glazing o~G* + ototG* r +fr(L/d ) [0)'-- 0~'1 + ( 1 - - f ) [80'-- #g(] + f r2 'a (--~Xx)[e'(X)--O,(]dx (7)

= %tOg'+ H* [0g -- #a ] -- n~* [ (0, + 00)]2 -- #o]-

Cell r dO

w*[ow(x)-oo]

dOI

l

K [a× dx x.o] +f[qT(×)-q'*(°)]

(8)

= 0

where

q* (X)

rgsG*zs(X) + r(X) [ #u4 - O0'] -- r ( L / d - X) [Op' - 8L']

=

-

fi:

~(x-x')Tx,d [O'(x')]ax'-~

L/a

"X

d, ~(x'--x)~[O'(x')]ax'

(9)

and K ~ =.t"K~* + (1 - - f ) K * .

(lO)

Porous bed W* ( ~ - OL) = fro:s ( L/d) G* + fz( L/d) 04 + ( 1 - f ) OL4- Op"

aol

- g?--r-I

axIx-Lm

$.E. V ~ 13 N o 2 . D

+/

:L,,

.to

r

f~--

t

a~[(L/a)_X]lo,(×)ax~

..

d[(L/d)-×] J

"

(11)

198

H. B U C H B E R G , O. A. L A L U D E and D. K. E D W A R D S

In addition to Eqs. (7), (8) and (1 1), another equation may be formed expressing the energy exchange by convection between the entering fluid and cell fluid and the wall edges, accounting for radiation exchange between the wall edge and entrance to the cell (treated as a black surface) and the glazing. Thus

K~,*doJlx-o

[0°20~

= w* [ o o - o,] +

Oo]+(1--f)[(Oo4--Og4)--Tg, G*]. (12)

Equations (7), (8), (1 1) and (12) were reduced to four algebraic equations through the following procedure: (1) Linearization of the 4th power terms by the approximation, 04- I+4(0--1);

(2) Quadrature by parts; (3) Assumption of finear temperature profile in the cell O(x) = Oo+ (On--Oo)

X L/d

or

dO

(4) Collocation at X =

1

(oL-Oo) £/d;

½(L/d ) and X = L/d.

Then

,

F(L/d

O , [ 4 + H * + H , +4,a] +0014f - ~ )

+ OL[4fr(L/d ) - - 4 f ~ ]

4--½H~*]

+ Op[-4flr(L/d ) ]

= o~o,G* + ao~G*~r + H*Oa +½H? +

3~a

(13)

O.[--4fr(L/d)] + Oo[4fr(L/d) --4f F(L/d)L/d - L/r;] +o L[[4~ F(L/d) K* ~ "I-~[d--W*--4]+Op[W*+4] (14)

Og[-4-H~'+4fr(L/2d)] +Oo{½W* + ~K*d +½H~' + 4-- 4fr(L/2d )

Performance characteristics of rectangular honeycomb solar-thermal converters

199

L:d [2F (L/2a ) -- F ( L/a) ] l + 0.{-- 4f[¢(L/2a) -- ~.(L/ a) ]} = O,[W* --½H~] + (1 --f)%sG* +f%sG* [1 --z,(L/2d )] [~g*

(15) q

0g[-- 4-- H~' + 4fr(L/d )]0, + 00[~-~ + ½H~*+ 4 - 4 f r ( L / d )J K*

OL{W* --L~d + 4f[1-~'(L/d ) ] } + O,{-4f[ l - r ( L / d ) ] } = O,[W*--½H*] + ( 1 - f ) r ~ G * +frosG* [ 1 - T s ( L / d ) ] .

(16)

Solution of Eqs. (13), (14), (15) and (16) for the unknowns, 0o, 00, 0L and 0p, was readily accomplished through a matrix inversion program on the IBM 360/75 computer, from which follows the glazing temperature To, the cell wall temperatures To and TL, and the exit fluid temperature Tp in accordance with the specified inlet fluid temperature T~. The cell transmittances zs and r, and the integrated transmittance F for the closed cell and transpiration systems, are dependent on cell geometry. Solar transmittance r, is additionally dependent upon the direction of the incoming solar beam with respect to the cell entrance plane. The methods used in calculating the cell transmittances are summarized in the Appendix. The directional solar transmittances of the Dazing were calculated from spectral normal transmittance measurements, given in Table 1, made on a sample of the actual glass used. An index of refraction of 1-5 and the Moon[8] spectral solar distribution, with air mass 2.0, were assumed for calculations of directional solar transmittance and absorptance given in Table 2. EXPERIMENTS

Experiments were performed with both closed cell and air transpiration systems to identify possible difficulties in the operation of such systems, to guide the formulation of a meaningful theory, and to provide measurements of performance that could be used to determine the validity of the theory. As shown in Fig. 2, all experiments were performed on the roof of Boelter Hall.*

Description of apparatus and instrumentation The apparatus comprised a test module mounted on a tiltable platform, weather instruments to measure ambient air temperature, humidity, velocity and direction, as well as direct and diffuse solar and total hemispherical irradiation, and the necessary multipoint strip-chart recorders. For the transpiration system, shown installed in Fig. 2, a filter and insulated duct is connected to the solar converter module intake while the exit plenum chamber is connected to the fan suction end through a flowrator. Variable flow was achieved by throttling the fan suction line. As shown in Fig. 3, the test module comprised a glazed plywood housing filled with *Engineering Building, University of California, Los Angeles, latitude 34"N and longitude 118*W.

200

H. B U C H B E R G , O. A. L A L U D E and D. K. E D W A R D S Table 1. Measurements of spectral normal transmittance of the solar c o n v e n e r glazing Wavelength (micrometers)

Normal transmittance

0-372 •40O •415 •430 •457 •481 •510 •540 •569 •600 •630

0.885 .887 •885 '887 .900 -902 .902 •907 .905 •907 .897 .886 '870 "857

•6 6 0

•700 •740 •790 •840 •900 0.970

"845

1-050

"832 -822 •809 •807

• 150

-802

•250

.811

•3 8 0 •5 8 0

-822

.847 "865 -855 •855 -845 0.840

•760 1.960 2-070 •260 2.420

Table 2. Calculated directional solar transmittance and absorptance of c o n v e n e r glazing Angle of incidence (def.)

Directional transmittance

Directional absorptance

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

0.862 .862 -862 .861 .860 .858 -856 '852 .846 "837 .823 "801 •768 .719 .646 .540 0"392

0-066 •066 •066 •067 •067 .069 .069 •071 .072 .074 •075 '078 .080 "081 -081 •082 0-081

Performance characteristics of rectangular honeycomb solar-thermal converters

ABSORBER~

RECTANGULAR HONEYCOMB" - ' ~ %

8

P'"-- GLASS O RCOVER O U S S

I/

20 ] IN. HOLES

P

,,,

L,,,,...o.,l L,.,o...,O.[c" EXIT PLENUM

I/2 IN. PLYWOOD

TEMPERATURE PROBE

,

UM

~

.oo..,o ~J I DE BAR

UMPLED ALUMI NIZED MYLAR

Fig. 3, Vertical section of the experimental transpiration test module.

Coming Foamglas insulation, accommodating a 12× 12 in. honeycomb-absorber combination. The transpiration module, shown in Fig. 4, was provided with perimeter plenums, to permit a distributed air inlet and exit through small passages drilled through the housing wall and insulation equally spaced around the perimeter. The experimental module designed without transpiration was similar to the module of Fig. 4, except that the air distributing plenums and thermocouple probe support structure were deleted. A ~ in. blackened copper plate, with ~ in. diameter brass tubes soldered to the back side between inlet and exit headers, comprised the absorber for most of the experiments. The honeycomb cells were essentially closed by the absorber plate on the bottom and the glazing on top. For both modules, a nominal insulation thickness of 6 in. on the sides and 8 in. on the bottom was provided. lron-constantan therrnocouples were installed throughout the insulation to provide a measure of conduction heat losses. Other instrumentation for the test modules included thermocouples to measure the glazing temperature, absorber temperatures, inlet and exit fluid temperatures, and the temperature distribution in the walls and space of a single honeycomb cell. The same honeycomb structures were employed in both systems. Measurements were made with three configurations: ( 1) L/d = 7-11, w/d = 3.4 for L = 1.5 in.; (2) L/d = 4-67, w/d= 3-4 for L = 1.0in.; (3) L/d=4.17, w/d= 2.1 for L = 1.5in. The initial intention was to construct test honeycomb structures from commercially processed plastic-coated paper, vacuum aluminized on both sides. After a search for potential suppliers of the basic material and possible fabricators, it became apparent that, in the interest of time, it would be most expedient to construct the cellular structures from a laminate consisting of commercially available aluminized 1 rail Mylar bonded to each face of 18 rail paperboard. The laminated sheet was then cut into strips 12 in. long with widths equal to the desired depth of cells. The strips were notched at the desired

202

H. B U C H B E R G , O. A. L A L U D E and D. K. E D W A R D S

intervals, assembled in "egg-crate" fashion to form a rectangular honeycomb, and then dipped in a clear polyurethane paint (Finch Co., Top Coat Gloss Enamel, 600 series) to obtain a coating thickness of approximately 1 mil. A photograph of a finished and instrumented honeycomb is shown in Fig. 5. To make the cells visible, back lighting was provided below the supporting glass plate. Also visible at the lower edge is a reflection from the glass plate. The honeycomb-porous bed combination was assembled as shown (Fig. 3). The porous bed absorber was made from a standard 1 in. fiberglass filter sprayed with black paint (Minnesota Mining and Manufacturing Co. Velvet Black) and compressed to ½in. when placed over a combination of blackened close weave fiberglass cloth supported by a ¼x ¼in. mesh wire screen and wooden frame. The fiberglass cloth served as a uniformly distributed flow resistance to help achieve uniform flow over the honeycomb. A {t in. gap was provided between the porous bed and bottom insulation, for an exit plenum communicating with the perimeter plenum through uniformly spaced in. holes. To prevent air leakage along the outer edges of the cellular structure, it was carefully sealed with silicone sealer adhesive to the foamglass insulation at the top edge. A single cell in the center of each honeycomb was instrumented to provide wall and space temperature profiles. Each laminated wall was split apart, allowing the insertion of 1 rail thermocouples at three depths on a centerline and then resealed. In addition, three 1 mill thermocouples were suspended between the cell walls at corresponding depths. Only one cell was instrumented in each test structure because of the intricacy of the procedure. As shown in Fig. 6, 12 30-gauge thermocouples were installed in the air gap between FLUID INLET PIPE

GUIDE BAR I0

II

12

I

TEMPERATURE

PROBE

, p

15

m~mamm~

I

2

14

I I t ~AL.r.~=.~.:J-a.

:3

E---~-W

12

N

II r

7

6.

.5

S

,

4 L

[~

z_PROK SLIDE PLATE FLUID OUTLET PIPE

N FRAME

NLET FLUID TEMPERATURETHERMOCOUPLES

Fig. 6. Top view of the experimental transpiration test module.

Performance characteristics of rectangular honeycombsolar-thermal converters

203

the top of the side insulation and the plywood cover. Each surface seen by the thermocouples was covered by a high reflectance aluminized Mylar sheet. The exit fluid temperature was measured by means of five 30-gauge thermocouple probes, movable in unison,just below the porous bed (Fig. 3) and spaced as shown (Fig. 6). The probes and slide mechanism are clearly visible on the right in Fig. 4. The probes were supported by bearing holes drilled through the wooden support frame, assuring translation in a straight line (see Fig. 6). For additional details regarding the apparatus, instrumentation and experimental procedure, the reader is referred to [9]. Experimental procedure

For all tests, the solar converter was oriented to face due south and the honeycomb located so that the long cell dimension (w) would lie along an east-west axis. Data collection always began after the test module was exposed for a minimum of 30 minutes. The system was kept running continuously during the day, during which time the angles of tilt and air flow were held constant. All environmental inputs were measured continuously, while the temperatures distributed throughout the test module were recorded at half-hour intervals. The angle of tilt used was generally that angle which resulted in the solar rays being approximately perpendicular to the entrance plane of the honeycomb at solar noon. In addition, several tests were conducted with an angle of tilt of 57", the position for normal incidence at solar noon during the winter solstice. During the transpiration experiments the outlet fluid temperature, measured with the five movable thermocouple probes, was recorded at 12 equally spaced locations, resulting in a matrix of 60 values at each time interval. Experiments were conducted with air flows of 6.7, 11.5, 15.0, and 20.0 lb/hr ft 2, making possible an exit air temperature range of about 100-250°F. Closed-cell results

The thermal performance of the honeycomb solar converter, operated as a closed cell system without transpiration, may be represented by an effective emittance ~e, defined as the ratio of the net power loss from the glazing to the net radiation exchange between infinite black parallel surfaces at temperatures equal to the absorber and glazing surfaces, respectively. Thus A

¢e = q L / ~ ( T A ' - - Tg4).

(17)

The net power loss from the glazing qL can be determined from measurements in two ways, viz. qL = ";g,G s -- qi -- qs -- qu

(18)

qL ---- ho ( Tg -- Ta) + egtrTg 4 - otg, G~ -- aglG,ur.

(19)

and

In either case, an accurate determination of qL is difficult. The insulation loss q~, power

204

H. B U C H B E R G ,

O. A. L A L U D E

and D. K. E D W A R D S

storage qs and power transferred to the transport fluid qu were determined as follows:

q~ = ~ (IqAf)ATi/Li

(20)

i=l

where n refers to the number of insulation blocks on the sides and below the absorber N

(21)

q8 = E (mjc~)dT~/dt J=t

N to the number of solar converter components that account for significant heat storage in the honeycomb, absorber, and side and bottom insulation

qu = p:VcpAT:

(22)

and ATe to the temperature rise of the fluid (a direct measurement). Some of the results are given in Table 3 for three honeycomb configurations, different angles of tilt, and for times near solar noon and displaced from noon. At each time, values are given for glazing and absorber mean temperature, single cell wall temperatures (top, middle and bottom) and the resulting effective emittance. The two angles of tilt used for each honeycomb configuration test represented the angle required to make the solar rays perpendicular to the entrance plane at solar noon, for the time of the experiment and for the winter solstice. As expected, the absorber temperatures were highest for the more favorable angle and for the largest w/d ratio or smallest value T a b l e 3. Non-transpiration s y s t e m p e r f o r m a n c e results H o n e y c o m b wall t e m p e r a t u r e s °F

Solar

To

T,

time

(°F)

(°F)

(1) w / d = 10.69 12" 14 12"65 15" 12 ( 1) w / d = 11"08 12"13 12"63 15"52 (2) w / d = 10"76 12"24 15"24 (2) w / d = 10.73 12.23 15-23 (3) w/d = 10-23 12-13 15"13 (3) w / d = 10.34 12"14 15.14

3.4, L i d 105 111 109 108 3"4, L / d 111 113 116 96 3.4, L / d 130 145 ! 18 3-4, L i d 144 146 123 2.1, L i d 118 129 122 2. l . L / d 126 131 122

Nu

To

To

TL,,

TL12

T,~

TL

E,

theory

exp.

theory

exp.

theory

exp.

exp.

~(exp.)

162 172 189 180

188 201 203 190

187 207 211 190

0-97 0.82 0.76 1.28

0"52 0"47 0.57 0-62

166 173 177 138

176 184 187 142

161 172 175 138

2.07 1.80 I '85 2" 17

1.09 0.96 0'95 0"90

215 246 217

229 265 231

231 269 234

0.78 0.74 0-70

0.67 0"61 0.64

229 251 217

244 271 231

239 267 228

0.79 0.80 1.00

0.93 0.77 0.71

183 222 211

197 245 228

177 251 230

0-81 0-74 0"92

0-83 0-67 0.77

196 227 208

219 248 227

204 247 220

0-94 0.83 1.05

1.14 0.85 0.88

= 7.11, 3' = 11°, m 192 1"39 223 1"56 224 1"62 187 1-29 = 7.11, 3" = 57 °. m 146 1"0 159 1"0 161 1"0 133 1"0 ==4"67, 3' ~ 35 °, m 245 1"35 291 1.39 246 1"53 = 4.67, 3' ----57% m 243 1- lO 278 1.32 235 1.39 = 4.17, 3' -----28 °. m 206 2"56 276 2"95 243 2"88 = 4-17,3" -----57 °, m 201 2"28 258 2"8 ! 226 2-71

= 3-3 lb/hr fl', 6/14168 139 150 159 140 158 167 146 161 179 162 157 177 = 3" 3 lb/hr ft 2, 6/15/68 179 154 177 181 159 183 185 163 186 146 129 144 = 0, 9/24168 167 202 202 184 229 231 167 200 203 = O, 9/25/68 202 220 224 208 237 244 181 204 208 = 0, 9/7/68 158 162 180 174 189 216 181 183 207 = 0, 9/9/68 191 177 206 198 196 226 194 182 212

E,(theory)

Performance characteristics of rectangular honeycomb solar-thermal converters

205

of d. An u n e x p e c t e d result is the slightly lower glazing temperatures with higher absorber temperatures for configuration (1). Detailed examination of the data at the time of highest absorber temperature, at 12.65 for 7 = 11° and 12.63 for y = 57 °, indicated that the net heat loss qL from the glazing was greatest for the higher T~ case. As shown in Table 4, values of qL based on Eqs. (I 8) and (19) c o m p a r e reasonably well. Whereas the glass temperature with an 11 ° angle o f tilt is less than the temperature for 57°, the heat loss qL is greater because o f the difference in environmental conditions. Also unexpected were the large values of Ce for configuration (1) when 3 / = 57°; however, they can be explained by the conditions of the experiment. F o r y = 57 °, during 15 June, the polar angle of incidence of the Sun is fairly high, causing many absorptions of the solar beam by the walls as it penetrates the cell and resulting in wall temperatures even higher than the cooled absorber, U n d e r these particular conditions, it is apparent that the h o n e y c o m b cells act as solar and long-wave radiation shields and cause an undesirable increase in the long-wave radiation loss. Table 4. Comparison of heat loss (qD based on Eqs. (18) and (19) 3' Deg.

Tg °F

TA qL B.t.u.Par °F Eq.(18) Eq.(19)

I1 57

109 116

224 161

152 139

148 124

T~ °F

u, ho Go mi/hr B.t.u./hrft2°F B.t.u./hrft 2

70 10"5 74 5"1

3"3 2.0

305 223

Transpiration results Both the detailed thermal response and gross performance was determined for the solar thermal converter operated as a transpiration system. The instantaneous glazing, cell wall, and inlet and exit fluid temperatures constituted the measures of thermal response permitting the calculation of thermal efficiency,as follows: ~1 =

dTpl / mc~,(T~,-T,)+ce"-~J/G,

(23)

and 13 = "0(1 -

TalTp)

(24)

for each time the temperature measurements were made. T h e inlet fluid temperature was an arithmetic average of the values recorded by the 12 thermocouples distributed in the inlet gap. Examination of the inlet temperature distribution for all tests indicated several trends. I. T h e r e was the possibility of instability,in the inlet gap, manifested in a tendency at low flows for temperatures to be slightly higher on the top side. 2. C o r n e r temperatures, shown as 1,4,7, and 8 in Fig. 6, were generally lower than for adjacent areas, probably due to increased velocity in those regions brought about by the narrowing of the flow channel toward the honeycomb. 3. T h e variation in inlet temperature with location was minimal at higher flows, less than about +_ 2°F from the mean value at 20 lb/hr ft 2.

206

H. B U C H B E R G , O. A. L A L U D E and D. K. E D W A R D S

The 60 exit fluid temperatures recorded over a period of about 15 minutes were synchronized with the inlet temperature employing parabolic interpolation, and were adjusted in accordance with area weighting and an estimated air velocity distribution based on an assumption of no recirculation in the cells. Heat storage was taken into account by means of the second term in the numerator of Eq. (23). The total converter heat capacity cc was determined by measuring the decrease in exit fluid temperature Tp, over a time interval At, after suddenly cutting off incident solar radiation. The solution of a simplified transient energy equation gives cc = ( U~ + mcp) /t{ ln[ ( Tp)l -

TaJl ( Tp)2- T,,I}-'

(25)

permitting the calculation of co. The only strong trend discernible in the distribution of exit fluid temperatures over the 60 locations was the relatively small variation along any east-west line and consistently large variations between probes 1 and 5 for all north-south locations, with the top (1) or north position always the highest. A careful calibration of the individual probes against each other ruled out any instrumentation error. The variation seemed to be greatest during times when the temperatures were highest. Differences in inlet temperatures at the same locations were insufficient to account for the large difference in the exit temperatures. Steeper angles of tilt seemed to accentuate the differences, but the temperatures did not always increase continuously along a north-south line toward the north (top). It was concluded that flow through the porous bed was not uniformly distributed. As mentioned previously, the temperatures were adjusted to account for the nonuniform velocity and it was assumed that the velocity was approximately inversely proportional to (Tp -- T0 at each location. To avoid duplication, experimental results are presented and discussed in the next section dealing with comparisons between theory and experiment. COMPARISONS BETWEEN THEORY AND EXPERIMENT In comparing the theory with experimental results, consideration will be given to the validity of certain postulates necessary to make the theory tractable and to the ability of the theory to predict the thermal response actually observed as well as the gross thermal performance. Closed-cell system The closed-cell theory rests on three important postulates: (1)heat-transfer modes are uncoupled; (2) the cell wall temperature distribution is determined only by radiation exchange; (3) wall emission varies linearly with cell depth. Postulates (1) and (2) are reasonable when natural convection has not been initiated in the cells. When the critical Rayleigh number has been exceeded, the theory will predict wall temperatures too low in the upper region of the cell. Natural convection in the cells increases wall temperatures toward the top, tending to flatten the gradients with depth causing deviation from postulate (3). Significant radiation exchange tends to increase the critical Rayleigh number[6]. Another factor omitted in the theory, that results in an underestimation of upper wall temperatures, is the solar absorption of the exposed edges of the cells which accounted for 9 to 13 per cent of the irradiated area. Inaccuracies in the solar and long-wave properties of the cell walls may also have a significant effect on the

Performancecharacteristicsof rectangularhoneycombsolar-thermalconveners

207

predicted wall temperatures. For example, underestimation of ~'s tends to cause predictions of wall temperature that are too high. This would be accentuated as the polar angle of incidence of the Sun increased, particularly in the upper cell area where the shape factor with respect to the glazing is the most favorable. Upon examination of Table 3, where a comparison between theoretical predictions and experimental results are given for three honeycomb configurations, each with two angles of flit, several inferences may be made. 1. For configuration (1), T = 57°, when natural convection was not initiated, predicted wall temperatures are high especially at the upper end of the cell. This is probably due to the exposed edges and an underestimation of rs which would be particularly visible when the polar angle of incidence 0 is large. The effect diminishes considerably as 0 becomes smaller, as is apparent for the other cases where 3' = 57°, the polar angles being smallest on 9/25/68. 2. The effect of free convection is quite apparent for the three configurations when 0 was approximately zero at noon. At these times, upper wall temperatures are considerably underpredicted. For configuration (1), 3' = 11°, it will be noted that as 0 increased from zero, at noon, to a value corresponding to 15-12, the predicted value of upper wall temperature went from 18° less than the experimental value to 5° greater. 3. In general, the theory seems to underestimate the effective emittance, which is what one would expect for cases of significant instability and top edge solar absorption. For all experiments, there was excellent agreement between the predicted lower wall temperature and the corresponding experimental value.

Transpiration system Several experimental observations indicated possibilities for simplifications in the theory used to describe the thermal behavior of the honeycomb-porous bed solar convener. For all measurements made at three depths with three different honeycomb configurations over many different times, the cell wall temperature distribution at each depth was uniform within 2°F on the average. The difference between cell wall and fluid temperature (T~.-Ts) was small, justifying the assumption of infinite convective conductance. On the average, (Tw-- Ts) was 3°F for w/d = 3.4 and 5°F for w/d = 2" 1. However, there was a visible tendency for l f, in the lower region of the cell, to exceed the wall temperature to a greater extent the lower the flow, indicating the effect of conduction from the porous bed and the possibilities of instability in flow. (Ts - Tw) was greatest for the lower w/d and L/d cells. Cell wall and fluid temperature predictions based only on molecular conduction in the fluid from the absorber were of the order of 25°F below the measured values. This discrepancy could not be explained by errors in calculations of solar and longwave honeycomb transmittances, leading to the conclusion that instability in flow, particularly at the lower transpiration rates, contributed to an increase in heat transfer from the absorber to the fluid above. An effective fluid conductivity, (Ks)e, is defined such that the ratio (ks)elks is a function of mass flow/unit area m and honeycomb geometry. From a solution of the conduction equation for the fluid column and experimental measurements, it was determined that (ks)e/ks = 1 + 10 m/(L/d ) was a reasonable basis for representing the enhanced heat transfer from the absorber. The sub-

H. BUCHBERG,O.A. LALUDE ~d D. K. EDWARDS

208

sequent cell wall or fluid temperature predictions never deviated more than 10 per cent from the measured values, with the exception of the lowest flow. For most observations, agreement is considerably better. The assumption of a linear cell wall temperature distribution with depth was fairly well substantiated, as shown in Fig. 7 for two cell configurations. All other temperature gradients measured were essentially similar to those shown. 160

•W

12.2 HRS---,~, f m=6.7LB/HR

w/d=2.I,L/d =4.17

b.

11.2 H

140

~

t

CC

5/15/68

n,. laoi M.I

.= ,ooi uJ

I'm=20.O L B / "=1 HR-FT2 T= 14" L5/20/68

I-.

12.7 HRS

.o

60

I

.o F

w/dr3.4, L/d =7.11

f m=6.9 LB/RH 1 -FT2

160

L3/3/oo p-2O.SLS,

I~- 120 w P" ..j..= I 0 0

t

~

< •

-FT2

y=15 o

RH_FT 2

/ T - 3e" 80

10.4 HR$ - - /

6o 0

11.gHRS

/

L5/10/68

I

I

I/2 L DISTANCE

L

Fig. 7. Measurements of cell wall temperature distribution.

The validity of the assumption of a semi-infinite porous bed was checked by calculations of exit air temperature based on grey, nonscattering, nonconducting theory [10] using a value of absorption coefficient, K = 2"54 in. -t, and a very large volumetric heat-transfer coefficient. For the range of air flows covered in this study, it was found that (Tp)o.d(Tp)= was approximately 0,98, where (Tp)o.~ and (Tp)® are the absolute exit air temperatures for a 0.5 in.-thick and an infinitely thick bed, respectively. Subsequent measurements of the 0-5 in.-thick blackened fiberglass medium used in the present experiments gave a value of K = 7.4 in. -~, which indicates that essentially total solar absorption occurs close to the upper absorber face. As mentioned previously, the calculation of ~'8 (details of which are given in the appendix) accounts for the spectral, directional and polarizing effects of the cell walls. It was of interest to determine the importance of accounting for directional and polarization effects, since calculations may be considerably simplified on the basis of spectral normal reflectance only. Accordingly, a comparison study was made of solar

Performance characteristics of rectangular honeycomb solar-thermal converters Table 5. Effect of polarization and directional variations on solar transmittance calculations

L/d

Cos 0

8 -----0-2 mil (%),, ~,

8 ==0.5 rail (%),, 7-,

8 = 2"0 mil (%),, %

0"340 •340 "340

4"0 7"0 10"0

0.448 "246 "135

0-351 "170 4)85

0.445 "244 "133

0"292 "124 "055

0"433 "232 "124

0"123 -030 "008

.459 -459 -459

4-0 7.0 I0-0

.609 .426 -296

.518 .327 .209

.607 -423 -293

.461 "268 -158

-596 "411 .281

.266 . 106 .045

.546 .546 .546

4-0 7"0 10-0

.723 -573 .448

.649 -479 -351

-721 -570 "445

.601 "420 -291

-713 -559 .433

.419 -229 "123

-783 -783 -783

4-0 7.0 10.0

.737 .572 -443

.670 .487 .357

"735 "570 -441

.625 -430 -297

-727 "559 .428

-446 -236 -127

-918 -918 •918

4.0 7"0 10.0

.860 .737 -633

.814 -OM .548

-859 "735 .631

.784 -615 .489

-854 "727 -621

"656 .427 -287

.989 -989 0.9e9

4"0 7.0 10.0

-952 .903 0.860

.938 .877 0"824

.951 "903 0"859

-928 .857 0-797

-950 .899 0-854

.887 .774 0"688

w/d I, 5"0, 6 1, 36" for various values ot 8. Table 6. D

~

aad polarization influence on solar transmittance calculations

w/d = 3"0 Cos 0

Lid

w/d = 5"0

w/d = 8"0

(%),

%

(~,).

%

(%).

%

0-340 -340 -340

4-0 7"0 10"0

0"394 "200 -100

0-246 "096 -039

0"445 -244 "133

0"292 "124 "055

0"479 -292 -160

0",~25 "158 "069

.459 .459 -459

4.0 7.0 10"0

.563 -359 .228

.414 .213 .113

.607 .423 "293

.461 -268 .158

-671 .464 -318

-536 -306 "175

•546 •546 •546

4.0 7,0 10-0

-671 .503 -363

.540 .351 .218

"721 -570 .445

.601 .420 .291

"759 .644 .499

-649 -505 -343

-783 "783 .783

4.0 7.0 10.0

.712 -560 -441

"595 .419 .297

"735 "570 -441

.625 -430 .297

"735 ,541 .441

.625 .398 "297

-918 -918 -918

4-0 7.0 10.0

-844 -735 -642

.763 -615 .526

.859 .735 "631

-784 .615 .489

.815 -735 .664

.721 -615 -527

.989 -989 0.989

4-0 7.0 10.0

-935 .903 0-873

.905 .857 0-817

.951 -903 0"859

-928 .857 0-797

-903 -903 0.903

-857 "857 0-857

8 = 0.5 mil, t# = 36° for various values ofwld.

209

210

H. BUCHBERG, O. A. L A L U D E and D. K. E D W A R D S

transmittance ~-,, accounting for the spectral, directional and polarizing effects, and 0"~). accounting only for spectral effects (by precalculating a solar-normal reflectance (Ps). based on 50 bandwidths of equal solar radiant intensity): (p.),, = ~ ~ p,,x,

(26)

P.~ = Pn~ + [P.~'x2 (1 --p.~:)]/[1 - p.~p.}=r~ 2] (see [111)

(27)

O.x~ = [ (n~,a- 1)2+k~x=]/[ (n~., + l ) Z + k ~ ]

(28)

iffil

where

= [ (n l - n,,.) 2 + le,,, ] / [

+

~'x = exp [--4,xk~lS/k].

+

(29) (30)

Subscripts 1 and 2 refer to the first and second reflections from the dielectric film and aluminized substrate surfaces, respectively. The results of the study are given in Tables 5 and 6 where (~-,), is compared to ~', for polar angles of incidence 0 from about 8 to 70 °, for various honeycomb configurations (L/d = 4,7, l0 and w/d = 3,5,8) and for several wall overcoat thicknesses (8 = 0.2, 0.5, 2.0). It is clear from the data that the error in the calculation of z,, when neglecting directional and polarization effects, increases as 0 increases, as Lid increases, as w/d decreases and as 8 increases. The errors can be very substantial; for example, on 22 June, for 8 ~ 0-5 rail, w/d = 5.0, L/d = 7.0 and for a polar angle of 70 ° corresponding to 08.00 with an angle of tilt equal to ( 6 + 15°) where the latitude is 36 °, (~-,),/r, is about 1.97. The ratio reduces to 1.32 at noon for the same conditions. With a more favorable tilt angle (6-- 150) on 22 June and with the same honeycomb configuration, (~-~),/r, is about 1.35 at 08.00 and 1.05 at noon. It was concluded that, for the present study, account should be taken of directional and polarization effects. The theoretical predictions of glazing, cell wall and exit fluid temperature were determined by solving Eqs. (13), (14), (15) and (16) for the identical conditions existing during the corresponding experiments. Comparisons between theory and experiment can be made only for two honeycomb configurations (w/d = 3.4, L/d = 7.11; w/d = 2. I, Lid = 4.17), because of the failure of the wind speed recorder and total hemispherical radiometer when the third configuration was under test. A comparison of predicted and measured glazing temperatures is given in Fig. 8 for w/d = 3.4, Temperatures as a function of solar time are shown for several rates of transpiration. Agreement between theory and experiment is good, with the exception of the lowest flow when the absorber temperatures are highest. Comparisons are not shown for w/d = 2" 1, because of apparent difficulties with the glazing temperature measurements for some of the tests; however, it can be reported that agreement of 1 to 5°F between theory and experiment was achieved for m = 6.7 lb/hr ft2. The essential prediction in connection with the thermal performance of the solarthermal converter is the exit fluid temperature for specified environmental conditions, inlet fluid temperature and flow. Figures 9 and l 0 display the predicted and measured exit temperature of transpired air as a function of solar time for several flows. Agreement between theory and experiment is excellent.

Performance characteristics of rectangular honeycomb solar-therraal converters

I10I00 ~

9O

~

8o

(0) m-6.9 LB/HR-FT 2, )'=37 e (3/14168) 0 GLAZINGTEMP (MEASURED) GLAZING TEMP (PREDICTED) D AMBIENTAIR TEMPERATURE

000000000 •

W Q.

~

~J I,-

/\/Xooo

Or--~

70

oOoOoooooooooo-

8O

,

50

I00 b.I

211

I

I

I

I

(b) m-ll.5 LB/HR-FT 2, y - 3 5 " (3/19/68)

90

0000

80 0 °

70 I--

I

60

.=-

I

I

I

I00 -(c) m,,16.0 LB/HR--FT 2, ),-51.5 e (2/3/68)

8O

i "° ?0

al

6O 9O

I

I

I

F ( d ) m,, 20.SLB/HR-FT2, y • 38 e 131101681 -

70

i

60 8

I =

I

I

I

I

I0

12

.14

16

18

SOLAR TIME, HOURS Fig. 8. Comparison of predicted and measured glazing temperatures (w/d = 3.4, Lid = 7.11, 6 = 1.0 mil).

The thermal performance of the three configurations tested is displayed in Figs. 11 and 12, in terms o f ~ and//, defined by Eqs. (23) and (24), as a function of (Tar - To)/Gs for all-air flows and times when measurements were made. As can be seen in Fig. 11, the mean curves for all data correlated are linear and ~ decreases only moderately with (Tav--T=)/G,, with a maximum variation from the mean of from +-5 to +-_8 per cent. The scatter reflects primarily the solar directional influence on performance, while the relatively small negative slope reflects the effective reduction in radiative losses at the higher absorber temperatures. Greater scatter is visible with respect to the mean curves f o r / / i n Fig. 12. It is of interest to note that/3 increases with (T=v- To)/G, and apparently reaches a maximum value at a point that corresponds to less than maximum "0- Figure 13 compares the mean theoretical predictions of ~ with the experimental mean curves. Also shown (in Fig. 13) are the data given in [12] for the corrugated air

212

H. B U C H B E R G , 220 '-

O. A. L A L U D E

a n d D. K. E D W A R D S

o

(b) rn = 11.5 LB/HR-FT 2 , y=35" (3/19/68)

0

~'- 200 uJ

0 o

ne

~- 180

O00

0 °

t~

~

160

~" 140 ,.~ 120

0

i'--

~ I00

.

~ (3/14/68)

-- (a) m.6.9 LB/HR_FT2 y=57 , I

8G

I

EXPERIMENT THEORY

\

N,O

)~)

I

I

I

I

I

~'. m o ,(c) m=16.0 LB/HR-FT 2,y=51.5 = (2/3/681

~6o

(d) m=20.5 L B / H R - F T 2 ,y= 38 ° (3/10/681

ne

UJ a. 1 4 0

"5 IM

P" 120

~tO0 I--

~u 8 0

9

l It

1

I

13 15 SOLAR TIME, HOURS

I 17

1 9

I

1

II 13 15 SOLAR TIME,HOURS

I 17

Fig. 9. Comparison of predicted and measured exit fluid temperatures ( w / d = 3-4, L / d = 7.11, 8 = 1-0 rail).

heaters I and II. The agreement between theory and experiment is excellent and it is apparent that the honeycomb air heater has some advantage over the corrugated plate sol,u" heater at the higher absorber temperatures. For a more complete understanding of the performance characteristics of the honeycomb-porous bed solar air-heater, it was of interest to determine the sensitivity of ~ and fl to variations in air flow, angle of tilt, and convective conductance at the external glazing surface. Figure 14 presents ~ and/3 as a function of air flow calculated for the honeycomb configuration and conditions specified. Note that for all of the times shown, fl reaches a maximum at about 3 lb/hr ft2 whereas maximum ~ is not approached until flows of about l0 lb/hr ft2 or greater. The optimum flow for a particular design application can be determined by minimizing costs per unit of energy utilized at the temperature level required, including the cost of moving air [ 14]. The effect on ~ and B of variations on angle of tilt y and the convective conductance hc is shown in Tables 7 and 8. It is apparent from Table 7 that the sensitivity of performance to variations in y is dependent on the time of year, being much greater at the solstices than at the equinoxes. Table 8 indicates that a 5-fold increase in hc causes a significant but small decrease in performance, greater as the difference in temperature ( T~- Ta) becomes greater. SUMMARY

AND CONCLUSIONS

Theoretical relationships that describe the performance of closed cell and transpiration honeycomb solar-thermal converters have been presented. Comparison of predictions with experimental results indicates that the closed-ceU uncoupled theory is

Performance characteristics of rectangular honeycomb solar-thermal conveners 220 la,.

213

- ( = ) m=6.7 L B / H R - F Tz, ?,-15" (5115168)

200

e.

_~

=so

~

iso

°

_

0 EXPERIMENT

,,o

w

120

-

~

I00 • 180

P

THEORY I

I

I

I

1

(b) m- 11.2LB/HR-FT 2 , ~'- I S* (5116/68)

160

,40 IAJ 120 l,,I

Io¢

1

I

I

I . 160 F (d)m,20.O LB/HR-FT 2, r-14* (5/20168) 140 F

tO0 L 8

I I0

I |2

....

I 14

I 16

I 18

SOLAR TIME,HOURS

Comparison of predicted and measured exit fluid temperatures (w/d = 2.1, L i d -ffi 4.17, 6 = 1.0 rail).

Fig. 10.

Table 7. Effect of tilt angle on converter performance 22 March Tilt T--15 ° T T + 15°

9 a.m. 0"55 "58 0"55

0-071 .078 0-070

22 June Noon

0.64 -67 0"64

0.116 .126

0.116

9 a.m. 0"56 "51 0-40

0.073 .060 0-038

Noon 0-65 "62 0:56

0.115 -101 0.079

T a = 7 0 ° F , T~= 100~F, h e = 3 - 0 B . t . u . / h r t t 2 °F, m = 10.OIb/hrfl=, w/d=5.0, L/dffi7.0, 8 = 0-5 rail.

adequate only when the critical Rayleigh number is not exceeded. When natural convection is initiated in the cells, it appears to be necessary to couple the radiationconvection-conduction modes to improve the prediction of cell wall temperature

S.E.Vdl3NoZE

214

H. BUCHBERG, O. A. LALUDE and D. K. EDWARDS v 6.7 A I 1.0 0 15.5 rl 20.0

'°f

¢. 0.6 |(a) >: o o.4--"

~

w/d=5.4

LB/HR- F'r'2 LB/HR-FT 2 LB/HR-FT 2 LBIHR-FT z

,L/d=4.67

I

I

l

I

I

I

i,

I

0.6

(b) w/d-3.4 I, L/d 7.11

~- 0 . 4 ~

I

,

(c) w/d 2.1 L/d 4.17 04~ i 0.05 0.10 O.U5 0.20 0.25 (Tav- Ta)/G=, * F / ( B t u / H R - FT z)

l-0.30

035

Fig. I 1. Thermal efficiency of experimented honeycomb-porous bed solar converters.

0"20 1 21 V 0.15 O 0 0.10

6.7 LB/HR-FT 2 I 1.0 LB/HR-FT 2 I 5.5 LB/HR-FT 2 LB/HR-FT2

(o)

w/d-5.4, L/d =4.67 ~7

d o.05 (b) w/d'3.4, Lid= 7.11 o,o

!ii0-

(c) w / d ' 2 . l , Lid=4.17

X? ~

-~-.---

.......~o-D.J

0.05

Fig.

12.

I O. I0

I I I 0.15 0.20 0.25 (Tav -T=)/G=, "F/(Bt u/HR- FT 2)

J 0.30

Availability factor for experimental honeycomb--porous bed solar convortol~.

distribution, which significantly affects the calculation of effective cell emittance. G o o d a g r e e m e n t was obtained between theory and e x p e r i m e n t for the air transpiration systems, permitting calculations to show the effect of variations in air flow, angle of tilt,

Performance characteristics o f rectangular h o n e y c o m b solar-thermal converters 1.0

i

0.8 0.6

UJ

. 0.4 ,¢[ :S

--

°'°'-...°.,°

,,¢, 0 . 2

(o)

-r

w/d=3.4 , Lid-7.11

I-

1

(;

I

I

I

1.0 E XPERIMENT

i

0.8

•- . -,..

----- THEORY

0.6 "''-.,.,.,

m 0 . 4 --

"-.....

..J

0.2

-

(b) w/d=2.1, Lid"4.17

t~J

~-

0 0.10

I

L

I

0.15

0.20

0.25

(Tar -1"= ) / ~ , ,

I 0.30

*F/(Bt'u/m-FTL~

Fig. 13. Comparison o f thermal efflciencies o f experimvntal honeycombs with predictions and with elf~ciancies o f heaters I and III offer. [12].

0.8

NOON- O.20

SCALE ~

0.6 ....

"~04

M" ......... l e e ".............. • emile• ¥ .... . ...................

0.2

(0) MARCH 22

NOON- O.IO B

me•melee

el

9A.~.,.,

,8 SCALE . . . .

I

0

0.15

,A,

.

I

l

I

O.8 -

0.05 0

-

O.20

NOON 0.6 -

O,15

................. NOON ~:.I feQo•ee ye ~ e ee~ • oee•eeeeoel • e ••e•eeemeee•eeeeeeee . . . . . . . . . . . 0.05 ,

O, 2 0

,AM2 o,o

........

~1 0.4

I

O

5

I

I

i

I0 15 20 FLUID FLOW, L B I H R - F T 2

0

25

Fig. 14. Effect o f fluid flow rate on thermalefficiency for Ta = 70°F, Tt = 100°F, he = 3-0B.t.u.lhr ft ~ °F, w/dffi 5-0, L/d= 7.0, 8 = 0"5 rail, y = ~b(d) - 36~1 solar inputs [ 13]).

215

216

H. B U C H B E R G , O. A. L A L U D E and D. K. E D W A R D S Table 8. Effect of glazing surface conductance (he) on converter performance 22 March h~ B.t.u./lb ft* *F

9 a.m.

22 June

T~ °F

Noon

9 a.m.

"0

B

•

/3

71

/3

"0

Noon /3

1.0 3-0 5"0 3' = 4 , - 15" 1"0 3.0 5-0

100 100 100

0.61 .58 .56

0.085 "078 .074

0.70 .67 "65

0-136 "126 . 122

0-53 .48 -46

0-061 -053 "051

0.66 "62 -60

0.112 . 101 .097

100 100 100

.59 "55 '53

.079 .071 .067

.68 .64 "63

-127 • 116 • 112

.60 .56 .54

'082 -073 .069

"68 .65 "63

.124 • 115 • 111

1.0 3.0 5"0 y = q~- 15" 1-0 3"0 5"0

80 80 80

.69 .68 -67

'083 -080 .079

.75 "73 "72

• 134 • 129 • 127

.63 .61 .60

.060 "056 "055

.72 .69 "68

•110 • 104 • 101

80 80 80

"67 "65 0"64

-077 .074 0-072

-73 -71 0.70

• 126 • 119 0" 116

-68 .66 0.65

.080 -076 0.074

.74 "72 0-71

- 123 • 118 0.115

~,ffi4,

w/d= 5-0, L/d= 7.0, 8 = 0-5 mil, Ta =~ 70°F, m ffi 10 lb/hr flz, 6 = 36°. a n d c o n v e c t i v e c o n d u c t a n c e at t h e e x t e r n a l g l a z i n g s u r f a c e , o n p e r f o r m a n c e p a r a meters. F o r d e s i g n p u r p o s e s , t h e r m a l e f f i c i e n c y m a y b e g i v e n as a l i n e a r f u n c t i o n o f t h e p a r a m e t e r (Tar-T~)/G, i n d e p e n d e n t o f a i r f l o w , a n g l e o f tilt ( f o r a s o u t h - f a c i n g c o n v e r t e r in t h e n o r t h e r n h e m i s p h e r e o r n o r t h - f a c i n g in t h e s o u t h e r n h e m i s p h e r e ) a n d d i r e c t i o n o f t h e s o l a r b e a m . O f t h e t h r e e h o n e y c o m b g e o m e t r i e s i n v e s t i g a t e d , w/d = 3.4 a n d Lid-- 7-11 ( F i g . 1 1 b) g a v e t h e b e s t p e r f o r m a n c e . W i t h t h e t h e o r y p r e s e n t e d , it is now possible to determine the best geometry, overcoat thickness, and values of o p e r a t i n g v a r i a b l e s , a i r f l o w a n d e x i t air t e m p e r a t u r e , t h a t will a c h i e v e m a x i m u m o u t p u t p e r u n i t c o s t f o r a p a r t i c u l a r a p p l i c a t i o n [ 1 4].

Acknowledgement-The authors acknowledge support from a National Science Foundation grant which made this investigation possible. NOMENCLATURE A area, ft z c specificheat at constant pressure, B.t.u./IbOF d width of a honeycomb cellopening, ft Eb~ spectral blackbody emission, B.t.u./hrflz~ m f ratioof honeycomb open cellarea to totalhoneycomb cross-section area F(x) integratedlong-wave transmittance of a honeycomb cell ffi f / ~ ( x ) d x G G* h H* k K* L m M

irradiation, B.t.u./hr ft z dimensionless quantity •ffi GfirTt" convective heat-transfer coefficient, B.t.u./hr fl: oF or specific enthalpy, B.t.u./lb dimensionless quantity *= hT:rT: thermal c°nductivity' B't'u'/ht ft °F °r abs°rptive index dimensionless quantity ffi kTdd~rT: depth of a honeycomb cell, ft fluid mass flux density, lb/hrft 2 physical mass, lb

Performance characteristics of rectangular honeycomb solar-thermal converters n q t T u U~ w W* x

217

refractive index heat flux density, B.t.u./hr ft ~ thickness, ft or time, hours absolute temperature, °R velocity, fl/hr overall insulation conductance, B.t.u./lu'ft z °F length of a honeycomb cell opening, ft dimensionless quantity == pcuTdoTi 4 distance measured from the top entrance to a honeycomb cell, ft

Greek symbols a absorpmuce avaihbility factor =. ~(1 - T, IT,) y converter ingle of tilt from the horizontal, de8 8 honeycomb cell wall dielectric overcoat thickness, rail • emittance effective emittance thermal efficiency ==mc( T~-- T~)/G, 0 dimensionless quantity == T/TI or polar angle of incidence, deg wavelength, micrometer p density, lb/ft a or reflectance Stefan-Boltzmann constant, B.t.u./hr fl 2 *R4 transmittance ~b latitude angle, deg X dimensionless quantity == x/d @ azimuthal angle of incidence, deg Subscripts a ambient A absorber b pom~b~tb~ckbody c hom~jcomb ceil c r m s - ~ o n or c o B v m e r top cover f au~ g glass cover i inlet to converter or insulation l long-wave L valueatx==L m mean or average quantity 0 value a t x =: 0 p temperature probe or outlet from the porous bed absorber r radiatiou quantity s solar or storage sur s ~ n d i n g s w honeycomb cell wall REFERENCES [1] V. B. Veinberg, Optics in Equipment for the Utilization of Solar Energy. State Publishing House of Defense Industry, Moscow (1959) (English translation). [2] G. Francia, A new collector of solar radiant energy-theory and experimental verification. U.N. Conf. on New Sources of Energy, Rome 4, 572 (1961). [3] M. Perrot et al., Ides structures cellularies antirayonnantes et leurs applications industrielles. Solar Energy 11, 34 (1967). [4] K. G. T. Hollands, Honeycomb devices in flat-piate solar collectors. Solar Energy 9, 159 (1965). [5] D. K. Edwards and R. D. Tobin, Effect of polarization on radiant heat transfer through long passages. J. Heat Transfer, Trans.ASME89, 132 (1967). [6] D. K. Edwards and W. M. Sun, Prediction of the onset of natural convection in rectangular honeycomb structures. Preprint paper No. 7•62, International Solar Energy Society Conference, Melbourne, Australia (1970). [7] W. M, Sun and D. K. Edwards, Natural convection in cells with finite conducting side walls heated from below. Paper presented at Fourth International Heat Transfer Conference, Paris, France (1970). [8] P. Moon, Proposed standard solar radiation curves for engineering use. J. "Franklin Inst. 230 (5), 583 (1940).

218

H. B U C H B E R G , O. A. L A L U D E and D. K. E D W A R D S

[9] O. A. Lalude, Thermal behavior and design of cellular matrix-porous bed solar thermal converters. Ph.D. Dissertation, School of Engineering and Appfied Science, University of California, Los Angeles, 1969. [ 10] D. K. Edwards and A. T. Leong, Collection of solar energy in a porous bed, Department of Engineering, University of California, Los Angeles, Report No. 64-14, 131-142 (1964). [11] A. Vasicek, Optics o f Thin Films. North-Holland Publishing, Amsterdam (1960). [ 12] C.L. Gupta and H. P. Garg, Performance studies on solar air heaters. Solar Energy 11, 25 (1967). [ 13] J. L. Threlkeld, Solarirradiation of surfaces on clear days. Trans. A S H R A E 69. 24 (1963). [14] A. O. Lalude and H. Buchberg, Design of honeycomb-porous bed solar air heaters. Preprint paper No. 7/63, International Solar Energy Society Conference, Melbourne, Australia (1970). [15] E. R. G. Eckcrt and E. M. Sparrow, Radiative heat exchange between surfaces with specular reflection. int. J: Heat and Mass Transfer 3, 42 (1961). [16] E. M. Sparrow et aL, An enclosure theory for radiative exchange between specularly and diffusely reflecting surfaces. Trans. A S M E , Series C, 84, 294 (1962). [17] D. K. Edwards and J. T. Bevans, Effect of polarization on spacecraR radiation heat transfer. AIA,4 J. 3 (7), 1323 (1965). [18] M. Czerny and A. Walther, Tables o f the Fractional Functions for the Planck Radiation Law. Springer-Verlag, Berlin ( 1961 ). APPENDIX

CALCULATION OF CELL TRANSMITTANCES Solar transmittance Calculations of T, (xo) were made for each cell configuration, assuming that the cell walls were constructed from a composite consisting of a metallic substrate with a dielectric overcoat. The necessary inputs were the cell dimensions (d,w,L), wall overcoat thickness, spectral optical constants of the metallic substrate and wall overcoat (given in Table 9), guographical latitude, solar time of day, angle of tilt and day of the year ( 1 January assigned the number 0 and 31 December 364). The incident solar radiation is assumed to be a beam of parallel rays striking the entrance cell cross-section at a polar angle 0 determined from the inputs. A fraction of the incident radiation emerges at the other end of the cell by a combination of direct transmittance between the two ands and one or more reflections at the honeycomb walls which are assumed specular. The cell was subdivided, into a pre-inlmted number of equal strips along its depth, and its entrance cross-section was subdivided into a grid of equal area elements which were as close as possible to being squares. Taking each area element in turn, the path of a monochromatic ray, parallel to the incident solar beam and passing through the center of the element, was traced down the cell by the method of images [ 15, 16] until it emerged at the exit cross-section. At each reflection, the incident radiant intensity matrix was premuitiplied by a reflectance and a transformation matrix[17]. Elements of the reflection matrix were determined from a programnd solution of the electromagnetic equations [ 11] for the composite cell wall. These elements were always precaiculated for the specified overcoat thickness and for several values of 0 between 0 and 90°. Thereafter, the desired matrix elements were obtained for any 0 by simple interpolation. The cell transmittance was finally determined by averaging, over all area elements, the transmittances determined for all rays. Thus 1 at,

(A 1 )

elemen~

where Na = number of cross-sectional area elements r,~ ----transmittance of thejth ray = [I l~lp.~/[D,._,,,.llp,._,,,~)[O,._,~,._,, l .. .. ID,aII~A[[Dt=IJp,=[ 0.5

[p~] = spectral reflection matrix for the nth reflection ID~,-I>.~[ -- transformation matrix after the (n - 1) th reflection.

(A2)

P e r f o r m a n c e c h a r a c t e r i s t i c s of r e c t a n g u l a r h o n e y c o m b solar-thermal c o n v e n e r s

T a b l e 9. O p t i c a l c o n s t a n t s of aluminum and polyurethane o v e r c o a t in t he solar w a v e l e n g t h s , a s s u m i n g refractive index o f t he o v e r c o a t == 1.3

Wavelength (pan)

0-375 •392 -418 •432

Polyurethane * absorptive index 7-5 x 10 -s .4 7"3 6-8

Aluminum Refractive Absorptive index i nde x 0.360 .384 .437 .463

4.20 "37 "67 4-80

•452 •462 •477 •488 •498

-I -3 "5 .4 "3

"520 .546 .598 "635 -663

5"03 -08 "30 "42 "50

•510 •523 •533 •546

"3 "2 "2 •1

-700 .745 .780 "820

-65 .80 5"88 6"01

•558 •571

•1 .0

.860 .900

• 15 .25

•584 •597 .610 •624 •637 •648 •662 •678 •690

.0 -5 "5 -5 .4 "3 •1 -0 -0

0.950 1.010 "082 • 160 .228 -285 -360 .440 "528

"38 "52 "65 "82 6"91 7-10 • 14 .09 -03

•704

.3

.6 -6

.575 .652

.O0

•720 •735 .752

•770 •788 •805 '824

•842 •862 •888

-918 •958 0.990

1.008 •028 •056 •097 . 164 •210 -248 .287 .500 -568 -647 1.960

.725

"02 .06

"5

.805

"14

"5 .6 .6 .6

.880 1.953 2.015 .064 .080

• 14 .08 .05

"5

.4 -4 .4 6-8

.030

.05

• 10 .26

2.000

"55

1-896 .700

7-96

7-3

.750

8-65 8.33

"3 "3 7.1 6.8 6-8

'750 .750 '750 .750 -800 .800 .800 .820 1-950 2.000 .000 2"300

9-00 .20 "33 9.70 10-30 .50 10.67 11.01 12.78 13-22 13.80 14-50

7.1 7"5 8.0 10"3 11.0 12-0 16-0

* F i n c h C o m p a n y , T o p C o a t G l o s s E n a m e l (600 Series), C l e a r C o a t No . 683-3-1.

219

220

H. B U C H B E R G , O. A. L A L U D E and D. K. E D W A R D S

The value of ~-,~ was evaluated at the moment that each ray crossed the lower cross-section of each of the strips into which the cell depth was divided. In this manner, ~-,~ (and hence r,D was determined as a function of cell depth. Fifty wavelengths were used in all calculations. These wavelengths ~ were determined such that

[f:' G~d~]/ [

f: G~dT~]~-0-01+0-02 (i-1),i~

1,2,3 . . . . . 50

where Gx ~ spectral solar distribution at sea level with air mass = 2-0 as given by Moon [ 18l. The solar cell transmittance was finally determined by summation, or ~', ~ ~ ~ ~',A-

(A3)

Long-wavetransmittances In the calculation of r(x), honeycomb cells were assumed to have specularly reflecting walls consisting of a metallic substrate with a dielectric overcoat. In each instance, T(x) was calculated from inputs of the cell length, cell width, a number of cell depths, spectral optical constants of both the metallic substrate and the dielectric overcoat given in Table 10, overcoat thickness and dimensions of a cell cross-sectional area element. Table 10. Optical constants of aluminum and polyurethane overcoat in the infrared, assuming refractive index of the overcoat ~- 1.6

Wavelength (ttm)

Polyurethane* absorptive index

4.{16 4.44 5"41 5"86 6"26 6-65 7-03 7"41 7"80 8" 19 8"60 9-04 9.48 10.00 10.50 11" 10 11"70 12-50 13-40 14"40 15"80 17-40 19-90 24-15 35-80

O- 14 × 10 -2 0-14 0"35 2" 10 2"80 2-94 3.08 3"23 3"39 3-59 3.77 3-05 2.66 3.37 3-95 1.79 1-78 2.40 3.45 3"48 4"20 3'98 3-85 3-25 3"62

Aluminum Refractive Absorptive index index 6"2 7-1 9-7 10"5 1 I-6 12-7 13"8 15"2 16"8 18"7 20"5 22' 1 23"9 25-9 27-8 30"0 32-0 38"0 48"8 54"0 61"3 69-5 81"5 I01"0 148"2

30"8 33"2 38"8 42"4 43"7 46.7 49"1 51-6 54"0 56-2 58.9 61"2 64.0 67.0 69-9 72.9 76.4 110.0 128.5 134"5 142"8 151"8 164"4 182"6 220-2

*Finch Company, Top Coat Gloss Enamel (600 Series), Clear Coat No. 683-3-1. The general calculation procedure was similar to that of Edwards and Tobin [5] except for the following differences: (a) cell cross-sections were assumed to be rectangular rather than square, (b) the effect of the wall overcoat had to be accounted for at each reflection. (c) a dhTerent shape factor kernel was used.

Performance

characteristics of rectangular honeycomb solar-thermal converters

221

Perfectly diffuse long-wave radiation was assumed to be incident on the inlet cross-section of the cell. A fraction of this ~dlation emerges at the outlet cross-section of the cell by a combination of direct transmission between the two cross-sections and one or more reflections up the cell. The fraction, ~(L) of the radiation which is transmitted beyond a distance, L from the cell inlet, was obtained by E c k e n and Sparrow's method of images [ 15,16]. The cell inlet cross-section was divided into a grid of area elements which were as close to being square as possible. As explained in [9], the sum of the reflection-weighted shape factors from any of the inlet area elements to all its images, as seen from the outlet end, represented the fraction of the radiation emanatin8 from the element, which is eventually transmitted to the outlet end of the cell. The transmittance of the entire cell was obtained by summing the contributions from all inlet area elements. Since the infinite number of images of each area segment cannot all be considered, the contributions of imases resulting from an increasing number of reflections at the cell wails were summed until the estimated error of neglecting higher order images was found to be less than 1 per cent. The spectral transmittance ~ ( L ) of a cell with depth L was obtained from the expression • ,(L)

=

I]]OOlp~,l

where I-

_ ~xAy ~ ] ~, ~, Fu~lp.~l ID(.-,,..I Ip,.-,,.I L •.

lP=,l ID,,,Ip,,[ 0i510"5 J1

~hz, &y==dimensions of a cross-sectional area element n = number of reflections suffered by a ray before it emerges at the outlet end of the cell N = hishest o n ~ of reflection image considered I~1 = spectral r d l e c t i ~ matrix for the nth reflection ID~.-I~,I = tnmsformat/on matrix after the ( n - - l ) t h reflection determined as outlined by Edwards and

Bevam[17]. F u ~ = shape factor for radiant transfer between an inlet area element at (x~,yj) and an imase element at

(xl,y=)

FiJ,~m

(L/Ru.b,) = •rR~.m "4"1.4Z~xAy(L/Rua m)

Ream = (xl--xl)=+ (y,n--yj)=+ L =. Calculations were performed for 25 wavelengths chosen such that

\ c, /

"~-'~-~[Eb,(T)]dX=O.02+O.04 ( i - l )

where i = 1,2.3 .... Eb~ (T) = Planck function = 2ca),-51 [e ~'~T- I ].

Czerny and Walther [ 18] have tabulated values of B* (~T/c=). Calculations were performed for T = 610*R, since values of • obtained were found to vary tittle in the range of 560-700"R. The desired value of • was finally obtained as the arithmetic average of the 25 spectral transmittances, 7d$ t-a