Solar Energy Vol. 48. No, 6, pp, 381-393. 1992 0038-092X/92 $5.00 + .00 Printed in the U.S.A. Copyright © 1992 Pergamon Press Ltd.

CALCULATION PROCEDURE FOR COLLECTORS WITH A HONEYCOMB COVER OF RECTANGULAR CROSS SECTION

W. J. PLATZER ~ Fraunhofer-lnstitute for Solar Energy Systems, OItmannstrasse 22, D-7800 Freiburg, Germany

Abstract--For a highly efficient collector with honeycomb cover an optimisation procedure has to take into account a lot of parameters like material choice, geometric dimensions, cell wall thickness, and aspect ratios of the honeycomb. An approximate, but comprehensive theoretical description of the optical properties and the heat transport within a honeycomb absorber system is given and discussed in the paper. Selective absorbers and air gaps with convecting air can be treated within the model. Solar transmittance and heat transport are treated with a consistent model, which in principle needs only film data and the geometric parameters as input. Theoretical results compare quite favourably with experimental data from honeycomb structures. The main innovative features of the model are its use of an effective optical thickness, and the inclusion of both, convective boundary conditions and solar absorption, in a coupled mode analysis leading to an analytical solution.

I. INTRODUCTION main interest in the past was the passive-solar collector

Thermal solar energy components in nearly all cases storage-wall with a cover &transparent insulation, only need two basic e lements-- the thermal absorber and a few structures are sufficiently temperature resistant some transparent cover--which reduces heat losses of for collector operation up to now, but other materials the absorber to the environment. Many efforts in the (e.g., glass) are being investigated now. Nevertheless a past have been undertaken in order to improve ab- method for optimizing absorber systems is available, sorbers: Selective coatings are being developed in order taking into account the coupling between different heat to reduce infrared radiation losses without losing their transfer modes (coupled or dependent mode analysis-- ability to absorb solar radiation. Another approach, DMA). Based on this knowledge, a fiat-plate collector also with some tradition, is the development of trans- with honeycombs and selective absorber has been built, parent covers, which restrain heat transport more ef- which is comparable with a vacuum tube collector ficiently than the conventional glazing of a collector. (50% efficiency at 800 W m -2 and 100 K temperature

difference to ambient) [ 12,13 ]. Convection suppressing devices have been first pro- posed by Veinberg in Russia as early as 1929 (noted by Tabor, 1969 [1]), the idea being picked up again by Francia [2] in 1961. These so-called honeycomb 2. SOLAR TRANSMIlq'ANCE OF HONEYCOMBS

structures should partition the space between absorber 2.1. Analyt ical model and cover in order to restrict heat transport by con- An analytical approximative formula for the solar vecting air. Convection suppression and infrared heat transmittance of honeycombs can be derived from a transfer have been treated by various authors in an summation of all individual rays transmitted or re- independent mode analysis [3-8] (an extensive liter- flected at the cell walls. For the beginning, scattering ature list is found in [24]). Symons et al. [9] in Aus- is excluded [14]. In a two-dimensional case for cell tralia, Hoogendoorn [10] in the Netherlands and Hol- walls much thinner than the cell diameter dj ,~ H, the lands [11] have done more recent work on thin film summation for one incidence angle 4~ is exact: structures such as honeycombs and slats with minimal thickness. This guarantees low transmission losses in M M! the solar spectrum, and on the other hand, allows easy I(c~) = Io ~ i ! ( M - i)! rl (3 )M- i .p1 (3) ' ,

i=0 combination with selective coatings because reemission of the plastic films, coupled to the absorber temperature with 3 = 7r/2 --- $ and

by air conduction or even by contact, is kept small. M = INT[A, tan(~b)] , (1) However, IR-absorbing materials for transparent in-

sulation, investigated at the Institute for Solar Energy where T~(3), 01(~3) are the single film transmittance Systems in Freiburg, seem to be more efficient as they and reflectance values. The sum simplifies to are able to reduce infrared radiation appreciably with and without selective coatings. Convection is sup- ~M(~b) = [ r l ( / 3 ) + p1(/3)] ~4= [ 1 - al(/3)] ~. (2) pressed when dimensions are properly chosen. As our

If one further considers the positions of the incident rays and the rays internally guided within the cell walls,

* ISES member, one comes to

381

382 W.J. PLATZER

Zhc(~b) = (1 -f~)*rc(cb) +fw*Zw(0), far IR), whereas for square cross section, even then there is little influence of polarisation [17]. For slit

rc(~b) = [fM*TM÷j(3)+ (1 --fM)*TM(/3)], structures it is possible to calculate the polarisation r,.(~b) = r](q~), (3) components separately, but for rectangular cross sec-

tions it is not. Averaged values may be used without where much loss of accuracy.

The treatment of scattering certainly is very difficult. fM := A ,tan(~b) - M, However, films used for honeycombs should certainly

have only little scattering. Hollands [15] therefore fw := surface fraction of finite cell walls, simply calculated rhc with the extinction coefficient in-

7'1 (4~) : single film transmittance for thickness D stead of the absorption coefficient and distribution 50% of the scattered radiation to the transmittance, 50% to

(usually negligible), the reflectance, assuming an effective isotropic scat- tering. An improved method was tried with the help

It was assumed that internally guided rays are re- of a three-flux approximation of isotropic scattering flected by total reflection. This equation also has been transport (Kaganer, 1969 [18 ], see Appendix A). If derived with other means by Hollands and co-workers the albedo of the film w, which is the relative strength [15] and used by Symons [16]. These authors, how- of the effective scattering compared to the extinction, ever, did not exactly take into account the variation is not equal to zero, rhc is first calculated with the total of absorption with incidence angle within the three- extinction coefficient ~ of the film. This transmittance dimensional honeycomb, does not include the scattered part and will be corrected

This will be another step of refinement in the model, by the quotient of two transmittances r , through a Figure 1 shows schematically the geometry of the ho- homogeneous layer with and without scattering, which neycombs. An incident ray, if not being parallel to one is equal or larger than unity: of the cell walls, hits walls vertical to each other. The number of hits and the corresponding incidence angles r/eff*D := 7"7, I (rhc(q~; ~7)/1 -- f , ) , (6a) at the cell walls therefore are

Thc(~); /7, 60)

Mx = tan(~b),cos(~)*Ax, (4a) := rhc(~b; r/)(r,(~efr, w)/r,(~efr, 0)). (6b)

My = tan(cb),sin(¢),Ay, (4b) 2.2. Monte Carlo simulation Monte Carlo methods are very versatile tools to

3x = arccos[sin(~b)*cos(~o)], (4c) simulate physical processes. The mathematical back-

fly = arccos[s in(~) , s in(~) ] . (4d) ground is described, e.g., in [19,20]. Computer-gen- erated pseudo-random numbers with known statistical

So the result for the honeycomb with rectangular probability distribution are used to simulate physical cross section (excluding scattering and polarisation ef- processes like reflection or emission of photons. In the fects) is transmission model used for this work equidistributed

random numbers within the interval [0, 1] from a con- rc(~b, ~p) = [1 - al(3~)]MX*[l -- oq(3y)]ML (5) gruential random number generator with periodicity

10 +14 were used. Other statistical distributions were

Polarisation becomes important for slit structures obtained by analytical or numerical inversion of the (A~ --~ 0) for strongly absorbing cell walls (e.g., in the cumulative distribution function, e.g., to obtain the

reflection properties of a Fresnel film. Photons were traced without permission of branching, only one sta-

, ~ . 1 , , tistically chosen direction was pursued at a time. Be- cl~./ /.,,,,.~ B- - -77 I ~ ' ~ / , / Z cause of symmetry the process can be confined to a

~ / z q ~ ~ , , ~ - / ~ single honeycomb cell, when transmission and reflec- tion through a cell wall is summed up as effective re-

l flection. The elements of the algorithm are as follows: • choice of a source point within a single honeycomb

cell (top area), • choice of direction of emitted photon, • geometrical calculation of hitpoint defined by the

cell geometry (vector algebra) • decision whether photon was scattered, absorbed or

regularly reflected, ¢P • no branching, and

• continuation of tracing the photon path until photon Fig. 1. Sketch of a honeycomb layer with incident ray (3: incidence angle for cell wall; superscripts s: specular, d: dif- hits the top surface (reflected photon ) or the bottom

fuse), surface of the cell (transmitted photon).

Collectors with a honeycomb cover 383

To obtain statistically reliable results, 20 runs with 1 ' i

about 2,000 photon starts were performed, each run ~ ~ ~ giving a transmittance and a reflectance by dividing .~ 0.8 the photon number reaching the top or bottom surface r '} ~ \ ~ of the cell by the total number started. The 20 runs =~ 0.6 ~ "6, ~ " are used to determine an average result with standard ~ deviation. This standard deviation is according to the ~. 0.4 i• o Tdif=0.579 I1 \ ' \ ~ central limit theorem an estimation of the statistical ~ ~ • Tdif=0.734 error of the procedure. ~ 0.2 ~ ,

The results of the analytical model compared very ~ i ~-, - well with results from Monte Carlo simulations, where 0 t . . . . . . . . . ~

0 15 30 45 60 75 90 the isotropic scattering was assumed to happen at the

incidence angle ]degree] cell wall surfaces. Figure 2 shows the influence of the albedo on the diffuse-diffuse (hemispherical-hemi- Fig. 3. Comparison of direct-diffuse solar transmittance de- spherical) transmittance. Also the dependencies on as- pending on incidence angle (lines: analytical theory, single pect ratios and incidence angle showed very good points: Monte Carlo calculations].

agreement between the two theories (Fig. 3).

2.3. Experiment and discussion Figure 5 shows results for identical geometry and film extinction for w = 0 (absorption) and t0 = 1 (scattering) Measurements of solar and visual direct-diffuse

transmittance have been made with a large integrating with and without corrugation (maximum ray deflection sphere (diameter 40 cm, sample port diameter 8 cm) from the ideal specular reflection due to corrugation using a solar simulator with a 2.5 kW halogen-metal was +_2.5°). Although experimental results indicate bulb, which has a spectrum closely resembling the that optical losses due to nonideal film properties are global AM 1.5 spectrum. The source irradiates a large important, still the simple model is valuable to describe sample of 40 cm × 40 cm homogeneously with nearly the material, e.g., within simulation tools. Moreover, parallel chopped light. The detectors were a pyroelectric the influence of honeycomb geometry (aspect ratios, radiometer and a photopically corrected photodiode, material thickness) is described well by the model. The connected to a lock-in amplifier. The whole sphere simplifications in the model reflect the experimental

uncertainties. In order to improve the model, angle- can be rotated by a motor drive so that transmittance for variable incidence angles can be measured. Details dependent-scattering properties as well as corrugation

of the cell walls have to be investigated more thoroughly of the experimental setup can be found in [ 8 ]. in the future. However, the transmittance model is used

The comparison of experimental data for one ma- terial with two different layer thicknesses shows that a quite successfully for radiation heat transport calcu-

lations as shown in the next section. single fit of the optical data for both thicknesses is not entirely satisfactory (Fig. 4). The reason is probably

the oversimplification of the geometry. Slight corru- 3. HEAT TRANSPORT gation of cell walls along the cell-axis (z-corrugation) leads to a lowering of the transmission, especially for 3.1. Radiation heat transport small incidence angles. On the other hand, corrugation Radiation heat transport within a honeycomb perpendicular to the axis (xy-corrugation) leads to a structure consists of radiation transmitted or absorbed distinct drop especially at incidence angles above 50 °, and reemitted by the cell walls. Within the structure as can be shown by Monte Carlo simulations [21]. there is an orientational preference for the radiative

1 ~ 1 - - , 1

'~ 0.6 - .~ 0.6

] b ] o D=6cm ] ] i i\]

o aspect5 ~ ~ \! _ 0.4 - 0.4 • - • aspect 10 ,~

~' [ ~ ~ [ • D=12cm[ ~ \ ", 0.2 ~ aspect 15 '~ 0.2 ~-

0 0~2 0.4 0.6 0.8 1 0 15 30 45 60 75 90 albedo incidence angle [degree]

Fig. 2. Comparison of diffuse-diffuse solar transmittance de- Fig, 4. Comparison of theoretical and experimental solar trans- pending on fraction of scattering for constant extinction (al- mittance. Polycarbonate honeycombs with aspect ratios A~ = Ay bedo) (lines: analytical theory, single points: Monte Carlo = 18 and A x = Ay = 36, the effective extinction has been fitted

calculations), once for both thicknesses D = 6 cm and D = 12 cm.

384 W.J. PLATZER

RA = eAo'T,~ + (1 -- ~A)[RB*r(D)

~ 0.8 D d r ( z ' ) k

- f o '~T4(z') dz---7-dz '] ' (8b) 0.6

.~ ] [ ~ RB = ~e~rT 4 + (1 - e~) RA*r(D)

~ 0"4 [ i!curvelTdif=0'755 " fo ] . curve2 Tdif=0.760

0.2 curve3 Tdif=0.600 ~ D d r ( D - z ') ..~ curve4 Tdif=0.645 + i - aT4(z ') - ~ - - - z 7 ) dz' . (8c)

0 , ,_ I , ~ I , , I ~ ~ , , ~

15 30 45 60 75 90 Equation (8a) can be made dimensionless, if one uses incidence angle [degree] the variables ~" = z / D , qg, = qr~/(RA -- RB) and • =

Fig. 5. Solar transmittance for different honeycomb cells ( aT4 - Rs ) / (RA - Rs); by partial integration one gets (Monte Carlo calculations): curve 1: absorbing film, no cor- rugation; curve 2: scattering film, no corrugation (isotropic ~r = r(~') + r(~'o - ~-)*~(~'o) - z(~')* ¢b(0) scattering within film); curve 3: absorbing film, sinusoidal corrugation of film; and curve 4: scattering film, sinusoidal f ~ d~ d ' fro d~ corrugationoffilm. The corrugation was perpendicular to the - ao r ( ~ ' - ~") ~~5 ~" + a¢ ~'(~"- ~')~Td~". (9)

cell z-axis with max. 2.5 ° deflection.

A linear ansatz for ~(~-) should be a very good as- transport, contrary to an absorbing-emitting gas which sumption, as this is the solution for a homogeneous has been thoroughly treated in literature. For the latter absorbing gas layer. The temperature distribution in the transport equation for a layer of thickness D can the direction z, however, will not depend much on the be solved with a number of methods (see, e.g., Siegel- angular variation of film emittance within the layer. Howell [22]). For the usual assumptions of diffusely This can be seen also from Monte Carlo results of the emitting boundaries, an approximate formula for the honeycomb radiation transport [24]. If eqn (9) will radiative heat transport is be used with the linear ansatz at three positions ( ~" =

a(T4A _ T 4) 0, ~" = 0.5, ~'o, ~" = s~o), the following solution is ob-

q r a d = ¢ ~ 1 + ¢ ~ 1 _ _ 1 + 4at0 ' (7a) tained:

a T 4 ( t ) = a T n - q r a d ( , ~ l - ½ + 3 t ) , (7b) ~° = 0"5"(1 + ¢I'1)' (10a)

where to is the optical thickness of the layer in the far 1 + r(~'o) - 2 , ~-(~'o/2) infrared. Tien and YiJen [23] developed an expression - 1 + ~'(~'0) - 2,r(~ 'o/2) + 2 / ~ o [ 2 F ( ~ o / 2 ) - F(~'0)], similar to eqn (7) for honeycombs, where 3 to has been (10b) exchanged by &(0.5* rd~0. This expression is not bad for honeycombs with rather small aspect ratios and q~a~ = r(~'0/2) + cbl[2/~oF(~o/2)

large diffuse-diffuse IR-transmittance rdif, but in gen- - r(~'o/2)], (10c) eral this formula underestimates grossly the radiative transport, where

Already in 1967, Edwards and Tobin [ 17 ] calculated the angular-dependent transmittance function of rec- I'~ tangular channels and also gave a procedure to deter- F(~') := J0 r( ~")d~", mine the radiation heat transport approximately for black end emissivities: The dimensionless transport and equation will be solved by parameterizing the temper- ¢I,(~') := ~o - ~ * ~'. ature profile. So the integral equation will be changed

to an algebraic one and can be easily solved. With this method, one may calculate the radiation Now this method can be extended to gray boundary transport depending on optical film thickness ~, dfand

emissivities and the analytical transmittance model of aspect ratios very fast. Other methods like Monte Carlo Section 2 will be included. The integral equation for- or numerical methods are much slower. To estimate mulated as a function of diffuse-diffuse transmittance the reliability of the results, for a wide range ofparam- (the index di f is missing for the sake of brevity) and etersthecaleulation has been compared to Monte Carlo with variable gray boundary emissivities is results. The Monte Carlo transmittance model of the

previous section has been extended for that purpose. q~d (Z) = RA * 7 " ( 2 ) - - R~* "r(D - z) From the hot top and the cold bottom area of the ho-

fo ~ neycomb cell photons with an energy ofeo are started, d z ( z - z') dz' the number of photons being proportional to the em- - aT4(z') d(z - z ')

issive power according to emissivity and temperature fD d r (z ' - z) of the end plates. The angular distribution of the pho-

+ a T ' ( z ' ) - ~ z ' - - - ~ dz', (8a) tons is isotropic, i.e., the boundary is assumed to be

Collectors with a honeycomb cover 385

diffuse. For IR-radiation transport it is important that 50 ~ - . . . . . . . . ~ - ~ , . ~ - . . . . the process is stationary; this means that each photon ~ eps=0.l [ 4' absorbed in the cell walls will be reemitted until it 40 L * " ~ " . . - - eps=0.3 ] * ~ * - " eps=0.5 eventually is absorbed by the top or the bot tom area. ~ ~ . • :g-_. -- eps=0.7

o.... 3 0 / ~o~ ' + "~"& - e p s = 0 . 9 i For this reason the modelling of the IR-transport needs ~ ~ o~ ~ + - :-,. much more t ime on the computer than the transmit- ~ ~ / ~ 4 I tance modelling. This is especially true for low plate g. 20 ~ "o~ o_ ~. i

l emissivities of the end plates where most photons are [ . . . . . . . "-v ? ? ,

again reflected back into the cell. The heat transport 1o - ~__~: . is calculated from the number of photons emitted by -- < - ' - ? the hot and cold plate L'A and EB, and the numbers o . . . . . . . . . . . . . .

0 0.2 0.4 0.6 0.8 I eventually absorbed by the end plates LA and LIdLA rel. position + LB = EA + EB). The radiation heat flux is propor- tional to the difference between absorbed and emitted Fig. 7. Comparison of temperature distribution within ho- photons: neycomb for different boundary emissivities (only radiative

heat transport) (lines: analytical theory, single points: Monte Carlo calculations).

EA -- LA qrad = q e m i t - qabs - - - - * c r * ( T 4 - T~). (11)

EA + EB tions calculated with both methods. The linear ansatz

It is also possible to determine position-dependent seems to be totally satisfactory. An advantage of the film temperatures with the Monte Carlo method: The analytical method is certainly that one gets fast results honeycomb cell walls are subdivided into a finite without any random error, which always are inherent number of segments; whenever a photon is absorbed, in Monte Carlo results. a counter for the segment is incremented. After corn- It has been shown now that the relevant parameter pletion of the Monte Carlo simulation these numbers for the radiation transport within honeycombs is not L, for each segment are used to calculate the radiative an optical thickness but the (diffuse-diffuse) trans- temperature at that position mittance and its integral. For practical reasons, how-

ever, an effective optical thickness may be introduced

T4 eo,L, (12a) by the following convention:

t0.~n ":= 3 t,~h~ -- 1). (13) eo = Atoo*a(T~ - T~), (12b)

With this convention formula, eqn (7) yields the where e0 is the energy of a single photon, A, is the area correct radiation heat flux and the formalism of the of the cell wall segment, and ~ the hemispherical emit- exponential kernel approximation (e.g., Cess, 1963 tance of the film. The segments should not be chosen [26]) is applicable. This will be used in the solution too small, because a statistically significant number of of the problem of coupling radiation and conduction. photon have to be absorbed and reemitted by each The concept of the effective optical thickness is appli-

segment, cable also to other absorbing structures with one-di- The random absorption and the emission of pho- mensional continuous temperature distribution in a

tons within the honeycomb structure used for corn- layer. Therefore, it is independent of the material type. parison with the analytical theory had the angular dis-

tribution as calculated by index of refraction and ab- 3.2. Conduction heat transJer sorption coefficient [25]. Figures 6 and 7 show Thin-walled honeycombs c o n d u c t h e a t w i t h i n t h e dimensionless heat fluxes and temperature distribu- cell walls and through the gas filling within the core of

the cells. If the aspect ratio is large enough (about 7 - 8 for 5-cm honeycombs and about 12 for 10-cm ho-

1 ~ ' - ' ~ neycombs with square cells), the convection is effec- | a s p e c t 5 ] - - aspect 10 2 tively suppressed, i.e., the Nusselt number is less than 0.8

~i~ [ aspect 20 [ 1.05 according to [ 27 ]. For the honeycomb structures .~ ~ | - aspect40 ] used in our collectors with A = 22 [12,13], the con- .~ 0.6 ~, vective heat transport due to the very slow convective .~ 0.4 ,+ L\ ~ ~ base flow does contribute less than one percent to the

! . ~- ~ conductive heat transport. This is true even for inclined i , • ~ ~ _ cells for usual collector conditions (AT = 80 K), pro-

0.2 " - ~_ * " ~ ~ ~ - ~ ~ ~ ° -~ vided one side of the honeycomb cells is closed to pro- 0 . . . . ~ . . . . . ~ _ _ • ~ hibit inter-cell convection rolls. The three-dimensional

0 0.5 1 1.5 2 behaviour of conduction may be neglected and as- film optical thickness sumed one-dimensional (HoUands et al. [28]) as a

Fig. 6. Comparison of dimensionless radiation heat transport very good approximation with an effective conductivity (lines: analytical theory, single points: Monte Carlo calcula-

tions). ~efr = J~" X,. + ( 1 -- f , . ) . Xg~. (14)

386 W.J. PLATZER

A simple addition, however, to the radiative part The nonlinear temperature term is linearised by the of the heat transport severely underestimates the latter least square method for selectively coated absorbers, whereas for black nonselective absorbers this additive or independent f~ mode analysis (IMA) is possible with acceptable results Q = do [ 04 - bp - cp* O] 2 dO minimal. (17)

2 [29,30]. Therefore one has to combine both modes in a coupled or dependent mode analysis (DMA), which Within the collector temperature ranges the error will be described now. of the linearisation is small (standard deviation 0.022

for TA = 300 K and TB = 400 K). So a linear differential 3.3. C o u p l e d hea t t ransport equation in the reduced temperature has to be solved,

With a coupled mode analysis not only the heat which can be done by standard methods. transport is calculated more exactly, but also the un- physicaljumps in the temperature distributions at the d20 (~ 4@.Cp) boundaries are removed and replaced by a continuous dt 2 + 0

physical profile. The starting point of the theory de- scribed here is the energy conservation within the ma- _ 9 ~* t + ( 3 ~7 ) terial in a stationary equilibrium with a one-dimen- 16N ~ - ~ b p - ~-~/ , (18) sional temperature distribution say between thermo- statically controlled heat source and sink with constant After a trivial but lengthy transformation one may temperatures TA and TB. eliminate the unknown constants faro, ~ and xI, with

the help of eqn (16). The coefficients for the general x , d T ( z ) + qrad(Z) solution functions then can be determined from the q ( z )

dz boundary conditions

= q c o n d ( Z ) q- q r a d ( z ) = 610 = const. (15) I. O(0) = 1, II. O(to) = 02, (19a,b)

Here qrad is the radiation flux of eqn (8) and qconO -~'-A --~-B the effective conduction. For homogeneous materials III. dO dO = SA, IV. = SB. (19c,d) the exponential kernel approximation is a well-known method and investigated extensively (e.g., [ 26 ] ). The transmittance terms of qrad are replaced by an expo- The resulting algebraic linear system in the reduced nential function. If the resulting expression is used in temperature gradients sA and s8 can be solved easily. eqn (15), this equation can be solved analytically with They determine the magnitude of the heat flux and appropriate boundary conditions by a transformation the temperature distribution as shown in Appendix B. into a nondimensional differential equation of second A similar solution method has been already used by degree: Marcus [ 3 l] for honeycombs, when he treated honey-

combs as an effective gas with re, as.eft(D) = rhc(D). But 4 d20 this underestimates the radiation transport. Instead the

N - - ~ - - 9NO(t) - 304(t) = 49-,~,t - rl, (16a) effective optical thickness of the honeycombs should be calculated from the transmittance function. By def-

4 inition the honeycomb material then has exactly the ,I, = { N l - O . - ~ , ( s A + sB)l

to + 3 ~ same temperature distribution and the same heat flux

1 + 3*(fA - fiB)}, (16b) as an equivalent homogeneous layer, for which the conduction-radiation coupling analysis has been de-

r/= 9 N + 3ffA - 6N*SA -- 1.5 .~ , (16c) scribed above.

fA = 1 -- ( 1 -- eA)/eA ( ~ + 4 N * s A ) , (16d) 3.4. Coup l ing ana lys i s wi th a i r g a p

If absorbing-emitting materials are used in close f~ = O~ - ( 1 - ¢B)/eB ( ~ + 4N*sB), (16e)

contact to selective absorbers, the coupling of radiation

with the dimensionless quantities: and conduction destroys the effect of the low-emissive coating. Therefore a reasonable use of selective ab-

dO sorbers with honeycombs needs a conductive decou- , piing of the absorber and the material. This is possible

si := - -~- with an air gap between absorber and honeycomb. to := K , D t := K,z, Hollands and Iynkaran [32] give a formalism to treat

this problem for quiescent air, but usually convection T will occur in collector systems. This problem may be xi t . = qrad q- qcon,J O := - -

aT4A ' TA ' tackled with the following treatment of the air gap. The basic idea is the continuity of the conductive

and the radiation-conduction-parameter heat transport at the boundary of the material surface. The situation is sketched in Fig. 8. In front of an ab-

X~rf* K sorber with temperature TA and emissivity eA there is N : = - -

4~rT 3 " an air gap of thickness de = Zo - ZA with a convective

Collectors with a honeycomb cover 387

AT[ Z=Z0 matical set-up of the device. The hot plate may be l ~ ~ heated up to 230oc electrically, whereas the cold plate

J is thermostatically controlled in the temperature range TA ~ I 5-60°C. The copper plates have an area of 40 cm ×

h o n e y c o m b 40 cm, the active measuring area is only 15 cm × 20 cm. For materials up to l 0-cm thickness the side losses to the environment may be kept small if the radiation is reflected by an aluminum foil covering all sides. The

~ h g a P l . ~ ~ ~ conduction losses are prevented by adding glass wool or a similar insulation material to the radiation shield.

~ ' ~ i ~ The apparatus can be tilted in the range 0-180° which makes it possible to investigate convection within the air gap T,B materials also. The heat flux meters have been cali-

¢ 1 ~ i ~ brated by comparing results for opaque standard sam- ] [ J pies with the measurements. The temperature depen-

ZA Z o za dence also has been,checked experimentally. If the side losses are negligible, the results from the heat flux me- Fig. 8. Schematical sketch of temperature distribution foran

absorber system with honeycomb and air gap. ters on hot and cold plate are identical. A single mea- surement for a light-weight insulation material takes about 40 min with temperatures and heat fluxes de-

heat transfer coefficient he. The honeycomb has surface termined every 5 s. The last hundred points in the time temperature T~, so the condition of continuity is series, where the experiment should be stationary, apart

from small fluctuations due to the temperature con- qco.d ( z = zo) = he*( T A - - Tz) trollers, are taken to determine the average values and

= Xhc,eff*~TT[:=~o+. (20) standard deviations. The results for a 5-cm sample be- tween 1.0 and 5.0 W / m 2 K is reproducible up to 2%,

As the temperature gradient within the honeycomb the absolute error being approximately 2% larger. A is known from eqn (17), Tz can be determined in this second hot-plate apparatus was also used for experi- way. The method is iterative: The coupled heat trans- ments with air gaps when convection could occur. An

active area smaller than the sample area may then lead port will be calculated for an estimated temperature 740) as boundary condition. Physically, of course, the to erroneous results as the locally dependent convection radiation heat flux qrad is determined by the absorber heat transport is not averaged over the whole element. temperature TA and the emissivity ~A- SO mathemati- The second apparatus therefore has an active area very cally with Tz substituted for TA an effective emissivity close to the sample area. The quantity measured is the te~ has to be introduced to correct the heat flux: heat conductance of material layers, defined by the

heat flux qo through the layer and the temperature dif- a*(T 4 - T~) ference of the plates A and B:

qrad = eA l + ~1 + 0.75*K*D

! a*(T 4 - T~) A. q0 . (23) = t~-d + ~ l + 0.75*K.D" (21) T~ - TB

As this is only a mathematical trick to express the For many materials not only the heat conductance uncoupled radiation heat flux as a function of Tz, vii- for different thicknesses, temperatures, inclinations, ues for re, larger than 1.0 are possible and useful. After and boundary conditions has been determined, but the determination of the temperature gradients and also the properties of the constituent films have been the (temperature-dependent) convective heat transfer coefficient, the surface temperature Tz may be cur- ~ - rected, if these two conductive fluxes differ: ~ D A ~ set point rollel - - F

T~ +l)=_ TA -- [(TA -- T~ )) +10"* ~'hc'eff* h cl *~r T ] q - 1 ~ computermi .... ~ ~ 1 ~] temperatu (22) ........ DV--I~~ ~

' e

The iteration is damped by a factor I" to prevent aluminum C l ~ over-shooting of the temperatures and an instability ~ foil + glass wool as a consequence of this. ~

heat flux 3.5. Experimental results _ ~ meters

For experiments a hot-plate apparatus using heat coolant pipes flux sensors has been built up. Figure 9 gives the sche- Fig. 9. Experimental set-up of hot-plate apparatus.

388 W.J. PLATZER

determined by a commercial spectrometer (Perkin- - - no air gap Elmer 330). Although real plastic honeycomb struc- 1 . 8 - - - a i r g a p 2 c m ( c o n v . ) ,

tures are never ideal in the geometric sense, as the cross - . . . . air gap 2cm (no conv.) sections are slightly distorted and cell wall thickness ~ 1.6 o no gap (exp.) • g a p 2 c m ( e x p . )

varies appreciably within one structure, a number of 1 . 4

honeycombs with aspect ratios ranging from 5 to 25 ~ have been used to test the theory. From volume, weight, ~ 1.2 and cross section the mean cell wall thickness and the mean aspect ratios were determined consistently. From ~ 1 IR-transmittance data of single films, the IR-extinction ~ 0.8 I . . . . . ~ . . . . . - ~ ' ~ coefficient for the mean temperature of the sample was = ~ ~ ~ ~ . . . . . . . . . . . . . . . . . . . . . calculated. Using these data, the heat transfer for the 0 . 6 L - ~ - ~ " _ " ' 7 " - ' _ ~ ' - " t ~ , J _ , _J , _ _

experimental boundary conditions was calculated. Due 3 0 5 0 7 0 9 0 1 1 0 1 3 0 a b s o r b e r t e m p e r a t u r e [ ° C l

to the uncertainty in the input parameters, the theo- retical values also were determined with error bounds. Fig. 11. Heat conductance for honeycomb collector with se- Figure 10 shows the theoretical r~ul ts plotted against lective absorber 10-cm polycarbonate honeycomb, small col- the experimental ones from a hot-plate apparatus. The lector model 40 cm × 40 cm (lines: theoretical calculations,

single points: measurements). case BB refers to both end plates having high emissivity (~ = 0.93), for the case SB the hot plate was covered with low-emissive a luminum (~ = 0.08). The results lie within a small range close to the line of perfect 4. TOTAL ENERGY TRANSMITTANCE

agreement. Because of solar absorptance in the plastic films, To show that the variation of the heat conductance solar irradiance will result in a temperature rise within

A with temperatures is determined correctly, Fig. l 1 the honeycomb material. This leads to a superposition plots results for a 45 °-inclined collector model using of a heat flux q~, towards the absorber, which is pro- a 10-cm polycarbonate honeycomb and a low-emissive portional, at least to a first approximation, to the solar absorber. Without air gap there is nearly a perfect input S. Therefore a total energy transmittance may match. For a 2-cm air gap the experimental values lie be defined by between the theoretical curves with and without con- vection within the air gap. This indicates that due to g := (rot)sol + qin/S. (24) the dimensions of the collector model (40 cm × 40

cm) the Nusselt number was lower than the predicted For collectors, this entity is commonly also called one by the formula used for real collector dimen- effective transmittance-absorptance product (rC0e(see,

sions [33]. e.g., Duffle and Beckman, Chap. 6.9, [34]). For mul- To show the difference between a collector with se- tiple glazings with discrete temperature distributions

lective and black absorber, the temperature distribu- within the materials, an approximate method for cal- tions for both cases are shown in Fig. 12 (due to con- culating g is given in [35 ]. The absorptances of the vection within the air gap the dotted line is only tic- glass panes or plastic films have to be known. Important titious). As a measurement of the honeycomb for calculating q~. are also heat resistances between temperatures in a collector design is not practicable, covers and to the boundaries (absorber, room, outside), the theory can be used to determine maximum ab- which are assumed constant. Ashrae gives a slightly sorber temperatures with regard to the thermal stability different procedure for calculation. Both methods apply of the plastic films.

2 . 5 - - ~ , ~ - n o n - s e l e c t i v e c a s e B B ' ' ' ' ' ~ 1 1 0 s e l e c t i v e c a s e S B

0 i 2 . 2 E 9 0 " ' ,

1 .9 ,

N n o n - s e l e c t i v e c a s e B B I 5 0 [ "~ 1.3 s e l e c t i v e c a s e S B

i d e a l m a t c h i n g 3 0 L ~ . . . . . . . . . . .

1 i i _ , , • -2 0 2 4 8 10 1 1.3 1 .6 1.9 2 . 2 2 . 5 pos i t i on l c m l

e x p . h e a t c o n d u c t a n c e [ W / m 2 K ]

Fig. 12. Calculated temperature distribution for a polycar- Fig. 10. Comparison of experimental and theoretical total bonate honeycomb collector aspect ratios Ax = Ay = 22, heat transport in honeycombs NB: heat conductance includes thickness D = 10 cm, air gap dg= 2 cm, no solar irradiation all heat transfer modes (case BB: both plates ~ = 0.9, case SB: (case BB: both plates ( = 0.9, case SB: hot plate ~ = 0.08, cold

hot plate ~ = 0.08, cold plate ~ = 0.9). plate = 0.9).

Collectors with a honeycomb cover 389

I ~ ~insulated The solution of the resulting system of equations is - - cab in analogous to the problem without absorption source,

doublefr°nt ~ t .o i l+_~ .~ ~ A' ~plate ~ - ' however, additional terms lead to a different particular glazing insulation_ 1 ) solution of the inhomogeneous differential equation.

. . . . . . ) - - heat t pipes The theory is outlined in Appendix B. For a calculation sample.x~ll ~ . flux

meter ~ of the g-value, the heat flux at the absorber with and halogen-metal J L solar temperature ~ ~ without solar radiation has to be determined. The dif- simulator sensors ~ - ~ ference then is the heat gain q i n . A s a reference the

solar intensity in front of the whole cover system So is ~:222222~ rotation i axis taken, whereas the source term in eqn (26) describes

IEC-Bus the intensity in front of the honeycomb, which may micro- c o m p u t e r be different with an additional cover glass:

scanner + DVM q ( S ) - q ( S = O)

Fig. 13. Experimental set-up for calorimetric indoor mea- g := (ra)sol + So (27) surement of total energy transmittance: variable incidence

angle, fast beat flux meter measurements. As an example of the application of the theory the

theoretical description of a honeycomb with square only for glazings with zero transmittance in the far cross section will be compared with measured data.

infrared. The calorimetric indoor measurement device using Hollands and Wright [36] developed an iterative heat flux meters is described in detail in [37], a sche-

method which can handle diathermanous films as coy- matical sketch is shown in Fig. 13. The solar simulator ers with variable boundary conditions. With the ex- is the same as for the solar transmittance measure- ponential kernel approximation of the last chapter it

merits, the incidence angle of the radiation is variable is possible to calculate g also for honeycomb systems between - 6 0 and +60 °. The problem for the calcu- with a continuous temperature distribution over the lation is to get reliable input data for solar absorption

material thickness, and effective (isotropic) scattering, as only the extinc- The method is an extension of the theory described tion can be easily measured and most of the scattering

in the last chapter. The starting point is the introduction will be forward scattering. In Fig. 14, the angular de-

of a source term in the equation of continuity ( 15 ), pendence of g and ( r a ) will be given for albedo w = which describes the solar absorption within the ma- 0.0, o~ = 0.5, and w = 0.75. Certainly for usual clear terial. This absorption can be described by a special plastic materials w = 0.75 is an upper bound, which transmittance function ra~(Z), where z is the position means that experimental results should lie above these within the material, which includes only the absorptive curve. The experimental points in fact are close to w losses of the solar transmittance r~o~(z), but not the = 0.5. To give an impression of the temperature dis-

reflective or scattering losses, tribution within an irradiated honeycomb structure,

for several diffuse intensity levels the theoretical results "gabs(Z) := "rsol(Z) + Pback(D) -- Pback(D -- Z) are plotted in Fig. 15. Increased temperature in this

f0 z range has been observed, but accurate experimental

= 1 - a ( z ' ) d z ' . (25) data are difficult to measure. This, however, is a point

Here 0b~ck is the reflected or scattered part of solar 1 ~ - , . . - , . . , - , radiation for a material thickness z. Instead of the heat ~ ~ flux, now a different quantity is a constant and inde- ~ 0.8 pendent of position '4

qs := -Xefr*dT(Z)dz + q~aa(Z) ~= 0.4 ~- - - g_m~mal ~ - ~ - - g scat50% \ 3 \

- g._scat75% ~,~, -- S * r ~ ( O - z ) . (26) e, 0.2 - ~ .g .(exp.value) \ \

For a general function ra~(Z) it is impossible to 0 . . . . . . . , , , , . . . . , . ~% give an analytical solution of the energy transport 0 ~5 30 45 60 75 9o equation, where the expression, eqn (8), is taken for incidence angle [degreel

q,~a(z). Within the exponential kernel approximation, Fig. 14. Calculated transmittance curves for a collector cover the IR-transmittance has been replaced by an expo- consisting of 10-cm polycarbonate honeycomb with a low- nential function. It seems natural to approximate iron cover glazing: g_minimal: no solar absorption (equivalent ra~(Z) also by an exponential function. This is ade- to solar transmittance); g__maximal: maximal absorption, no

scattering; g__scat50%, g__scat75%: 50% resp. 75% of cell wall quate also because solar absorption in good materials extinction due to isotropic scattering; single points: calori- should be very small, metric measurements.

390 W.J. PLATZER

130 ' ~ ' ' ' ' - ' ' ' ,~ fw volume / surface fraction of honeycomb (eqns [ 3, 14 ]) '~:i. ~... l - - - S= OW/m2 IJ fN fractionofcellarea, where incident ray hits N + 1 cell

110 ~- ~\ ~ I - S=300 W/m2 I~ walls (eqn [3]) / " L \ ~ [ - - S=600 W/m2 [1 g total energy transmittance

°~ ~" " ~ " ~ l h heat transfer coetficient, W/m2 K o 90 = q heat flux, W m -2 ~ " .~ . ~ j s dimensionless temperature gradient (eqn [ 16], and

o~ 70 ~a i r "~..~ ~ ~ AppendixB) I ~.-.. t optical thickness (eqns [7, 13], and Appendix B)

50 ~ ~ " ~ i x coordinate ~_ , ~ ~ . y coordinate

z coordinate 30 - - , A aspect ratio (length divided by width)

-2 0 2 4 6 8 10 B width of cell (x-direction) position [cm] D length of cell (z-direction), honeycomb thickness

H width of cell (y-direction) Fig. 15. Calculated temperature distribution for a selective I intensity polycarbonate honeycomb collector; aspect ratios Ax = Ay = N radiation-conduction-parameter (eqn [16], and Ap- 22, thickness D = 10 cm, air gap d e = 2 cm; solar diffuse pendix B)

irradiation levels 0, 300 and 600 W m -2. R boundary radiosity (eqns [8, B3]) S solar radiation, W m -2

to be raised in favor of the usefulness of the theoretical T temperature k fit constant for exponential approximation ofrab~ (Ap- description, as with reliable input data maximal tern- pendix B) peratures for certain absorber configurations can be kj constant describing extinction in absorbing-scattering calculated, which is important for plastics with a low layer (Appendix A) melting point. E, L Monte Carlo events

M number of transversed cell walls by an incident ray ai, b~, ci, d,, e~, p~, r~, w~, ui are coefficients used for

5. CONCLUSION the algebraical manipulations of Appendix B. They have no physical meaning.

A complete method to determine the optical and

thermal properties of absorber systems with honey- Greek comb materials has been presented, which is able to a Boltzmann's constant determine the impact of parameter variations like film ~ angle thickness, thickness, aspect ratios, and material change ~ azimuthal incidence angle for a real material. Also temperature dependencies of ~ polar incidence angle

u cosine of incidence angle the U-value, angular dependencies of solar and total albedo (fraction of extinction due to scattering) energy transmittance can be calculated as well as tem- ~ solar extinction coefficient, cm -~ perature distributions within the materials. As the K infrared absorption coetficient, cm -~ methods are also very fast, they can be programmed ~, conductivity, W m -~ K -1 easily on a PC, being a valuable design tool for honey- h conductance (A = X/D), W m -2 K J

a absorptance, absorptivity comb design and for collector optimisation. The cal- E emissivity culation can be done on first principles with acceptable p reflectance accuracy. All properties can be calculated from spectral r transmittance film data and from geometric dimensions of the struc- O dimensionless temperature (eqn [16]) ture. However, the problem of getting the appropriate • dimensionless heat flux (eqns [9, 16])

dimensionless emissive power (eqn [9]) input parameter in the solar range can be avoided by ~- dimensionless coordinate (eqn [9 ]) using the method to fit experimental data for one real ~ integration constant (eqn [16]) material. This seems an appropriate way to investigate 7 matrix coefficients (eqn (B7)) Variations of an existing honeycomb, as the structures F iteration damping factor (eqn [22])

produced are very often far from being regular and ideal. Therefore even input parameters such as film Subscript

0 initial or total value thickness cannot be defined exactly. It cannot be ex- I single film pected that a theory of ideal structures can describe c cell that exactly, f film

hc honeycomb

Acknowledgments--This work was supported by a grant from w cell wall the Alfried Krupp von Bohlen und Halbach-Stiftung, Essen. x x-direction Part of the work was supported financially by the German y y-direction ministry of research and technology under research contract z z-direction 03E-8411-A. abs absorptive

dif diffuse-diffuse (hemispherical-hemispherical) eft effective

NOMENCLATURE cond conductive tad radiative

bp, cp coefficients for linearising the dimensionless emissive sol solar power (eqn [17], and Appendix B) A endplate A (hot)

df film thickness B endplate B (cold) dg air gap width H homogeneous layer

Collectors with a honeycomb cover 391

REFERENCES

1. H. Tabor, Cellularinsulation(honeycombs),SolarEnergy 19. J. M. Hammersley and D. C. Handscomb, Monte Carlo 12, 549-552 (1969). methods. John Wiley & Sons, New York (1964).

2. G. Francia, Un nouveau collecteur de l'energie rayonnante 20. Y. A. Schreider (ed.), Method of statistical testing--Monte solaire--theorie et verifications experimentales, Confer- Carlo method, Elsevier, New York (1964). ence des Nations Unies sur les sources nouvelles d'energie, 21. W.J. Platzer, P. Apian-Bennewitz, and V. Wittwer. Mea- E/conf. 35/S/71 (1961). surement of hemispherical transmittance of structured

3. K. G. T. Hollands, Honeycomb devices in flat-plate col- materials like transparent insulation materials, Proceed- lectors, Solar Energy9, 159-164 (1965). ings SPIE-ConF 1272. March 12-13 1990, The Hague

4. K. G. T. Hollands, Natural convection in horizontal thin- ( 1991 ). walled honeycomb panels, Z Heat Trans,,r, 430-444 22. R. Siegel and J. R. Howell, Thermal radiation heat trans- (1973). jbr. Hemisphere Publishing Co., New York (1980).

5. I. Catton, Natural convection in enclosures, Pro~,edings 23. C. L. Tien and W. W. Yfien, Radiation characteristics of ~)[ the 6th Int. Heat Tran,~/~,r ConiC, Toronto, Vol. 6. 13- honeycomb solar collectors, Int. Z Heat Mass Transfer 31 (1978). 18, 1409-1413 (1975).

6. W. W. S. Charters and L. F. Peterson, Free convection 24. W.J. Platzer, Solare Transmission und W~irmetransport- suppression using honeycomb cellular materials, Solar mechanismen bei transparenten W~irmed~immaterialien, Energy 13, 353-361 ( 1972 ). Doctoral Thesis, Albert-Ludwigs-Univ., Freiburg ( 1988 ).

7. H. Buchberg, O. A. Lalude, and D. K. Edwards, Perfor- 25. M. Rubin, Infrared properties of polyethylene terephtalate mance characteristics of rectangular honeycomb solar- films, Sol Energy Mater. 6, 375-380 (1982). thermalconverters, Solar Energy l3, 193-221 (1971). 26. R. D. Cess, The interaction of thermal radiation with

8. H. Buchberg and D. K. Edwards, Design considerations conduction and convection heat transfer, Adv. Heat for solar collectors with cylindrical glass honeycombs, Tran~/~,r 1, 1-50 (1964). Solar Energy 18, 193-203 (1976). 27. R. L. Cane, K. G. T. Hollands, G. D. Raithby, and T. E.

9. J.G. Symons and M. K. Peck, An overview of the CS1RO Unny, Free convection heat transfer across inclined hon- project on advanced fiat-plate solar collectors, ISES Solar eycomb panels. J. tteat Transfer 99, 86-91 (1977). World Congress, Perth (1983). 28. K. G. T. Hollands, G. D. Raithby, F. B. Russell, and

10. C. J. Hoogendoorn, Natural convection suppression in R.G. Wilkinson, Coupled radiative and conductive heat solar collectors, In: S. Kakac, W. Aung, and R. Viskanta transfer across honeycomb panels and through single cells, (eds.), Nat ural convection, Hemisphere Publ. Corp., New Int. J. Heat Mass Tran.~'fer 27, 2119-2131 ( 1984 ). York (1985). 29. K. G. T. Hollands, G. D. Raithby, and F. B. Russell,

11. K.G.T. Hollands, C. Ford, K. Lyn Karan, and E. Brun- Methods for reducing heat losses from flat plate solar col- drett, Manufacture, solar transmission and heat transfer lectors phase Ill, Final Report for Period May 1, 1977 to characteristics of a compound-honeycomb, Proceedings January 31, 1979, University Waterloo, Waterloo (1979). ISES Solar World Congress 1987, Hamburg, Pergamon 30. S.J.M. Linthorst, Natural convection suppression in solar Press, Oxford, pp. 646-650 (1988). collectors, Ph.D. Thesis, Delft University of Technology,

12. A. Pfltiger, W. Platzer, and V. Wittwer, Transparent in- Dutch Efficiency Bureau, Pijnacker (1985). sulation systems composed of different materials, Pro- 3 I. S. L. Marcus, An approximate method for calculating the ceedings ISES Solar World Congress 1987, Hamburg, heat flux through a solar collector honeycomb, Solar En- Pergamon Press, Oxford, pp. 636-640 ( 1988 ). ergy 30, 127-131 ( 1983 ).

13. M. Rommel and V. Wittwer, Flat plate collector for pro- 32. K. G. T. Hollands and K. lynkaran, Proposal for a com- cess heat with honeycomb cover--An alternative to vac- pound-honeycomb collector, Solar Energy 34, 309-316 uum tube collectors, Proceedings ISES Solar World Con- ( 1985 ). gress' 1987. Hamburg, Pergamon Press, Oxford, pp. 641- 33. K. G. T. Hollands, T. E. Unny, and G. D, Raithby, Free 645 ( 1988 ). convective heat transfer across inclined air layers, Z fh, at

14. W.J. Platzer, Solar transmission of transparent insulation Tran~[~,r 98, 189-193 ( 1976 ). materials, Sol. Energy Mater. 16, 275-287 (1987). 34. J. A. Duffle and W. A. Beckman, Solar engineering ~)[

15. K.G.T. Hollands, K. N. Marshall, and R. K. Wedel, An thermalprocesses, John Wiley & Sons, New York, Chap- approximate equation for predicting the solar transmit- ter 6.9 (1980). tance of transparent honeycombs, Solar Energy 21,231- 35. Draft international standard ISO/DIS 9050, Glass in 236 (1978). building--Determination of light transmittance, direct

16. J.G. Symons, The solar transmittance of some convection solar transmittance, total solar energy transmittance and suppression devices for solar energy applications: An ex- ultraviolet transmittance, and related glazing properties, perimental study, J. Sol. Energy Eng. 104, 251-256 ISO(1987). ( 1982 ). 36. K.G.T. Hollands and J. L. Wright, Heat loss coefficients

and effective tau-alpha-products for fiat-plate collectors 17. D. K. Edwards and R. D. Tobin, Effect of polarization with diathermanous covers, Solar Energy 31, 211-216

of radiant heat transfer through long passages, 3~ Heat (1983). Tran.~;[i,r 89C, 132-138 ( 1967 ). 37. B. Jacobs, Bestimmung des Gesamtenergiedurchlal3grades

18. M.G. Kaganer, (l~Ibersetzg. G. Wahl), Untersuchung der transparenter W~irmed/immsysteme, Diploma Thesis, Ausbreitung von Licht im streuenden Medium durch die University Freiburg (1989). Methode der diskreten Ordinaten, Optika und Spektros- 38. E. M. Sparrow and R, D. Cess, Radiation heat lran,sJbr, kopika, Vol. XXVI/3, 443-449 (1969). McGraw-Hill, New York (1978).

APPENDIX A

Formulae for the calculation ofthetransmittance through o = A-i -- Al*a - (1 - a2)*F, (A2) a homogeneous absorbing-scattering layer with the help of the three-flux approximation ( Kaganer [ 18 ] ). where:

: cosine of incident beam radiation,

~" = (At + 1)*exp(-to/tXo) - Al*exp(-kj*to) a*[exp(kl*to) exp(-kl*to)]*F, (A1) F= [A_~*exp(-to/l~o)- Al*a*exp(-kl , to)] (A3)

- exp(kl*to) - a2,exp(-kl , to) '

392 W.J . PLATZER

09 A.~ = (A4) k~ = ~ffl*V(l - w) / ( l - w * ( l - a l ) ) , (A6)

4*(go ~- U,)*( 1 - w*(9tto: - l ) / (9gg - 4)) '

a = (1 - k l * g ~ ) / ( l + k ~ * u l ) , (A5) a ~ = ~ and Ul =2,3. (AT)

A P P E N D I X B

With dimensionless quantities 'I% = - 4 N ~ - ~s + ~Aexp(--1.5t0)

t . l t B

t = K * z , t o = K . D , (Bla )

f[° 3 exp(-1 .5 to) O%xp( I .5 t )d t - ~ , , (B5b) X * d T / d z q,ol - - qrad ql c = -- - - kO, = ( B I b ) 2 % = aTA4 ' e~ ¢TA4 ' ~TA4 '

d 2 0 3 kO = ~I'c + q , , (BIc) 0 = - 4 N --d'~-" A -- ~ [¢.4 + ~Bexp(- l .5 to)

0 = T / T A , (Bid) 9 ~,o 2 + kXVsTo] - ~ 0 4 e x p ( - l . 5 t ) d t , (BSc)

~ef f * g N - - - (Ble) 4~TA 3 '

4 d20 - ~ [ ¢ s CAexp(-l .5to) 2 + k f f ' , ] Tabs(t) := rabs(z(t)), (Blf ) 0 = - N " ~ - s + -

£o one can write the energy conservation equation, eqn (26): _ 9 exp(-1 .5 /o) O4exp( 1.5t)dl, (B5d)

4

q% := ~c + ~r - ~ , * Tabs dO one gets the constants ~ and ~o:

= - 4 N -~- + ~ - qG* T, us(to - t). (B2) 3[

= 9 N - 6NSA + 3~A -- ~ ~I'o -- 'IqTo , (B6a)

The radiation heat flux from eqn (8) in dimensionless form i , se e0W,, t e0imen ,on,ess oon a a io itie , O one gets the central equation (for homogeneous materials): ~o = 1 s 3 3 (~A -- ~B)

qG __ T o + k ( l + To)]) q/o = - 4 N * dt + 2tPA*E3(t) 2¢s*E3( to - t) ~ . [1 . (B6b)

f + 2 0 4 ( l ' ) E 2 ( t - l ' )d t ' • The radiosities Cx can be determined from eqn ( 8 ) using eqn (B3) again to eliminate the integrals. £0

- 2 0 4 ( t ' ) E 2 ( t ' - t ) d t ' - q/~*l~bdto -- t), (B3) (1 - ~A)

CA = 1 -- - - [qo + 4N*sA + kO,To], (B7a)

RA R e ~A with CA= aT--'~' C s = a t e ' (1 - ¢ s )

~B = 0 4 - - - [~o + 4 N * s s + ~ ] . (B7b)

Ei : Exponential integral functions. ~B

With the exponential kernel approximation (e.g., [38]) and After linearising the nonlinear term O4(t) , which is good for an approximation of the absorptive transmittance r ,~ ( z ) with O~ ~ 1, the resulting linear differential equation of second an exponential function one gets after trivial, but extensive order can be easily solved and the constants determined by manipulations (two partial differentiations, subtraction from the boundary conditions for the temperatures. With the ab- original form, and integration of remaining terms): breviations

d20( t ) 9 4 N - - - - ~ - 9NO(t) - 304( t ) ~ (~o + ~s) *t bp = 4u3u5 -- 5U2U6 10UlU6 -- 6U2U5 - - = 20UlU3- 15U2Uz' Cp 20UtU3- 15U2U2

3 gG, T , ~ ( t o _ t ) , ( l - k z) for u i := 1 - O ~ , , + 2 k n, (B4)

9 3cp 9 3 1 - k z)

where T~bdt) = T o e x p ( l . 5 * k * t ) and T o - T(to) .

With eqns (B4) and (B3) taken at the boundaries A a n d B r, = ( 4 b p - n), = s~ = SA, = S2 = SB,

9p2 e~ e~ ~o = - 4 N + ¢'A -- ¢,sexp(--1.5to) ao = 16Nr~ ' a, 4p~r~ ' a2 4p~r~ '

A

3(bp - 1) - 9N + e~p: + ~ T o ( e l - ~ k -1) 23 £to O ~ e x p ( - I . S t ) dt _ ~IqTo, (BSa) bo = 4 N q

Collectors with a h o n e y c o m b cover 393

( 8a, ' 9 r lp j ' 3"1,3"22 - 3"n3"2, ' 3"1,3"22 - 3",23"2~ '

3'I ' ,( 1 - k 2) 3(2 - cA) 3(2 - EB) 'Iio = 16Nr2 Co 2 N k ( 9 r ~ _ 4 k 2 ) , e l - 2 ,~ - - -~ ' e 2 - 2 , ~ " - - - ~ ( a o + a l s l + a 2 s 2 ) . (B8b)

Pl = to~4 + (el + e2)/9, The resulting temperature distribution is given by the function:

P2 = [ N ( I - Os ) + (1 - O ~ ) / 3 - t s ( 2 e ~ T o + 2e2

+ 3(1 - T o ) / r l ) / 1 8 ] / p l , O ( t ) = d ~ e x p ( r . t ) + d 2 e x p ( - r l t ) - ~ [ t cosh( r l t ) r l [ rl

_ _ ] 4r3 w~ = exp ( r~ to ) , w2 = e x p ( - r l t o ) , r4 (1 + s inh ( r l t ) ) ] 4- 9 k 2 _ 4r-----~ T ( t o t ) , (B9) 3"1o = [ - r l - ao bo + coTo(r l + 1.5k)]wl r2

+ [rl - ao + bo - coTo(r l - 1 .5k )]w2 + [2a0 - 3c0k], with coefficients

3'11 = [1 + al + bl]w~ + [1 + al - b~]w2 - 2a~, d l - 0.5[r~ + s~ - coTo(r~ + 1.5k)]

712 = [a2 + b2]w~ + [a2 - b2]w2 - 2(I + a2), + ao + al + a2 + bo + b~ + b 2 } / r ~ ,

3"20 = [ - r l - ao - Do + coTo(r l + 1.5k)]wl d2 = 0.5[r l s l - ~ b T o ( r l - 1 . 5 k ) ] } / r l .

+ I - r 1 + ao - bo + coTo(r~ - 1.5k)]w2

+ [2bo + 2rj(OB + aoto - c0)k], The d imens ionless coupled heat flux is then t r÷c := t 0 + t , , To for S 4: 0, for S = 0 one has to calculate again with

3'2~ - [1 + a~ + b~]w~ - [1 + a~ - b t ]w2 - 2(a~r~lo + b~), t , = 0. The result ing equat ions in this case correspond to the result of Marcus [31], who had a mispr in t in his algebraic

3"22 = [a2 + b2] Wl - - [a2 - b2] w2 - 2 (a2rdo + b2), system, however. A limit k --~ 0 for nonabsorb ing materials canno t be formed with the results, but taking k = 0 in the

one gets the solut ions for t empera tu re gradients and heat flux: equat ions leads to the equivalent case t s = 0.