NASA TECHNICAL NOTE

CO

NASA TN D-7384

—— BB m

C j ^k H i ^ r

O P Y

NET RADIATION METHOD FORENCLOSURE SYSTEMS INVOLVINGPARTIALLY TRANSPARENT WALLS

Robert Siegel

Lewis Research CenterCleveland, Ohio 44135

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • AUGUST 1973

https://ntrs.nasa.gov/search.jsp?R=19730020183 2018-05-12T11:28:17+00:00Z

1. Report No.

NASA TN D-73842. Government Accession No.

4. Title and Subtitle

NET RADIATION METHOD FOR ENCLOSURE SYSTEMSINVOLVING PARTIALLY TRANSPARENT WALLS

7. Author(s)

Robert Siegel

9. Performing Organization Name and Address

Lewis Research CenterNational Aeronautics and SpaceCleveland, Ohio 44135

' " * » ( • • > • •

Administration

12. Sponsoring Agency Name and Address

National Aeronautics and Space AdministrationWashington, D.C. 20546

3. Recipient's Catalog No.

5. Report DateAugust 1973

6. Performing Organization Code

8. Performing Organization Report

E-7363No.

JO. Work Unit No.

770-181 1 . Contract or Grant No.

13. Type of Report and Period Coveted

Technical Note14. Sponsoring Agency Code

15. Supplementary Notes

16. Abstract

The net radiation method is developed for analyzing radiation heat transfer in enclosure systemsinvolving partially transparent walls. One such system is an enclosure with windows in it. Theconventional net radiation method was developed by Hottel, Poljak, and others for enclosureshaving opaque walls. If a partially transparent wall is present, it will permit radiation to enterand leave the enclosure. The net radiation equations are developed here for gray and semigrayenclosures with one or more windows. Another system of interest, such as in a flat plate solarcollector, consists of a series of parallel partially transparent layers. The transmission char-acteristics of such window systems are obtained by the net radiation method, and the techniqueappears to be more convenient than the ray tracing method which has been used in the past.Relations are developed for windows consisting of any number of parallel layers having differingabsorption coefficients and differing surface reflectivities, and for systems composed ofparallel transmitting layers and opaque plates.

17. Key Words (Suggested by Author(sl)

Net radiation methodSolar collectorsWindow transmittanceRadiation enclosure theory

18. Distribution Statement

Unclassified - unlimited

19. Security Classif. (of this report)

Unclassified20. Security Classif. (of this page)

Unclassified21. No. of Pages

32

22. Price*

$3.00

'For sale by the National Technical Information Service, Springfield, Virginia 22151

NET RADIATION METHOD FOR ENCLOSURE SYSTEMS

INVOLVING PARTIALLY TRANSPARENT WALLS

by Robert Siegel

Lewis Research Center

SUMMARY

The net radiation method is developed for analyzing radiation heat transfer in enclo-sure systems involving partially transparent walls. One such system is an enclosurewith windows in it. The conventional net radiation method has been well developed inthe literature for enclosures with opaque walls. When one or more of the enclosuresurfaces are transparent or partially transparent, radiation can directly enter or leavethe enclosure. The enclosure equations are developed here to account for these effects.Specifically treated are gray and semigray enclosures. Another system of interest is aseries of parallel partially transparent layers such as used in flat-plate solar collec-tors. The characteristics of such a system are obtained here by the net radiationmethod which appears to be more convenient and systematic than the ray tracing tech-nique that has been used in the past. Overall transmission, absorption, and reflectionrelations are developed for windows consisting of any number of parallel layers with orwithout spaces between them, and that have differing absorption coefficients and differ-ing surface reflectivities. Also considered are the characteristics of combinations ofparallel transparent layers and opaque walls.

INTRODUCTION

The net radiation method has been well developed in the literature, such as in thetextbooks (refs. 1 to 3), for calculating radiation heat transfer in enclosures. The-wallsof these enclosures have been opaque. The object of this report is to show how the netradiation method can be modified and applied for systems involving walls that are par-tially transparent for radiant energy. Such systems would include enclosures with one

The term "partially" transparent means that there can be absorption of some ofthe radiation traveling within the wall with the rest being transmitted through the entirethickness.

or more windows, or devices like flat plate solar collectors that consist of a series ofparallel partially transparent plates with an opaque absorber surface behind them.

The transmission characteristics of windows in the form of single or multiple glassplates have been derived by Whillier (ref. 4) using ray tracing techniques. It will beshown here how these results can be developed by use of net radiation concepts thatprovide a convenient systematic approach.

SYMBOLS

A surface area

a absorptance for path within layer

F radiation configuration factor

^0-X T fractional emission of blackbody in wavelength interval 0 to ACc

m number of opaque surfaces in enclosure

N total number of surface areas in enclosure

q . heat flux

r surface reflectivity r at m surface of n layer

T absolute temperature

a overall absorptance of layer or system of layers

^at plate' a^SOTT?^a-nc^ at P^te in presence of transparent plate system

/3 angle measured away from normal of surface

e emittance of partially transparent layer; emissivity of opaque surface

X wavelength of radiation

X cut-off wavelength for partially transparent material\f

p overall reflectance of transparent plate system

a Stefan-Boltzmann constant

T * overall transmittance of transparent plate system

Subscripts:

e externally incident

k k surface area

i incoming radiation .

I loss to outside

0 outgoing radiation

s for entire system

Superscripts:

1 for radiation incident on inside surface

I long wavelength radiation X > \L*

(n) for system of n layers

o for radiation incident on outside surface

s short wavelength radiation A < \

ANALYSIS

Radiative Characteristics of a Single Layer

To illustrate the transmission, absorption, and reflection characteristics of a win-dow for incident radiation, consider a single layer as shown in figure 1. The reflectivity

%,2 Absorbinglayer a

«i,3 '

Figure 1. - Net radiation quantities for ana-lyzing overall performance of partiallytransparent single layer.

at each of the interfaces is r and the absorptance of the layer is a. The relations be-tween the outgoing and incoming energy fluxes at each interface can be expressed interms of the reflection at the interface and the transmission across the interface. Usingthe net radiation concepts, the outgoing flux is written for each side of each interface togive

The internal q/s and q^s are related by using the absorptance of the layer to give

(2a)

The overall radiative behavior can be studied by considering energy incident on only oneside of the layer, and superposition used when radiation is incident on both sides. Sinceit is the fractions of incident radiation transmitted, absorbed, and reflected, that aredesired, the incident flux can be set equal to unity without any loss of generality. Thenany q in the calculation will be a fraction of the incident flux. Hence, using q. 1 = 1i, iand q. A = 0 along with equations (2), the q. 's are eliminated from equations (1) toi, 4 iyield

q0j ! = r + (1 - r)(l - a)q0> 3 (3a)

Q = 1 - r + r(l - a)q (3b)

(3d)

These are solved simultaneously to yield the reflectance p and transmittance r of theentire layer as

rfl + (1 - 2r)(l - a)2]=_L - =1

l - r 2 ( l - a ) 2

l - r 2 ( l - a ) 2

(4a)

(4b)

The absorptance is then obtained from the overall heat balance

a = l - p - r = a^ " r) (4c)1 - r(l - a)

Equations (4) can be directly applied when the incident energy is from a singledirection. The r would be evaluated for that incidence angle and the a would be basedon the path length traversed within the medium (allow for change in direction as radia-tion crosses interface of layer). If the incident flux is diffuse, the average p is obtain-ed by averaging over all angles as (ref. 1)

' fV/2p = 2 / p(/3) cos |3 sin 0 d/3 (5)

•'0=0

where /3 is the incidence angle measured away from the surface normal. The p(/3)is given by equation (4a) with the r and a inserted as a function of |3. The T fromequation (4b) can be averaged in the same manner and the average a then found as1 - p - T.

In the enclosure theory that follows, the p and T for a layer refer to expressionssuch as equations (4a) and (4b), that is, relations that provide the overall performanceof a window for incident energy. If the window is hot enough to radiate appreciably, theemittance of the window is also needed in the enclosure heat balances. According toKirchhoff's law, the e can be found by evaluating a for incident radiation having thesame spectral distribution as blackbody emission at the temperature of the window.This assumes that the window configuration is such that a uniform temperature can beassigned to its entire volume.

Enclosure Theory Equations for Gray Surfaces and Windows

The net radiation method in the literature will now be generalized to include enclo-sure boundaries that are partially transparent to radiation. The usual restrictions ofthe enclosure theory apply which include diffuse emission and reflection, uniform fluxes

*over each surface area and a uniform temperature over each area. A window such as asmooth glass plate will reflect primarily in a specular fashion so that the assumption ofdiffuse reflection is violated. However, within an enclosure there are extensive multi-ple reflections and the directionality of each reflection loses its importance in contribut-ing to the heat fluxes on the boundaries. Hence, the assumption of diffuse reflections is

usually satisfactory. The enclosure heat balances include possible emission from eachwindow. The formulation here is only general enough to account for a window that canbe characterized by a single temperature throughout its thickness. The formulationwould require extension to account for a multilayered window with each layer at a differ-ent temperature. No extension is necessary, however, if the multilayered windows arekept cool by convection for example, so that emission from the layers is small relativeto the transmission through them.

A typical enclosure containing one or more partially transparent windows is shownin figure 2. In general, let there be a total of N surface areas with the k area beinga typical area. There are m opaque surfaces and N - m partially transparent sur-faces.

Outside

Figure 2. - Enclosure with N surface areas, somebeing partially transparent Inside

Figure 3. -Typical nonsymmetric window.

Each window can be nonsymmetric about its center plane as shown in figure 3. Thesuperscripts o and i refer to the outside and inside surfaces. The a° and p° arethe absorbed and reflected portions of the radiation incident from the outside. The e°is the emittance toward the outside and the r° is the fraction of the outside radiationthat is*transmitted all the way through the window. A similar notation is used for thequantities at the inside surface (it will be found later that, for composite windows,r° = 7*).

For the m surface areas that are opaque the conventional net radiation equationscan be written as:

Overall heat balance:

6

% = V k - % k U * k * m ) (6a)

Outgoing energy consists of emitted and reflected portions :

Vk^k^k+^-^k <!***») (6b)

Incoming energy in terms of outgoing fluxes and view factors:

N

N„ (6c)

These equations can be combined in various ways to provide other interrelationsbetween the variables that are sometimes useful. Combine equations (6a) and (6c) torelate q and qQ's,

N

Combine equations (6a) and (6b) to relate q, T and n

d ^ k s m ) (7b)

Combine equations (6b) and (6c) to relate the ^'s and T,

N(7c)

Now consider a partially transparent surface. Using the quantities in figure 2 anoverall heat balance yields

The ch, could be supplied for example by heating wires within the semitransparentmaterial or by convection on the outside surface. For cooling the q^ is negative. Theradiative flux leaving the inside surface of the window consists of emitted energy, re-flected incoming energy, and transmitted externally incident flux,

Similarly the flux lost from the outside surface of the window is given by

(8c)

The incoming flux q. t is given by the same relation as equation (6c)1, K

(8d)

Equations (8) are valid for m + 1 ^ k < N.Equations (8a) and (8c) are combined to eliminate q, to give

Some additional relations are found by combining some of the previous equations. Combine equations (8d) and (9) to yield

Combine equations (8b) and (9) to yield

where the relation 1 - r^ - p^ = ot^ - e^ has been used for gray surfaces. Combineequations (8b) and (8d) to yield

N

, jFk-j

The previous relations relate the qk, T., q ^, and n k for.each of the enclo-sure surfaces. By writing these equations for each surface there are sufficient equa-tions to solve for the unknown values. For an enclosure with opaque walls this is dis-cussed in detail in references 1 to 3. One convenient approach is to eliminate theq^ , 's thereby obtaining a set of equations relating the surface temperatures T,'s, thesurface heat flux additions qi.'s, and the externally incident heat fluxes q^'s.

As a first step the ci > is obtained as a function of Tk and q^^ from equa-tions (7b) and (11)

4 1 - ek<*o k = k -- ~ qk (1 - k ~ m) (13a)

ek

(m + 1 =s k < N)

(13b)

Equations (13) are then inserted into equation (7a) and into equation (10) to eliminate theN

n . 's. The summation £ has to be made in two parts and equations (13a) and (13b)

used in their appropriate ranges. After simplification this yields for the opaque sur-faces,

j=m+l

(1 < k == m) (14a)

and for the partially transparent surfaces;

Nzj=m+l

(14b)L ej J "

j=m+l

.• J

j=m+l

Writing equation (14) for each k yields the final set of N simultaneous equations relat-ing the N surface temperatures, the N surface heat flux additions, and the N - mexternal fluxes on the windows. If 2N - m of these quantities are specified, the equa-tions can be solved for the remaining unknowns.

The enclosure theory formulation contains within it the transmittance, reflectance,and emittance of each window. If a window is kept cool by convection so it does not emit

10

appreciably, then the window emission is not of significance in the analysis. The windowcan then be multilayered as the overall transmittance and reflectance can be obtainedfor such a system as will be shown in subsequent sections. If a window is hot as a re-sult of absorbing radiation or being convectively or internally heated and is emittingsignificantly, then the emission term involves the window temperature which is specifiedas uniform. This would restrict the formulation to simple windows for which a uniformtemperature can be assigned. The net radiation method can be applied for more com-plex situations than those discussed here, including enclosures with multilayered win-dows, with each layer emitting and at a different temperature.

Semigray (Two Band) Method for Enclosure Analysis

The transmission of most window materials is very wavelength dependent havinghigh transmittance in the visible and a portion of the near infrared and low transmittanceat longer wavelengths. There is often a rather sharp transition between the high andlow transmission regions. The cut-off wavelength at which this transition occurs can betypically at about 2. 5 micrometers for ordinary window glass or 25 micrometers for asodium chloride crystal. . .

In the semigray method (ref. 5) or two-band approximation the property variationsare accounted for by using two sets of property values, one uniform value for wave-lengths shorter than the cut-off and a second value for wavelengths longer than the cut-off. The notation es, ps, and so forth, will be used for the short wavelength region0 < X < A and e , p , and so forth, will be used for the long wavelength region

\s

X < X < °° . A common application of the semigray method is in dealing with devicesL/

exposed to solar energy. The incident solar energy is predominantly at short wave-lengths characteristic of visible radiation, while the reradiation is at long wavelengthscharacteristic of the device surface temperature . In the present discussion it will beassumed that the externally incident fluxes and their spectral distributions are known;

s Ihence, q^ and q^ are known.If all the surface temperatures are known, then equations (14a) and (14b) can be

solved after small modifications. Two equations are written for each surface, one for4

X < X and the second for X > X . In each of these equations the oT is replaced by\j \s

the blackbody emission at T within the respective wavelength intervals 0 - Xn and4 4X - °°. These quantities are given by oT FQ_X T and oT (1 - FQ_^ T) where FQ , T

c c cis the fractional emission of a blackbody in the range 0 - Xc for a blackbody at T; thefunction F_ depends only on the product XT.

Since all quantities on the right of each equation are given, this yields 2N equa-tions that can be solved directly for the unknown N values of q? and N values of q

11

Then the heat flux being supplied to each of the k surfaces or windows by means otherthan radiation is

qk = qj + q£ (15)

The set of equations for two wavelength bands is given in the appendix.A complication arises in the simultaneous solution when it is the equilibrium tem-

perature of a surface that is to be calculated when the q,, for that surface is specified.s ZIn that instance the qk and qk are not known individually, and it is only known that

their sum is q, . This introduces an additional unknown in the simultaneous solutions isince the T^ must be found with the separate values of qk and q, still being involved

in the equations as unknowns as it is only their sum that is known. The equations givenin the appendix still apply. There are equations (14a) and (14b) written for each Xrange which gives 2N equations, and equation (15) written for each of the N surfaces.In general, let there be p surfaces that have qk specified; hence N - p surfaceshave T, specified. The total of 3N simultaneous equations can then be solved for thefollowing unknowns: p values of Tk, N - p values of q, , N values of q{? and Nvalues of q£. These equations contain, however, the term crT F«_^ T and the tem-

perature is not contained in FQ_^ j, in a simple way. Hence, an iterative procedureL*

is required to determine the unknown T values.If the property variations with wavelength are such that many more than two wave-

length regions are required to adequately account for the property variations, then thenet radiation equations can be written in spectral form and the fluxes then integratedover wavelength, or a summation made over a finite number of wavelength bands. Thisis an extension of the two band procedure and has been described in textbooks such asreference 1 (chapter 10); hence, the procedure will not be further detailed here.

Transmission Relations for Various Types of Windows

Single layer with different surface reflectivities. - A geometry that provides a use-ful building block to obtain results for more complex windows is a single layer with adifferent reflectivity at each surface. A dual subscript notation will be used for thesurface reflectivities: For a group of several parallel layers, the first subscript desig-nates the layer number and the second designates the upper or lower surface of thatlayer (the layers will be drawn horizontally). Thus, r,_9 would refer to the second or

tVilower surface of the k layer. Referring to figure 4 the net radiation equations for

12

K Xqo 1—V—_/\

,2 ^<*o,2

"i,3

rll

If, 4

Figure 4. - Single absorbing layer withdifferent reflectivity at each surface.

the first of several possible layers are written as

The absorptance of the medium is used to relate the internal q^'s and c^'s to yield

(17a)

To consider the interaction of the layer with radiation incident on the upper surface,let q. -, = 1 and q. 4 = 0. Using these relations and equations (17) to eliminate theif i i> q. 's from equations (16) give

(18a)

(18b)

(18c)

(18d)

13

The solution yields the overall transmittance and reflectance for radiation incident onthe upper surface as

(1 - rn)(l - r12)(l - a,)

- 2rn)(l -a , ) 2

U

The subscript 11 indicates that the T« .. and p« .. are for radiation incident on the firstsurface of the first of several possible layers. The overall absorptance for the layer isobtained from the energy balance

Substituting the previous expressions for TJJ and p^*, this simplifies to

12(l - a,)]a r n = — - — - - - (19c)

1 - rnr12(l - a t)2

If radiation is now considered incident from below (that is on surface 12), the onlychange in the analysis is that r and r12 are interchanged. The overall transmit-tance, reflectance, and absorptance are designated by T-JQ, p*2, a^ an(* they aregiven by

(1 - r12)(l - rn)(l - a,)r12 = - — (20a)

1 - r12rn(l - aj)

r12 + rn(l - 2r12)(l - a^2

(20b)

- r!2rll(1 " al>

+ ri1 "a i-* (20c)1 - r12rn(l - a/

14

This yields the relation T-, j = r-,* so that the second subscript for r can be omittedand the overall transmittance of the layer called TJ. The emissivity of the layer canbe found according to Kirchhoff's law by evaluating a using window properties for in-cident radiation that has the same spectral distribution as blackbody radiation at thewindow temperature.

Two differing layers with all reflectivities different. - The behavior for a singlelayer can be used to obtain the performance for two layers as shown in figure 5. The

1i.ll 1o.ll

Layer 1

r!2

1o,12

%, 21

Layer 2

r21

r22

Figure 5. - Two differing absorbing layers withall surface reflectivities different.

qQ's are written in terms of the qj's using the overall characteristics of single layers,

qo,

, 12 = \ 12P12 + <k, 11T1

(21a)

(21b)

(21c)

(21d)

To obtain the overall characteristics of the two layer system for radiation incident from

above, let q^ 11 = 1 and qi 22 = 0< Also> from figure 5

\ 12 = %, 21

15

and

^,21 =%,12

These relations are used to eliminate the q^s from equations (21) to yield

% 11 =pll + (lo 21T1 (22a)

%, 12 = %, 21P12 + Tl (22b)

V 21 = V 12P21 (22c)

V 22 = %, 12^2 <22d>

Now introduce the notation T «, TO and pgi, Pgo- The subscript s refers tothe entire system. Hence, pgj and pgg are, respectively, the reflectances for radia-tion incident on the first and second surfaces of the entire system. The simultaneousequations (22) are solved to yield

TiTo(23a)

P12P21

0 0' P12P21

Tsl - psl = 1 - pll - (T

By changing subscripts to reverse the order in which the radiation passes throughthe layer surfaces, the overall system characteristics are obtained for radiation inci-dent from below on the two layer system,

-— (24a)- "21*12 -

16

Ps2 =T2p12 (24b)

2p12) (24c)

A comparison of (23a) and (24a) reveals that Tgl = Tg2 so that the system transmittancecan be called simply T_, .

o

Two layers with no space between them. - Two adjacent layers with differing prop-erties are shown in figure 6. The reflectivity at the common interface between the two

Layer 1

) Layer 2

rl-2

r?2

Figure 6. - Two differing adjacent absorbinglayers.

media is called r^_ 2 - The configuration in figure 5 will correspond to this case if welet r j2 = r j_o and roi = 0 (°r using r,,.. = r.._2 and rj2 = 0 will yield the same finalresult). The overall transmittance of the system can be computed by using equation (24a)

Ts = (25)

where from equations (19) and (20),

- rn)(l -(26a)

- a )

- T22^ - a2> (26b)

17

(26c)

P21 = r22(1 - (26d)

The expressions in equations (26) can also be used in equations (23b and c) and (24band c) to compute p ,, a «, p 2* and aS2 f°r the system of two immediately adjacentlayers.

A general system of N layers. - The overall characteristics of one layer wereused to build up the overall behavior of a two layer system. This procedure can becontinued and the behavior for one layer and two layers can be combined to yield thebehavior for three layers; two layers and two layers can be combined to yield the be-havior of four layers; etc. To accomplish this it is desired to have relations for deter-mining the performance of a combined system of n layers and k layers, wheren + k = N.

Referring to figure 7 let Tgn+ ' be the system transmittance for the entire system

and let T_n ,. T_ ' be the transmittances for n and k plate systems, respectively.

The quantity pQJ is the reflectance for a system of n layers by itself for energy in-fa]cident on the second (lower) surface of that entire n layer system. Similarly, p'

Layer 1 aj

1 Layer 2 a2

: : :Layer n an

'11

r21

r22

rmr,,9

, n plates

Layer n + 1

Layer N = n + k an+k

JUILI r(rul)2

rNl

rN2

k plates

Figure 7. - System of N = n + k plates.

18

is the overall reflectance for a system of k layers by itself for energy incident on thefirst (upper) surface of that entire system.

Comparing figures 7 and 5 indicates that the results for the two layer system canbe applied if we use the overall properties of the separate n and k plate systems infigure 7 to replace, respectively, the properties of the first and second single platesin figure 5. Then from equations (23a), (23b), and (24b),

->+*)_.s

n

4Tk) = p(si + — ^ — <27b>1 oH)fe)1 Ps2psl

(27c)

If the spacing between two plates is zero, say the m and m + 1 layers are in per-feet contact, then let rm2 = rm_(m+1) and let r(m+1)1 = 0.

As an example, compute T for a three layer system. Letting n = 2 and k = 1in equation (27a) gives

T(2)T(DT(3) _ Ts Ts

S 1 fl(2)D(1)1 ' Ps2psl

19

where

T,T1T2

21

T(D _ TTs ~ T3

(2) _Ps2 -

2^ 'T2p121 - P21p12

Then

T1T2T3

„ n' P22P31

2P12P31n

" P21P12

T1T2T3

- P22P31> ' T2P12P31

(28)

where the required characteristics for single layers are given by equations (19) and (20)as

pml =ml +

Jm2m2 +

m = 1, 2, 3

As a special case if there is no absorption in any of the plates, a = 0 and, if in addi-tion, all the surface, reflectivities are equal, then equation (28) reduces to

20

T (3)_ T3

s ~

where

T _ (1 - r)2 = 1 - r

1- r 2 1 + r

_ r + r(l - 2r) 2r

1-r 2 " J + r

(0\

If T and p are inserted into the expression for T;, ' it simplifies too

T(3) -5r

This agrees with the general result in reference 6 for a series of n layers with a spacebetween each of them and with all surfaces having the same reflectivity r

(30)(2n -

Systems of Parallel Partially Transparent Layers and Opaque Plates

A system of partially transparent layers with an absorbing surface below them. -This is a typical solar collector geometry and is illustrated in figure 8. It is desired toknow how much of the incident solar radiation is absorbed at the surface N. For a unitflux incident on the first surface of the first layer, q. .- = 1, the fraction absorbed ati, 11the plate can be obtained from

aat plate = q i ,Nl " VN1 (31)

The net radiation method with q. in = 1 yields the following relations:'

21

Layer 1

Layer 2

rllr!2

r21

Layer N-l aN_j

Absorber, qUN-l>2/ \q0>

s n plates

(N-112surface lo.Nl

r(N-l)2 .

rNl

Figure 8. - System of n absorbing layers followed by absorberplate.

qo,Nl ~ rNlq i ,Nl

qi,Nl = qo,(N-l)2

where n = N - 1 layers. Solving for q. .tion (31) yield

"VN1

and a

(32b)

(32c)

(32d)

and substituting into equa-

- rraat plate (33)

r" r nNlps2

As a simplified case, if there are n nonabsorbing layers all having the same surfacereflectivities r, then using equation (30) and the fact that p = 1 - T for a nonabsorbing

system, yields,

1- r(2n -

(n) _ fl(n) _ i (n) _c9 -Pol - -1 To -

2nr

1 + (2n - l)r

22

If these relations are substituted into equation (33), the fraction absorbed by the platebecomes

a(1 - rN1)(l - r) (1 - rN1)(l - r)

at P^te ~ 1 + (2n - l)r - rN12nr " (1 - r) + 2nr(l - rm)(34)

where n = N - 1 layers.Some additional results can be obtained quite easily for the systems shown in fig-

ure 9. For the situation in figure 9(a), T^,n' = T12 = T21 from equations (19a) or (20a)

rn

r!2

(a) With space between layer and surface.

-Opaque surface

^-Opaque surface

rn

T1-N

(b) No space between layer and surface.

Figure 9. - Single absorbing layer inter-acting with opaque surface.

and Pg2 = Pio ^rom equation (20b). Making the substitution in equation (33) yields

(1 - rm)(l - r12)(l - rn)(l - ax)

If

aat plate- rNlr!2

~ r as a sPecial case, equation (35) reduces to

(35)

aat plate- rm)(l -

- rN1r(l - 2r)(l -

If in addition the layer is nonabsorbing, a.. = 0, the result simplifies to

(36)

23

(1 - rN1)(l - r)"at plate = l + r\ f ' (37)i + r - ^rjjjr

For the situation in figure 9(b) the a ^ p^te can be obtained from equation (35) byletting rj2 - r i_N and r\n = 0- Tnis yields

(1 - r i_N)(l - rn)(l - a,)"at plate = ~ " T^ (38)

1 - 'l-N*!!*1 - al>

Transparent layer between two opaque plates. - As a further illustration of the ap-plication of the net radiation technique, consider the effect on the heat exchange of plac-ing a transparent nonabsorbing layer between two parallel plates as shown in figure 10.

i-Opaque

surface qi,l

1i.

Opaquesurface-,

Figure 10. - Effect of intermediate transparentlayer on exchange between two opaque par-allel plates.

The net radiation equations are written as follows:

At plate 1

qi, 4 + * = %, 4 (39a>

qo,4 = e l f f T l+ (1 ' el)(li,4 (39b)

24

At plate 2

(40a)

(40b)

For the transparent plate

QQ 2 =qt 3r+ qi 2p (41a)

q0 3 = qj 2r + qj 3P (41b)

where p and T are the overall reflectance and transmittance of the transparent plate.The connection between the q.'s and q*s is given by

(42)

The relations in equations (42) are used to eliminate the q.'s from equations (39) to (41)to yield

25

The cu 2 an(* QQ 3 ai*e eliminated to yield

These relations are solved for q which is the heat flux being transferred from the low-er plate maintained at T.. to the upper plate at Tg. The result is simplified by usingthe relation p - 1 - r valid for a nonabsorbing layer to yield

q= 1-i Z— (43)1 I 1 „— + — + 2

el e2 T

For a single transparent layer with surface reflectivities r, equation (4b) yields

1 + r

so that by substituting into equation (43) the final result is obtained that

q =

- r

The 2r/(l - r) term is the contribution provided by the transparent plate.

CONCLUSIONS

The net radiation method has been generalized to situations where partially trans-parent layers are present, such as an enclosure having one or more windows. Theutility of the method is further illustrated by applying it to determine the overall trans-mission and reflection characteristics of systems of parallel plates with or without par-

26

allel opaque plates. These configurations arise in solar collectors, solar stills, and incoated surfaces where the coatings are thick relative to the wavelength of the radiationso that interference effects do not have to be included. The method is found convenientto apply and provides a systematic approach that is not difficult even for complex sys-tems. Most partially transparent plates have spectrally dependent properties. The netradiation equations can be applied for various wave length, regions and the energy quanti-ties then summed over wavelength. The semigray (two band) approximation has beenspecifically discussed here which utilizes two spectral regions in each of which theproperties are constant. The utilization of multiple wavelength bands proceeds in thesame fashion as has been well documented in the literature for systems with opaquewalls.

Lewis Research Center,National Aeronautics and Space Administration,

Cleveland, Ohio, May 17, 1973,770-18.

27

A P P E N D I X - ENCLOSURE EQUATIONS FOR TWO BAND APPROXIMATION

The equations for the short wavelength region are for the opaque surfaces,

m N

E €I "' *|V ' >€) Fk-H. . £S'

j o-xcTj

j=m+l

N

(1 sk .sm, X < X C ) (Al)

and for the partially transparent surfaces,

N

x aT?Fn

.i=m+l

(A2)

28

The equations for the long wavelength region are for the opaque surfaces,

m

j=m+l ]

/T T^ /1 T?

'< T T ' '1-F0-XCT J

Z i I, o Zx o I, j"' "V '

j=m+l

T? n M •< lr •< TV* A ^ \ \

(A3)

and for the partially transparent surfaces,

£N

1 - e? V^ l' m

£d k-jqj + > jk pj k-j ^ , ) Z - Z^ k-j j

j=l j=m+l

(A4)

j=m+l

29

The total heat fluxes are found by summing the fluxes for the two wavelength regions,

qk = q£ + q£ (1 < k < N) (A5)

30

REFERENCES

1. Siegel, Robert; and Howell, John R.: Thermal Radiation Heat Transfer. McGraw-Hill Book Co., Inc., 1972.

2. Hottel, Hoyt C.; and Sarofim, Adel F.: Radiative Transfer. McGraw-Hill BookCo., Inc., 1967.

3. Sparrow, E. M.; and Cess, R. D.: Radiation Heat Transfer. Brooks/Cole Publ.Co., 1966.

4. Whillier, A.: Solar Energy Collection and its Utilization for House Heating. Sc.D.Thesis, Dept. Mech. Eng., Massachusetts Inst. Tech., 1953.

5. Bobco, R. P.; Allen, G. E.; and Othmer, P. W.: Local Radiation EquilibriumTemperatures in Semigray Enclosures. J. Spacecraft Rockets, vol. 4, no. 8,

__ Aug. 1967, pp. 1076-1082.

6. Hottel, H. C.; and Woertz, B. B.: The Performance of Flat-Plate Solar-HeatCollectors. ASME Trans., vol. 64, no. 2, Feb. 1942, pp. 91-104.

NASA-Langley, 1973 33 E~7363 31

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