TINTED AUTOMOBILE WINDSHIELDS

during the day, and clear windshields of the highestattainable transmittance. In closing, the author wishesto endorse strongly the recommendation made byHeath and Finch,5 namely that the 70 percent minimumtransmittance requirement for windshields in the Ameri-can Standard Safety Code Z26.1-1950 be reconsidered.

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

ACKNOWLEDGMENT

The author wishes to express his grateful appreciationto Professor J. H. Mathewson, Assistant Director ofthis Institute, for many valuable discussions and sug-gestions regarding this paper.

VOLUME 45, NUMBER 6 JUNE, 1955

Interreflections in Rooms by a Network Method

PHILIP F. O'BRIENDepartment of Engineering, University of California, Los Angeles 24, California

(Received November 15, 1954)

A network method for the prediction of the luminous-emittance distribution after interreflections inrooms with specified geometry, initial division of light flux to the room surfaces, and luminous reflectances ofthe room surfaces is presented. Potentials in the network are initial and total (i.e., after interreflections)luminous emittances while the surface reflectances and shape factors define the resistances. Uniform illu-mination of all surfaces and diffuse light-flux emission and reflection are postulated. Algebraic solutions tothe networks describing infinite and cubical rooms are presented. The average illumination of any plane ofarbitrary orientation, such as the working plane, can be computed by a floating potential node technique.Luminous-emittance distributions in a cubical room as obtained by Moon's integral-equation solutions arecompared with data obtained by the network method.

THE rational design of room lighting systems mustbe based on a prediction of the total luminous

emittances of the room surfaces when all interreflectionsof the light flux are considered. Without these luminousemittance data, the calculation of glare parameters,shadow conditions, illumination distribution over thework, and other important properties that influence theperformance of visual tasks cannot be convenientlypredicted. Because economic and visual performancerelationships dictate room surfaces of high luminousreflectance, the interreflections of luminous flux withinthe enclosure must be recognized.

Investigations of light-flux interreflections withinenclosures have been undertaken by several researchersbut the experimental work of Harrison and Andersonand the analytical work of Moon2 have been presentedin forms most available to the lighting practitioner.The outstanding limitation of the Harrison and Ander-son data is (1) the average luminous emittances of theroom surfaces are not available, and (2) the range ofsurface reflectances, particularly of the floor, is notfitted to modern practice.

A network method for the description of the inter-reflections of radiant flux within an enclosure has re-cently been delineated by Oppenheim.' AlthoughOppenheim was concerned with radiant heat-transfersystems where the wall temperatures are sensitive tothe radiant-flux transfer, the method may be applied tolight-flux systems with minor changes. While flux

I W. Harrison and E. A. Anderson, Trans. IlIum. Eng. Soc. 11'67 (1916).

2 Parry Moon, J. Opt. Soc. Am. 31, 374 (1941).3 A. K. Oppenheim, The Network Method of Radiation Analysis

(Heat Transfer and Fluid Mechanics Institute, University ofCalifornia, Berkeley, 1954).

transfer within an enclosure may be regarded as asteady-state field problem, the field system can beapproximated by a lumped parameter network suchthat the relative geometry and radiant properties definethe impedances. A network that can be employed todescribe light-flux interreflections within an enclosure isdefined in the following section, and some applications ofthe method to infinite and cubical rooms are outlined.

DEFINITION OF THE LUMINOUS-FLUX NETWORK

Consider an opaque surface, forming one element ofan enclosure, from which light flux is streaming awayand at which light flux is incident. The net flux transferaway from the surface can be expressed as the differencebetween the total flux streaming away (regardless ofwhere it entered the enclosure originally) and the totalincident light flux. The net flux is a rate difference thatrepresents the net luminous power lost or gained at asurface and is a quantity often used in reasoning relatedto power balances and the first law of thermodynamics.

qnet=AdiHi-Ds], (1)

where qnet-net light flux from surface 1 (lumens),A 7-area of surface 1 (feet2), H-total luminousemittance of surface 1 after all interreflections (lumens/ft2), and Di-total illumination of surface 1 from othersurfaces after all interreflections (lumens/ft2). But thetotal luminous emittance, H,, is the sum of the fluxwhich originates at surface 1 and the flux (regardless oforigin) that is reflected from surface 1.

H1= Hol+piDl, (2)

where H 0 -the original or initial (before interreflec-tions) luminous emittance of surface 1 (lumens/f t2) and

419June 1955

PHILIP F. O'BRIEN

By equating Eqs. (3) and (5):

n~~m Ai(1 -pO) H o,E A F r-[HHI-.]I= ( -H1n=2 Pi (1-pI)

(6)

H.,(I-P,)

FIG. 1. Network describing the luminous-flux transferwithin an enclosure defined by three surfaces.

p 1 -luminous reflectance of surface 1 (dimensionless).Equations (1) and (2) can be combined to yield anexpression for the net flux transfer from surface 1 interms of the initial and total luminous emittances.

Equation (6) indicates that the net flux from surface 1after all interreflections is equal to the product of apotential difference and a conductance that is a functionof the goemetry (e.g., AIF,,,) and the surface reflect-ance (e.g., (1-p')/pi).

A network of a form described by Eq. (6) is depictedin Fig. 1 for a system of three surfaces that define anenclosure.

The simultaneous equations that represent a solutionto the network depicting the luminous transfer are

A,-Ho10 =Pi

A,+- Hi-F 1. 2A 1 H2 -* * -Fj FmA iHm,

Pi

Ai (-pp]i) H 1P1 [=(1)P1) A2(3) -HO2 = - F2 1A 2 H,

P2

A 2+- H2* - F2 ...,A 2H.,

P2

The total light flux A 1Di incident on surface 1 is the sumof the light fluxes from all surfaces (forming the en-closure) seen by surface 1.

A 1Di= A lFl 2H2 +A lFl3H3+. -A iFilmHm,

n-1n4 (4a)ADi= EAlFl-.H.,

where F 1 an-the shape factor' of surface n with respectto surface 1 (dimensionless) and Ha-total luminousemittance of surface n after interreflections (lumens/f t2).Similarly, the total flux A HI leaving surface 1 is thesum of the fluxes initially intercepted by each surfaceseen by surface 1.

A H =A F 1 2 J11+A F1 3H1+..A F ,Hi,

n-m (4b)A Hl = 57 A F1 bnIl.

n=2

By the combination of Eqs. (1), (4a), and (4b), andconsidering the reciprocity relationship of shape factorsand noting that the sum of the shape factors of allsurfaces seen by a given surface is unity, the followingrelationship may be written:

n=mqnet= E A2F1 nEH,-Hn].

n=2(5)

4 The shape or view factor F1,n is defined as the fraction of theperfectly diffuse flux leaving surface 1 which is initially interceptedby surface n. A detailed definition and the special properties of thisfactor that apply to all diffuse radiations are given by H. C. Hottel,"Radiant-heat transmission" in W. H. McAdam's Heat Transmis-sion (McGraw-Hill Book Company, Inc., New York, 1954),Chap. 4.

Am*Ho. =- Fm.-iA mH -Fm - 2A mH2 . . .

Pm

Am+- Hm.

Pm

(7)

During the development of Eq. (6) and its associatednetwork, several postulates necessary to the completedefinition of the idealized luminous transfer-systemhave been omitted in order to facilitate the presentationof this derivation. These postulates must, however, beclearly understood when the luminous network, whichaccurately describes an idealized system, is applied tothe design of real systems. A list of these postulates is asfollows: 1-. Surfaces are opaque. 2. Emission and reflec-tion of flux are diffuse. 3. The flux incident on a surfacefrom any other surface is uniformly distributed over thesurface. 4. The surfaces are grey (i.e., nonselectivereflectance according to wavelength). 5. A surface doesnot see itself. 6. Steady-state conditions exist. 7. The

CEILING H0

_Ho, /0 How P °

Il-p.)A | W00 / (l-PWP-rt(-p)

(/-P3)A H03FLOOR I-Np,) PS

FIG. 2. Network for the luminous-flux distributionwithin an infinite room.

420 Vlol. 45

June1955 INTERREFLECTIONS IN ROOMS BY NETWORK METHOD

enclosure contains no absorbing, reflecting, or scatteringmedia.

The most important limitation of the application ofthe luminous network is probably reflected in postulate(3). That is, unless the mean distance between the sur-faces is large compared to the characteristic dimension ofthe surfaces the incident flux will depart from uniformdistribution. As the number of surfaces composing theenclosure is increased the flux distribution approachesthe condition of uniformity at any surface. The utilityof the network method, however, may be lost whenmany surfaces are employed because the number ofcircuit elements increases according to the relationship:

n~n+ 1]R= 2 (8)

2

where R-number of resistances in the network, andn-number of surfaces composing the enclosure.

APPLICATION OF THE LUMINOUS NETWORKTO THE INFINITE ROOM

The network method may be applied to the fluxdistribution within an infinite room which is defined asan enclosure whose ceiling and floor are parallel planes,infinite in extent. A particular advantage of the infiniteroom is that postulate (3), above, is exactly met. Thecircuit for flux transfer within an infinite room is shownin Fig. 2.

Some important properties of an infinite room are:

A2=A3> > >Ai;Fl-3= Fl-.2=:2; (9)F2-3= 1.0; F3, B= F21 1= 0

where A,--area of walls (feet2 ), A 2 -area of ceiling(feet2), A 3 -area of floor (feet2), F 1, 3-fraction of fluxleaving the walls and incident on the floor (dimension-less), F 2 -fraction of flux leaving the walls andincident on the ceiling (dimensionless), and F2 .3 -

fraction of flux leaving the ceiling and incident on thefloor.

Note that by the reciprocity relationship for shapefactors,

F1,3A1 = F3 .1A 3,

F2 3A 2 = F3-.2A 3, (10)

F ,-)2A1 = F2 .1A 2.

INDIRECT LIGHTING IN AN INFINITE ROOM

In the case of indirect lighting all flux initially entersthe infinite room via the luminous emittance, H02, of theceiling. Because all the flux originates at the ceiling, theluminous emittances of the wall, Ho,, and the floor, H0 3,are equal to zero. By reference to the simultaneousEqs. (7), the total luminous emittance, 12, of the ceilingafter all interreflections may be written in determinant

form as:

A, HoiA +- + -F1. 3 A 1

pi P1

H 0 2 A 2-F2...A 2+- -F2 -3A2

P2

H 03 A3 A 3-F3 ,,A 3+ +-P3 P2

A,

Pl-F1 +2Al-Fl 1 3Al

A 2-F2P1A 2+-P2

- F2 -+3A 2

A3-F 3 1A 3-F3 -+2A 3+-

P3

A,

Pl

I0 -

o o

102- -1P2

10 0 +-

P3

Al

Pi0

10 +-

P2

0

- 1

1O - 1 +-

P3

H02

1 -P2P3.(11)

In the same manner the total luminous emittance,H3, of the floor can be computed and is listed in Table I.

The total illumination, Dwp, of the working plane isalso a useful quantity. The working plane may be de-fined as a surface parallel to the floor and one-quarter theheight of the room above the floor. It is convenient interms of the network to regard the working plane asexhibiting zero reflectance and absorptance but a trans-mittance of unity. Because the working plane sees onlythe walls and ceiling of the infinite room the illumina-tion, Dwp, of the working plane is

D. 1,= HIFw., 1+H2Fw.p 2- (12)

By reference to the circuit of Fig. (2),

H2-Hwp Hwp-H1

1 1

Fwp-+2Awp Fwp y)lAwp

(13)

TABLE I. Total luminous emittances of the ceiling, 2, and thefloor, H13, and the average illumination, Dwp, of the working planein an infinite room.

Indirect Direct Diffusedlighting lighting lighting

2 102 P21 103 H02+P2Ho31P2P3 1 P2P3 1-P2P3

P3H1 02 H03 H03+P3HO2

1-2P3 I-P2P3 1-2P3

D ~ 1 Ho2 H03 H03±P3H021-2P3 p3[-P2P3] P3[1-P2P3]

. =

421

PHILIP F. O'BRIEN

By operating on Eq. (13)

HFw,,p *i+H2Fwp-+2Hap=

DIFFUSED LIGHTING IN AN INFINITE ROOM

In this case all the flux enters the infinite room via(14) H02 and H03. By the same method the total luminous

emittance, H2, of the ceiling is

But the sum,

Fwp- +Fwp-2 = 1.0. (15)

By Eqs. (12), (14), and (15), the illumination, Dwp,and the total luminous emittance, Hwp, are shown to beidentical. Referring to the circuit of Fig. 2, it is apparentthat Hwp is equal to H2. The illumination of the workingplane is see Eq. (11)]

H02

1 -p2P3(16)

DIRECT LIGHTING IN AN INFINITE ROOM

In the case of direct lighting all flux initially entersthe infinite room via the luminous emittance, Ho3, of thefloor. Paralleling the method outlined above, the totalluminous emittance, H2, of the ceiling for floor or directlighting is

A,

Pi

0

0

H2=Al

Pi

0 0

0 -1

H03 1

P3 P3

0

10 +-P2

0

- 1

10 -1 +-Pa

Table I contains expressions for H2, H3, and Dwp fordirect, indirect, and diffusing lighting systems in aninfinite room. These expressions are identical to thosederived by Moon' using a ray-tracing and infinite-seriesmethod.

P2Ho3

1-P2P3(17)

When direct lighting is employed, all of the light fluxabout to enter the interreflection network passes theworking plane on the way to the floor. Therefore, in thecase of direct lighting

H03D.v= H2+-* (18)

P3

Introducing Eq. (17) into Eq. (18)

H03Dtop= P3[1-P2P3] (19)

APPLICATION OF THE LUMINOUS NETWORKTO THE CUBICAL ROOM

In the infinite room the walls did not influence theinterreflections, but the walls of a cubical room are asignificant factor in determining the light-flux distribu-tion. Using the network method, expressions for thewall, ceiling, and floor luminous emittances are com-puted for the case of ceiling or indirect lighting. That is,all flux enters the cubical room via the initial luminousemittance, H02, of the ceiling. The cubical room is con-sidered to be formed by six surfaces (i.e., the four walls,the floor, and the ceiling). As in the infinite room, theceiling properties are designated by the subscript "2"and the floor by a subscript "3", while the walls aredesignated by the subscripts "la," "lb," "," andld."A useful property of a cubical room is that the shape

factor of a surface of the cube with respect to any othersurface is identical. This is expressed by

F~s3= Fg.S1a= Fool a= F2.. 1 = F~e1of= 0.2. (21)

5Parry Moon, Ligkting.Design (Addison-Wesley Press, Inc.,Cambridge, 1948), pp. 165-239.

A,

Pi0 0

H020 +-

P2

H030 +P3

-1

P3H2:

H02+P2Ho3

1 -P2P30 .0

(20)Al

Pi

0

0

1

P2-1

1-1 +-

P3

422 Vol. 45

June1955 INTERREFLECTIONS IN ROOMS BY NETWORK METHOD

The total luminous emittance, H3, of the floor afterall interreflections are included is

1

P.

-0.2

-0.2 -0.2 -0.2 -0.2

1

P1k

-0.2 -0.2

-0.2 -0.2 -0.2

1+- -0.2 -0.2

Pie1

-0.2 -0.2 -0.2 +-Pld

-0.2

1-0.2 -0.2 0.2 -0.2 +-

P2

-0.2 -0.2 -0.2 -0.2 -0.2

1

Pla-0.2

1-0.2 +-

Plb

-0.2

0

0

0

0

H 0 2

P2

0

-0.2 -0.2 -0.2

-0.2 -0.2

1-0.2 -0.2 +-

Pi,

-0.2

-0.2 -0.2

1-0.2 -0.2 -0.2 +-

Pid

-0.2 -0.2 -0.2

-0.2

-0.2

-0.2 -0.2

1-0.2 +-

P2-0.2

1-0.2 -0.2 -0.2 -0.2 -0.2 +-

P3

H 0 2

L(- 0.04)

p2p3) 1 1 1 1

0.2 P2P3 Pia Plb Pie P1 -

Eq. (23) for meaning of primed reflectances]:

(--0.04)H02 _ _

L 0.2pla'P3' Pla' Plb'

Hla =

p- p 1

P2P3' ( 0.04 ) 10.2p2 /p3' Pia

H 02

PIe

(25)

1 1 1

Pb PIZ Pld -

(2P3 ) 1 1 1Pla P2P2 _ __ .. _

0. 2 P2'P3' Pla Plb Plc Pid-

(26)

Expressions for Hlb, Hi,, and Hid are symmetrical toEq. (26) by interchanging Plb', Ple', or PId with Pla'.

The luminous emittances, Hi, H2 , and H3 , for a cubicalroom with ceiling lighting as computed by Eqs. (22),(25), and (26) are listed in Table II. Note that the dataof Table II were computed for a floor reflectance, P3

= 0.30, and for a ceiling reflectance, P2= 0.80, while thereflectance of the four walls, pi, was equal to 0.80, 0.50,0.30, and 0.10. The starred entries in Table II are com-puted using Moon's data5 based on the integral-equationmethod. Because the wall was lumped into a single unitfor these network calculations and the integral-equationmethod allows for integration along the height of thewall, the network data are not identical to Moon's butcertainly are the same order of magnitude and constitutea check of the network method.

Equations (22) and (26) can be combined to revealan interesting property of a cubical room as defined in

,(22) this section. That is, the ratio of luminous emittanceof the floor to that of the walls is independent of theceiling reflectance, P2. This relationship may be writtenas

H3/H1= Pl'/P3a- (27)

where

1pn '=-+0.2. (23)

Pn

A check of Eq. (22) may be obtained by consideringthe cubical room with all black surfaces. In this case[i.e., Pi=P2=P3=O]-

H3- 0.2H0 2=F 3 -2H02=D 3 .P3

(24)

Expressions for H2 and HIa are as follows [note: see

TABLE II. The luminous emittances of the walls, Hi, ceiling, H2,and floor, H3, with respect to the initial luminous emittance of theceiling, H02, in a cubical room with ceiling lighting. The luminousreflectance, pi, of the walls is varied from 0.80 to 0.10 while theceiling reflectance (P2=0.

80) and the floor reflectance (p3=0.30)

are constant. Numbers marked with an asterisk are computedfrom Moon's data (reference 5).

pi =0.80 pi =0.50 pi =0.30 pi =0.10

1 0.465 0.184 0.084 0.0231102 0.485* 0.174* 0.080* 0.021*

H2 1.33 1.18 1.11 1.031102 1.42* 1.17* 1.08* 1.03*

H3 0.191 0.115 0.084 0.0671102 0.169* 0.087* 0.062* 0.047*

t3=

H3=

=

: -

423

PHILIP F. O'BRIEN

OA5,

0.40

0.35

0.30

X 0.25

. 0.20

. 0.15

0.10

0.05

0

3 WHITE + I BLACK WALLS

2 WHITE + 2 BLACK WALLS

yl WHITE + 3 BLACK WALLS

\4 BLACK WALLS

0 0.1 0.2 0.3 0.4 05 0.6 Q7 0.B 09 1.0

CEILING REFLECTANCE, P2

FIG. 3. Ratio of the luminous emittance of the floor, H3, to theinitial emittance of the ceiling, H02, as a function in the ceilingreflectance with various combinations of black (=O) and white(pl= 1.0) walls. Data describe flux transfer in a cubical room withindirect or ceiling lighting. The floor reflectance, p3, is constant at0.30.

The importance of the wall reflectance, , in thedetermination of the total luminous emittance, H3, ofthe floor may be determined by means of Eqs. (22), (25),and (26). In Fig. 3, the luminous emittance ratio,H3/HO2, is plotted as a function of the ceiling reflectance,P2, for various numbers of black (i.e. pi=0) and white(Pi= 1.0) walls. The floor reflectance, p3, is constant at0.30. With a ceiling reflectance of 0.90 the total luminousemittance of the floor is reduced 60 percent with theaddition of one black wall. Figure 3 should be particu-larly useful to estimate the influence of clear-glass wallson the flux distribution within a cubical room.

SOME POSSIBLE APPLICATIONS OF THELUMINOUS NETWORK

The network method seems to be applicable to anextended range of radiant-flux transfer problems. Somepossible applications of the method are:

1. The transfer of flux through a scattering andabsorbing media, such as the earth's atmosphere that is

characterized by a nonuniform spatial distribution ofradiant properties but might be lumped in sections ofnearly uniform property for solution by the networkmethod.

2. The modification of the spectral energy distribu-tion of flux in an enclosure due to interreflections fromnongrey surfaces might be considered by a step by step(according to wavelength) solution of the luminousnetwork.

3. The flux incident on any imaginary plane withinthe enclosure might be computed as suggested by Eqs.(12), (14), and (15). This procedure may be called the"floating potential-node method."

4. The introduction of daylight flux into a roomthrough a light-maze (e.g. louvers) might be repre-sented by the network.

5. Specialized analog computers can be constructedto facilitate the use of the network method for fluxdistributions.

6. Flux transients can be considered by the intro-duction of capacitors at the proper positions in thenetwork.

CONCLUSION

A network method for the calculation of luminous-flux distributions within rooms of any relative geometryand luminous reflectance combinations has been out-lined. Digital and/or analog computers can be em-ployed to speed the solution of these networks.

The over-all nature of the luminous-flux distributionwithin a room can be seen by inspection of a singlecircuit diagram, and the effect of various parts of thecircuit, or their lack of effect, relative to other parts ofthe circuit can be quickly observed. These are the out-standing advantages of the network method.

ACKNOWLEDGMENTS

The suggestions of R. Bromberg and F. E. Romie ofthe Department of Engineering, University of CaliforniaLos Angeles, were particularly helpful to the develop-ment of this paper.

424 Vol. 45

4 WHITE WALLSI