INTER-REFLECTIONS BETWEEN WALLS, CEILING AND FLOOR

Some idea concerning the bearing of reflection upon illumination can be gained readily from a brief study of the values derived from the above absorption problem.

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The total incident flux on the ceiling, walls and floor equal 2000 + 4200 + 6000 12,200 lumens, whereas the lighting units are required to produce only 7900 lumens. The "mean effective absorption coefficient" of the room as a whole is, therefore, 7900 ÷ 12,200 = 0.65. Of the total of 12,200 lumens incident upon the surfaces only 7900 come directly from the lamps, 12,200 - 7900 = 4300 lumens being attributable to inter-reflection between the surfaces.

Since only 2000 lumens are directed toward the ceiling (where 400 are absorbed and 1600 are reflected), whereas 6000 are directed toward the floor, it is apparent at once that the room selected is lighted by lamps which produced considerably more light in the lower than in the upper hemisphere; that is to say use is not made of the indirect system of lighting.

For sake of comparison, consider now the same room with the same absorption coefficients with the same total amount of incident flux upon the floor and walls but with such an amount directed toward the ceiling that the reflection therefrom equals the amount absorbed by the floor. In other words assume that, in effect, use is made of the "totally indirect" system-so far as the ceiling and floor are concerned.

The light flux reflected from the ceiling (with its 0.20 absorption = 0.80 reflection) must equal the 5400 lumens absorbed by the floor. Hence, 5400 0.80= 6750 equals the flux incident upon the ceiling. The tabulation will then be as follows:

Total lumens absorbed = 8850.

Total mean spherical candle-power 8850 ÷ 4 = 705.

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The total incident flux is equal to 6750 + 4200 + 6000 16,950 lumens, as compared with the former 12,200 lumens. Thus with

an increase of 11.9 per cent. in the candle-power of the lighting units, there is an increase of 16,950 12,200 = 4750 or 39 per cent. in the total incident flux in the room, with an increase of 1350 400 = 950 or 237 per cent. in the ceiling illumination.

In referring above to the change in the system of lighting equipment use was made of the term "totally indirect," in order to concentrate ideas on the immediate problem at hand rather than to describe the system actually required to produce the results indicated. With only 6750 lumens incident upon the ceiling which absorbs 1350 lumens, and a total of 4200 lumens incident upon the walls which absorb 2100 lumens, it is evident that the lighting units must supply considerable flux directly to the walls, and hence a "totally indirect" system of lighting would not produce the results required.

As already stated, in actual practice conditions are not so readily defined as assumed above, and the absorption method cannot be applied practically with the degree of simplicity that might be inferred from the above examples, but it can be looked upon as a most reliable check upon the more complicated methods of calculation and as an invaluable aid in solving problems connected with the illuminating of reflecting surfaces, investigating quantitatively the effect of inter-reflection between surfaces, and ascertaining the limits in the distribution of light flux between illuminated surfaces.

UTILIZATION FACTOR

In actual practical problems in illumination design it has been found quite convenient to make use of the direct relations between the so-called "total lumens utilized" and the lumens produced by the lighting sources, because the former can be considered to be the known quantity and the latter the unknown quantity in one phase of the practical illumination problem. The "lumens utilized" are assumed to be equal to the mean illumination (in, say, foot-candles) over the reference plane (say 30 in. above the floor) multiplied by the area of the floor (in square feet). The ratio between this quantity of lumens to the lumens produced by the source is called the "utilization factor," or "coefficient of utilization."

Referring to the two examples given above it will be seen that (if the illumination on the reference plane be assumed to be equal to that at the floor level) the utilization factor in the so-called "direct lighting" problem would be 6000 7900 0.76, whereas in the ÷ "indirect" problem it would be 6000 ÷ 8850 = 0.68.

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A study of the above problems in the light of the above definition. will show that the "utilization factor" depends on not only the system of lighting and the absorption by the ceiling and walls but also on the absorption by the floor. The fact of the matter is that with highly reflecting floor, walls and ceiling the "utilization factor" would have a value greater than unity. This condition would seldom be reached in practice but would be closely approached in the case of a dining-room decorated in light colors, with a wide expanse of table linen and light floor covering. The value of the utilization factor depends upon the character of the lighting units, relative dimensions of the room, color and material of the ceiling, walls and floor. Utilization factors, as determined by actual tests under service conditions, will be discussed fully in other lectures, and need not be dwelt upon herein.

ILLUMINATION BY DAYLIGHT

Mention has already been made of the simple solution of problems that would otherwise prove quite complex by means of certain solid angular relations. This statement applies with particular force to problems relating to the illumination from either artificial or natural sky-light through either ceiling or side-wall windows.

In view of the fact that as a lighting source the sky is located at an indefinite, if not infinite, distance from the objects illuminated, it is obvious at once that resort cannot be had to the method of calculation based upon the so-called "inverse-square law." For purposes of calculation the sky can best be considered as an extended surface source of undefined shape at an indefinite distance from and completely surrounding the observer, being visible (except for local obstructions) throughout the upper hemisphere above the horizontal plane occupied by the observer. The first and most important step is to establish the relation between the illumination produced at any chosen point by such a source and the solid angle subtended by the source when viewed from that point; or rather first to show that the solid angular relations are in strict agreement with the "inverse-square law" and that by basing the calculations exclusively on the former the latter may be eliminated.

Referring to Fig. 13, consider the perfectly general case of a small section (dA) of a surface lighting source of any shape or inclination (a) situated at any distance (R) from any chosen point (P). Let c be the normal emitting density (here used as "apparent candle

power per unit area") of this source. The illumination produced at point P, from the inverse square and cosine laws, is,

Consider now the illumination that would be produced at the same point P, by a surface source (da)—at the circumference of the imaginary enclosing sphere-subtending the same solid angle as

Fig. 13.-Photometric relations based on equality of solid angles.

(dA) and having an equal normal emitting density c. The illumination at the central point, P, would be

From simple geometrical relations, the correctness of which will be appreciated at once from a glance at Fig. 13, it is seen that the areas (da) and (dA) bear to each other such a ratio that

Combining equations (3) and (2) and comparing the result with equation (1), there is obtained

Equation (4) shows that when dealing with surface lighting sources (such as the sky, artificial windows, or indirect lighting systems) the illumination at any chosen point is fully defined when the emitting density of the source and the solid angle subtended by the source as viewed from the point chosen are known. Upon this relation can be based some extremely simple graphical solutions of problems relating to illumination by daylight or by surface lighting sources in general.

From the relations derived above it will be seen that in calculating the illumination produced by a surface source it is unnecessary to know either the candle-power of the source or the distance of the source from the point of observation, provided only that the solid angle subtended by the source and the emitting density (expressed preferably in lumens per unit area) are known. It is obvious therefore that, so far as calculations are concerned, any surface source of indefinite shape, size or location (such as the exposed sky surface) can be treated as equivalent to a definitely located source of definite shape and size provided only that such values are assigned to the dimensions and position of the substituted surface source that the solid angles are the same as before and the assumed emitting density of the substituted source is identical with that of the original. Hence in day-lighting problems it may be assumed that a plane surface source of sky-value emitting density having the exact dimensions of the exposed area of either a ceiling or a side-wall window can safely be substituted for the sky.

From all points within a room receiving an unobstructed view of the sky through a window, the window itself can be treated as the surface lighting source having an emitting density in lumens per unit area exactly equal to that of the sky. At point where the sky is partly hid from view through the window, the solid angle is correspondingly reduced for the full sky density, and a lower density must be assigned to the remaining portion of the original solid angle in accordance with the relative reflection coefficients of the obstructing areas on the side exposed to view through the window.

CIRCULAR SKY-WINDOW SOURCES

The above described method of substituting a surface source of known dimensions and location for some other source of more complex dimensions and uncertain location is invaluable in determining the illumination produced by the light received through ceiling