As is well known, according to the so-called "inverse square law" the illumination (or luminous flux density) on a plane at any chosen distance from a "point-source" varies inversely with the distance from the source. If it were possible to obtain a true point-source, it would be possible to produce infinite illumination by bringing the plane within an infinitesimal distance from the source. With a spherical surface source the "inverse square" law holds true provided only that the distance from the source is measured from the center thereof. In this case the minimum distance from the source is equal to the radius of the sphere. With a spherical surface source 1 cm. in radius producing 100 c.p. uniformly in all directions the maximum illumination (at minimum distance) is equal to 100 ÷ r2 = 100 lumens per sq. cm. This means that the maximum possible illumination in lumens per sq. cm. is equal to the “brightness" of the source expressed in "lamberts." This relation holds true for surface sources of all kinds and shape, being absolutely fundamental. Any assumption that would lead to results contrary thereto can be said not to be in accord with the physical fact. In order always to have before one a correct mental picture of the true physical conditions of lighting sources, it is best always to assume that the so-called "point-source" is in reality a spherical surface source (having finite dimensions), and to base all calculations on the surface source rather than point-source conception. That is to say, it is not necessary to employ the "point-source conception" in order to take advantage of the "inverse square law" and similar relations developed and employed on the basis of the assumed "point-source," because the same relations are applicable even more accurately and completely to the spherical surface source. Moreover, there are certain relations between the output density of the surface sources and the illumination (flux density) produced on surfaces illuminated thereby, which can be utilized immediately when all calculations are based on the surface source conception but which must be ignored in effect when the point-source conception is used. This fact is becoming of increasing importance as the indirect or semi-indirect system of lighting is being substituted for the direct. FLUX-SUMMATION ON MEAN SPHERICAL CANDLE-POWER DIAGRAM Reference has already been made to the Rousseau diagram for representing by means of an area the total flux produced by a light source of which the candle-power distribution curve is known. As a matter of actual practice in illumination calculations use may be said always to be made for the purpose indicated of either the Rousseau diagram or some one of several modifications thereof that have been developed for eliminating the necessity of a planimeter for determining the area or its equivalent. Figs. 8 and 9 have been drawn to show one of the methods employed for representing the equivalent of an area by means of a straight line. The irregular curve Xbey of Fig. 8 is a candle-power Figs. 8 and 9.-Linear and area representations of zonal flux. distribution curve of which Fig. 9 is the corresponding Rousseau, or flux-summation, diagram. Consider the small area ABCTPS of Fig. 9. If such a section be so selected that its mean width is equal to PB then the small area ABCTPS is equal to the product of AC (the height) by PB (the width). The problem is to select some one line which, by geometrical construction, is proportional to the product of AC and PB. In Fig. 8 such a line is shown by A'C', which by construction, bears to AC (of Fig. 9) the direct ratio of Ob to OP (of Fig.8). That is to say, it is proportional directly to the area ABCTPS, the proportionality constant being dependent upon the linear candle-power scale and the diameter of V' the circle of reference, or rather the enclosing sphere. The summation of all the various part-areas of Fig. 9 as indicated by A'C', E'F', etc., of Fig. 8, would produce a single linear dimension directly proportional to the total area of Fig. 9; that is, directly proportional to the total flux from the source of which the irregular curve of Fig. 8 shows the space distribution of the candle-power. One can easily define the proportionality constant by applying the method here outlined to the determination of the total flux from the candle-power curve of a "spherical surface" source producing equal candle-power in all directions. It will be seen at once that the total of the vertical lengths (corresponding to A'C' and E'F', etc.) would then equal twice the length chosen to represent the uniform candle-power of the "spherical surface" source; now the total flux is equal to 4 I whereas the summatio length is 2 I and hence the proportionality constant is 2. That is to say, independent in every respect of the irregularities of the candle-power curve, the linear summation method outlined above gives at once a value equal (if sufficiently small sections are selected for summation) to twice the mean spherical candle-power of the source, measured on the candle-power scale, and this value multiplied by 2 equals (with the same degree of accuracy) the total flux from the source expressed in lumens. It will be noted that, contrary to the relations involved in the Rousseau diagram, the diameter of the circle (or sphere) of reference cancels out from the proportionality constant in the linear summation of Fig. 8, whereas it appears as a direct factor in the area summation of Fig. 9. LINEAR SUMMATION BY GRAPHICAL CONSTRUCTION In Fig. 11 is reproduced the candle-power curve of an infinitesimal cylindrical surface source of which the Rousseau flux diagram (elliptical) is shown in Fig. 12, identical except as to dimensions with the elliptical diagram in Fig. 7. In Fig. 10 is shown a graphical method for adding together the vertical linear equivalents of the separate 30 degree areas in the Rousseau diagram of Fig. 12, the equivalents in each case being determined by the geometrical method already outlined in connection with Fig. 8. It will be noted that the 30-degree, 60-degree and 90-degree angle lines have been so transposed, while retaining their equivalent lengths, that the corre sponding vertical distances are directly added one to the other to produce at once the total length of QQ', which (according to the proportionality constant derived above) is equivalent to twice the mean spherical candle-power represented by the candle-power distribution curve of Fig. 11 or the Rousseau diagram of Fig. 12. The linear summation diagram briefly outlined in connection with Fig. 10 was developed by Dr. A. E. Kennelly, past president of the Illuminating Engineering Society, and is known as the Kennelly Diagram. Figs. 10, 11, and 12.-Kennelly linear summation diagram; candle-power curve of cylindrical surface source; Rousseau area summation diagram. ABSORPTION-OF-LIGHT METHOD OF CALCULATION One of the most convenient and an absolutely reliable method of calculation in illumination problems is that based on the law of conservation. According to this law the total flux (lumens) of light absorbed by the illuminated surfaces within any chosen enclosure of any size, shape or character is exactly equal to the total amount of flux (lumens) produced by the sources of the lumination. This law is fundamental and calculations based upon it give absolutely accurate results when the assumptions as to absorption, etc., are correct. That is to say, by adding together the value of the lumens separately absorbed by the various surfaces illuminated one obtains at once an exact measure of the lumens produced by the sources of light. In order to determine the absorbed flux, it is necessary to know only the value of the incident flux and the absorption coefficient; the product of these two represents accurately the lumens absorbed. Any error found in applying this method is to be attributed to the inability to determine either the value of the incident flux, or the absorption coefficient, or both, but not to the method itself. For example, assume a room 25 ft. wide, 80 ft. long, 10 ft. high having a white ceiling with an absorption coefficient of 0.20; light walls with an absorption of 0.50; and a dark floor with an absorption of 0.90, to be so lighted that the incident illumination on the ceiling is I foot-candle, that on the walls 2-foot candles and on the floor 3 foot-candles. The following summation shows the amount of lumens absorbed: Total mean spherical candle-power equals 7900 ÷ 4′′ = 630. This method is not approximate; it is absolutely exact. However, it should not be assumed that results in practice can be obtained with such ready facility as here indicated, because the absorption coefficients of ceiling, wall and floor materials are not known to a high degree of accuracy; various surfaces in addition to those here considered intercept and absorb much of the light, and the light is not uniformly distributed over the various surfaces. In regard to the last mentioned limitation it is worthy of note that the mere lack of uniformity in the distribution of light flux does not affect the accuracy of the absorption method provided only that the true mean effective values of the incident illumination and of the absorption coefficient are assumed in each case. The actual distribution of the incident flux can be approximated by means of some of the point-by-point methods of calculations, while the absorption coefficient must be based on the results of tests relating to the materials composing the absorbing surfaces. Values for such coefficients will be given in connection with other lectures dealing with the practical application of the methods of calculation herein described. |