Performance CalculationsDirect Visual Signaling as a Means for Occupant Notification in Large Spaces 
13. ANNEX: Performance Based Calculations
13.1. GeneralNFPA 72, Section 7.5.4.3 permits strobe system design using performance based calculations in lieu of the prescriptive requirements. The code states that “Any design that provides a minimum of 0.4036 lumens/m2 (0.0375 lumens/ft2) of illumination at any point within the covered area as calculated for the maximum distance from the nearest visual notification appliance to any point within the covered area shall be permitted …”
Inverse square law calculations must be done for each of the vertical and horizontal polar distribution angles in ANSI/UL 1971, Standard for Safety Signaling Devices for Hearing Impaired, or equivalent. The calculations are based on the inverse square law equation:
where E is the illumination in lumens per square foot, I is the intensity in candela (candela effective, cd eff. for strobes) and d is the distance.
There are several different factors that result in designers using different approaches to solve the performance based calculations.
For ceilingmount applications, most designers assume that the strobe is on the ceiling (or a horizontal plane below the ceiling) in the center of a large, square room. Figure 18 is an elevation view showing the angles and distances involved in the calculations. Figure 19 is a plan view of the same single strobe coverage area.
One variable that is often chosen differently for the calculations is the distance, d. Looking at the plan view, some designers perform calculations only for vertical planes that are perpendicular to the surrounding walls (where they exist) or imaginary walls (for coverage of large spaces). Thus, looking at the elevation view, the distance d at an offaxis angle of 90 degrees would be one half of the room width, W. This is the most common method used in the industry for calculating strobe coverage in large spaces even when the actual strobe coverage area has no walls.
However, this ignores the fact that the chosen coverage area is a square with a diagonal that is greater than W/2. This is similar to the circle of coverage permitted by NFPA 72, 5.6.5.1.1(2) for initiating devices. Figure 20 shows a plan view of a large space broken down into smaller square coverage areas. The circles show that strobes in the center of the square coverage areas actually form a pattern of overlapping circular coverage areas. The worst case distance in plan view is not the distance measured perpendicular to a wall. It is the distance, r, to the far corner of the required coverage area. This is the second most common method used in the industry for calculating strobe coverage in large spaces even when the actual strobe coverage area has no walls. Figure 18  Strobe Calculations (Elevation View)
Figure 19  Strobe Calculations (Plan View)
Figure 20  Large Space Showing Square and Circular Areas of Coverage
Regardless of whether calculations are done for a plane that passes through a wall center or through a wall corner, a common point of confusion and error often noted in performance based calculations is the calculation of the distance to the floor or wall in a vertical section. Looking at the elevation view in Figure 18, the onaxis (A = 0^{0}) distance (d) is equal to the strobe mounting height H. As the offaxis angle (A) increases, the distance a light ray travels to reach a surface (first the floor) increases until it reaches the point where it starts to travel up the vertical surface. The angle, A’, at which the light ray transitions from the floor to the wall of an enclosed space varies depending on the height and width (or radius) of the coverage area. Note that equations are presented for both types of solution discussed above: 1) calculations for a plane through a wall center or 2) calculations for a plane through a wall corner.
For offaxis angles up to A = A’:
For offaxis angles greater than A’:
The calculation of the distance, d, for each inverse square law calculation is further complicated by the fact that the strobes are often being used in large open plan spaces without walls or in warehouses and “super stores” with racks and shelving. Figure 21 shows a typical warehouse store with strobe coverage providing both direct and indirect signaling to the occupants. Figure 22 is the same diagram highlighted to show the surfaces where one of the strobes provides illumination. These highlighted surfaces define the points at which NFPA 72 would require calculation of the illumination, E.
Figure 21  Direct and Indirect Strobe Coverage in Racks
Figure 22  Actual Strobe Penetration in Racks
It would be reasonable to not provide calculations for the top surfaces of the stock on the racks since that is not a surface that occupants see during the normal use of the space. In most cases calculations for a square space using the distance to the corner will be conservative. Nevertheless, performing calculations for the actual illuminated surfaces and coverage distances may be warranted in some instances and may result in in either fewer strobes or lower strobe intensities being required.
The actual illumination of a surface must be adjusted for the angle at which the light ray strikes the surface. Lambert's Cosine law states that the amount of reflected light is proportional to the cosine of the angle of incidence. Commercially available calculation programs do not include this correction and designers rarely make the correction.
Figure 23 shows that a light flux striking a surface at an angle (other than 0^{0}) illuminates an area larger than the plane normal to the ray. Since the quantity of light flux is distributed over a larger area, the resulting illumination is less. Using basic trigonometry it can be shown that the adjustment factor is the cosine of the angle of incidence. At 0^{0}, the correction is Cos (0) = 1.0. That is, no correction onaxis. At 90^{0} the correction is Cos (90) = 0. That is, no illumination for a ray that is parallel to the surface. Therefore, the correction factor will always vary from zero to one. This is sometimes referred to as the Cosine Cubed Law. This is because if all calculations for the distance (height, off axis angle and angle of incidence) are combined, the equation will include the cosine of the off axis angle cubed. Note that in Figure 18 as the light ray reaches the corner and turns to trace up the wall, the calculation of the angle of incidence (B) changes. Up to the corner, B = A. On a vertical surface, B = 90 – A.
Figure 23  Lambert Cosine Law
The final complicating factor affecting strobe visibility is that people do not see the illumination that a strobe light creates on a surface. It is the light that is reflected off of the surface in the direction of the eye that is perceived. The reflectance of the surface, the angle of incidence and the angle of to the observer’s eye are all factors affecting what is actually perceived. The National Fire Alarm Code does not require consideration of these factors.
Another error sometimes made in the calculations occurs when software spreadsheets are used. Generally, trigonometry functions in spreadsheets require angles to be input in radians not degrees. Failure to use the proper units results in errors.
For each of the three test sites, basic calculations were done using nominal ceiling heights and strobe spacings and the reported strobe intensities. Two simplified sets of calculations are presented. The first is for an imaginary square coverage area with the plane through the center of the wall. The second is for the vertical plane through the corner of the square. For each, the direct calculation result is presented along with the value corrected using Lambert’s Cosine Law. 13.2. Home Depot, Reading, MAThe following calculation summary is based on a strobe covering an imaginary square. The calculations do not model the actual illumination on stock in racks at distances closer to the strobe than the imaginary coverage square. The calculations assume a strobe intensity of 75 cd eff. even though one strobe was found to be set at 15 cd eff. It is not known whether other strobes may also have not been adjusted to the design requirement of 75 cd. eff. Also, even though the spreadsheet notes some locations as being below the required 0.0375 lm/ft^{2}, some of these values may be at heights where indirect signaling is not needed. For example, the most commonly required calculation method is for a plane through a wall center without correction for Lambert’s Cosine Law. The calculations show illuminations less than 0.0375 lm/ft^{2} at 85 and 90 degrees off axis. A ray traveling 45 ft at 85^{0} off axis would be at an elevation of four feet below the strobe (45 ft / TAN(85^{0})), which is 19 ft above the floor.
13.3. Home Depot, Danvers, MAThe following calculation summary is based on a strobe covering an imaginary square. The calculations do not model the actual illumination on stock in racks at distances closer to the strobe than the imaginary coverage square. Also, even though the spreadsheet notes some locations as being below the required 0.0375 lm/ft^{2}, some of these values may be at heights where indirect signaling is not needed.
13.4. Wal*Mart, Kissimmee, FLThe following calculation summary is based on a strobe covering an imaginary square. The calculations do not model the actual illumination on stock in racks at distances closer to the strobe than the imaginary coverage square. Also, even though the spreadsheet notes some locations as being below the required 0.0375 lm/ft^{2}, some of these values may be at heights where indirect signaling is not needed.

