# Calculation of daylight coefficients

The daylight coefficient D as the relation of interior illuminance to exterior illuminance based on Equation (8.1) is defined as standard for two measuring points at a height of 0.85 m, and 1 m away from the side walls. The daylight coefficient consists of a skylight proportion Dsky, a proportion of diffuse reflection of shading obstructions Dsh and an interior reflection proportion Dr.

D = + Dh + Dr (8.29)

Since about 1920, graphic methods have been used to determine the proportion of the sky Dsky seen from the work surface and of the shaded proportion Dsh, which only reflects light. The so-called Waldram diagram contains the projection of the sky dome onto a horizontal surface and takes into account the luminance increase to the zenith. Shading buildings obstructing the horizon are included with their solid angle and reflection coefficient. The more complex inter-reflections in the interior (interior reflection proportion Dr) were only later included in the calculation of the daylight coefficient and are today usually calculated in a simplified way by the so-called “split flux method”, in which only sky light reflections from the interior floor and the lower wall sections plus reflection of ground reflected light from the ceiling and the upper wall sections are regarded separately.

The daylight coefficient is determined first in dependence on the room geometry from the raw dimensions of the window openings (index r) and afterwards multiplied by light – reducing factors (window transmittance T, a framework proportion factor kj, a dirt factor k2 and a correction factor for non-vertical incident k3).

D = ( + DhKr + Drr ) k3 (8.30)

For the calculation of the sky light proportion seen from the point of reference P, first the effective elevation angle aw of the window’s upper edge and the lateral delimitations of the window by a left and right azimuth angle ywl and ywr have to be determined.

To determine the effective elevation angle aw, first determine the maximum elevation angle awmax from window height hw (upper edge) and the shortest distance between the window and point of observation, and then calculate the lateral reduction of the elevation angle with the azimuth y.  aw = arctan(tanawmax cos/)

To take obstructions into account, the obstruction elevation angles ash(y) must be known as a function of the azimuth. The illuminance on a horizontal surface in the unshaded exterior is Eho = LZn7/9, based on Equation (8.27). The illuminance on a horizontal

surface in the interior can likewise be calculated, taking into account the reduced height and azimuth angles of the window:

 ysm2 (9,) 9, ,2 лп/2 + 1 dy i—nl! ( 2 A —2 cos3 (9Z) 9Zj2 A 2 J 9z, i 3 J 9, і 2,1 )

The zenith angle is replaced by the window height angle aw = P2 — 9z, with

cos(9, ) = sin(aw), sin(9, ) = cos(aw) and sin2(9, ) = cos2(aw ) = 1 — sin2(aw).

The delimitation angles for the luminance integral are the left and right azimuth angle of the window opening ywi and ywr, the lower elevation angle is the obstruction elevation angle ash (corresponding to the larger zenith angle 9z 2) and the upper elevation angle is the window height angle aw (corresponding to 9z1).      Thus the daylight coefficient results in:

d, = E    sky’r e = 7П J V2 (sin3 ( (y)- sin3 ash (Y))+) (sin2 a, (y)- sin2 ash (y)   The externally reflected proportion Dsh r results as a function of the obstruction angles and the reflection coefficient _sh of the obstruction (typically 20%), by integration from the elevation angle a = 0 up to the obstruction elevation angle ash as well as over the azimuth angles of the obstruction yshj and ysh, r- In the German standard DIN 5034 the external reflection proportion is reduced at a flat rate by a factor of 0.75.

The interior reflection proportion calculated using the split flux method is calculated depending on the surface-weighted reflection coefficient of the floor and of the wall lower part pfw (without window walls, wall lower part to height of window centre) as well as on the corresponding surface-weighted reflection coefficient of the ceiling and wall upper sections pcw (likewise without window walls). In contrast, the average reflection coefficient of the room p includes all walls.

Dr. r =JW W —__2(p_fw + f0w_cw ) (8.36)

Aroom 1 _

Aroom: total room confinement surface [m2] bw, hw: Window width and height [m]

The upper window factor fup describes the integrated luminance of the Moon and Spencer sky model on the vertical surface, depending on an average obstruction angle a
(obstruction elevation angle in arc measure, measured from the window centre). The lower window factor fiow takes into account the diffuse radiation reflected by the floor. fup (a) = 0.3188-0.1822sina + 0.0773cos(2a)

fkw (a) = 0.03286cosa’-0.03638a’ + 0.01819sin(2a’) + 0.06714

with a = arctan (2 tan a).

Example 8.6

Calculation of the daylight coefficient of a side-illuminated room without obstructions, with window transmittance T = 0.65, a glazing proportion of 80%, a dirt factor k2 = 0.9 (low contamination) and a factor k3 = 0.85 to take account of the non-vertical incidence angle of the irradiance.

Room geometry:

Width B: 4 m

Depth T: 6 m

Height H: 3 m

Height of window upper edge hw: 2.5 m

Height of window bottom edge hwb: 0.85 m

Width of window bw: 4 m

 Reflection coefficients of the surfaces: pfloor: 0.3 pceiling: 0.7 pwall- 0.5 pwindows: 0.15 From this results a surface-weighted reflection degree
 PJloorAfoor + PceilingAceUing + P wallAwall + P window^rnndow + P wallAw

P = 0.3 x 24m2 + 0.7 x 24m2 + 0.5 x 48m2 + 0.15 x 6.6m2 + 0.5 x 5.4m2
108m2

Without obstructions the window factors are fup = 0.3961 and fow = 0.1. The interior reflection proportion thus becomes Drr = 0.008, less than 1%! 0 1 2 3 4 5 6

room depth [m]

Figure 8.19: Daylight coefficient of the side-illuminated room.

If the reflection coefficient of the walls is increased to 0.7, the interior reflection proportion rises to 1.28%. Raising the window’s upper edge to room height (3 m) increases the interior reflection proportion further to 1.7%.

If the reduction in the daylight coefficient by transmittance, framework proportion etc. is taken into account (factor 0.4!), at a depth of 4 m a daylight coefficient of about 1% is obtained.

 =I/ (прАсеІІ)

v = V/ns

 п/2 П/2 2 п/2 П/2 ^2

= Lz (І I sin 6zd6z cosydy + l I 2cos0z sin 0zd0z cosydy)

Updated: August 21, 2015 — 9:36 am