The following methodology for calculation of daylight illumination in building interior is applicable to both sidelighting and toplighting using skylight. When dealing with reflected light in the interior, the underlying assumption is that the interior surfaces are diffusive.

Consider the situation of daylight reaching a point on a work plane in a building interior in Figure 3.4.1 . Three components of light fluxes contributes to illuminance at point P in the figure. These are the sky component (SC) , the externally reflected component (ERC) , and the internally reflected component (IRC) .

Figure 3.4.1 The three components of daylight illuminance.

Sky Component (SC)
This is the total light flux from the part of the sky which is seen by the point P   through the opening(s). Under a changing sky this component will vary directly with the variation in the luminance of the sky. Even under clear sky, the traverse of the sun with the passage of time will change the luminance pattern of the patch of the sky seen by the point through the opening(s). The sky component diminishes in the deeper interior of the building.

Externally Reflected Component (ERC)
This is the total light flux reflected from external surfaces reaching the point through the opening(s). The primary source of light on the external surfaces(s) can be sunlight, skylight or both. Usually skylight illuminance on a surface outdoor is much smaller than the illuminance from sunlight. An external surface is usually assumed diffusive. This is true for opaque surface, but not quite so for buildings facade comprising a large proportion of glazings. When the external surface is highly reflective, the problem is more complex, as the specular reflection is also dependent on the movement of the sun. The methodology in this section cannot deal with sunlight and its reflection on a specular surface. This component also diminishes in the deeper interior of the building.

Internally Reflected Component (IRC)
This is the total light flux reaching the point from the light flux reflected from all internal surfaces. In this section all internal surfaces are assumed diffusive, and the reflected fluxes are also diffuse.

The internally reflected component can be further identified to comprise two smaller components, IRC1 and IRC2. Figure 3.4.2 illustrates the two components of IRC.

Skylight which enters the window and falls on the floor and walls is reflected. This reflected skylight eventually contributes to illumination at point P. This component is labeled IRC1. Daylight reflected from the ground in front of a window can enter the window and be reflected from the ceiling and the walls. The ground-reflected component reaching point P   is label IRC2 in the figure.

The internally reflected component is dominant in the deep interior space of a building, as its magnitude in the space is relatively uniform and does not change much with distance from windows

Figure 3.4.2 The two components of IRC. .

The total daylight illuminance E_DL at point P comprises all the components as follows.

E_DL    =   E_SC   + E_ERC   + E_IRC1   + E_IRC2         (3.4.1)

Daylight Factor (DF)

In locations dominated by overcast or cloudy sky, the ratio

DF  =   E_DL / E_kh                                                       (3.4.2)

which is called the daylight factor is independent of the position of the sun in this sky. In such a situation, the daylight factor is a function of the geometrical configuration of the point in the room only. For such locations and sky type, the daylight factor is very useful. Once it is evaluated for any location in interior, it remains valid even sky luminance changes, as on gas the sky is relatively uniform or overcast. The daylight illuminance at the point can be found simply from (3.4.2) if the exterior sky illuminance is known.

The use of daylight factor is widespread in the United Kingdom and northern Europe where the sky is predominantly overcast.

However, under clear or partly cloudy sky, the ratio in (3.4.2) evaluated for a given time is no longer valid for another time.

Isoclines of Horizontal Illuminance
In the United Kingdom where the use of daylight factor is common, a graph of isoclines of horizontal sky illuminance plotted against the time of a day in different days of a year is used. Such a graph is shown in Figure 3.4.3

The graph in the figure is produced and published by the Building Research Establishment, a government agency, for use in the United Kingdom. Accompanying such a graph is a graph of the probability a given value of horizontal skylight illuminance will be exceeded, an example of which is shown in Figure 3.4.4.

In practice, the daylight factor in a space is first determined. The mean daylight illuminance can then be obtained from the daylight factor and the information obtainable from Figure 3.4.3. To determine the probability an illuminance level will likely be exceeded, the information in Figure 3.4.4 may be used.

Figure 3.4.3 Mean horizontal diffuse illuminance from an unobstructed sky showing

variation with time of day and month of year at Kew. (LAT = Local apparent time which approximates to GMT).

Figure 3.4.4 Sample graph: Percentage of year a given value of horizontal skylight illuminance is exceeded, with data from Kew and Bracknell, United Kingdom.

3.4.1 Calculation of the Sky Component
Figure 3.4.5 illustrates a typical situation in which skylight enters a window and reaches a point P   in the interior.

Figure 3.4.5 The configuration of light flux from a sky patch da passing through window and reaches P .

The point P   is at a height H_p  .  It is lower than the window sill by H_d  .  The distance from P   to the plane of the window is D .  Flux of skylight from an incremental area da reach P   through the window.  The altitude angle from P  to da is a  , the corresponding zenith angle and azimuth angles are f and q respectively.  The distance from da  along the plane of the window to the projection of P  on the window is w  .

The angle q, a and f are related to the geometrical position of the point P   as follows:

              q     =   tan^-1 (w / D ),                                               (3.4.3a)
          a     =   tan^-1 [(h+H_d  )cosq /D  ]                                     (3.4.3b)
          f      =   p /2 - a                                                                   (3.4.3c)

The luminance of the sky is L ( q,f,x ), where x is the angular distance between the incremental area da  and the sun.  From (3.3.24) the angle x can be obtained as

          x     =   cos^-1[sina_s sina + cosa_s cosa cos (q - g_s )],            (3.4.4)

where   a_s  is the altitude angle of the sun, and

               g_s  is the azimuth angle of the sun.

The two angles a_s and g_s are referenced to the rectangular coordinates with its center at P.  The horizontal illuminance at P  due to the incremental area da  with luminance distribution L(q, f, x ) is then given as

dE_sc =L (q,f,x )sinf cosf df dq                                  (3.4.5a)

       = L (q,f,x )sina cosa da dq                                  (3.4.5b)

The limits of integration for q  is from the left edge of the window to the right edge.  For a , the limit is from the lower edge of the window to the upper edge.  Based on the dimensions in the figure, these limits' can be written as

        tan^-1[(H_d cosq)/D] < a <  tan^-1[((H_w + H_d )cosq)/D]    

and       tan^-1(W_L /D) < q <  tan^-1(W_R /D)                      (3.4.6)

When the luminance distribution function L ( q,f,x ) is complex, (3.4.5) cannot be solved analytically.  Many graphical methods have been devised to solve the problem.

Use of BRE Table
The Building Research Establishment (BRE) publishes a table to assist in the evaluation of the sky component at point P . The Table is prepared for an overcast sky , based on the luminance distribution (3.3.18). The result is given in terms of the value of the daylight factor (DF) at the point P . Table 3.4.1 shows an abbreviated sample table.

Table 3.4.1 Sample abbreviated table of BRE table for obtaining a value of DF for the sky component.

The Waldram Diagram is a diagram prepared for evaluation of the extent of the sky seen through the window. Again the overcast sky is assumed. The diagram comprises a grid which represents 50% of the total sky hemisphere. Droop lines in the diagram represents the edges of rectangular window(s). An appropriate area identified in the diagram represents the effective portion (sky luminance distribution is already accounted for) of the skylight seen through the window at the given point. Figure 3.4.6 illustrates the features of the Waldram diagram.

Figure 3.4.6 Features of the Waldram diagram.

As the whole diagram represents 50% of the hemisphere, and the diagram comprises 50 enclosed blocks, each block effectively represents 1% of sky light seen at the point. Here, the azimuth and altitude angles from the point to the edges of the window are used to determine the size and location of the window projection on the diagram.

Obstruction from external object can also be projected on the diagram. The BRE also produces sets of protractors as aids to obtain the values of skylight in building interior.

Geometrical Approximation
A geometrical approximation can be made to the configuration of the window in Figure 3.4.7.  Figure 3.4.7 illustrates the substitution of a the rectangular plane of the window by a curved area representing the equivalent hemispherical surface as seen by a point directly in front of the window.

To obtain the value of illuminance due to skylight at P, equation (3.4.5) still applies, but the limits for the variables f (or a) is now independent of q.  If the Luminance distribution function L ( q,f,x ) comprises functions which allow independent (analytic) integration of each variable, then (3.4.5) can be analytically evaluated.  This condition is satisfied by the distribution function of uniform and overcast skies.  The limits of integration for both cases are given as

(3.4.7)

Figure 3.4.7 An approximation of the rectangular window by a hemispherical surface.

For uniform-sky luminance , the horizontal illuminance from the sky component is obtained as

                                                     (3.4.8)

For overcast-sky luminance, the horizontal illuminance is obtained as

(3.4.9)

These results are valid only for a point directly in front of the window. For the situation where the point is not directly in front of the window, such as that in Figure 3.4.8, the projected width of the window can be used in the expressions (3.4.9) and (3.4.9) instead of the full width. The projected width of the window in Figure 3.4.8 is given as
                                                        W_wp   =   W_w  cosq_w,               and

  q_w      =     tan^-1(W_d  /D )          (3.4.10)

The window width W_wp  in (3.4.10) should be used in place of W_w  in (3.4.8) and (3.4.9).

Figure 3.4.8  The situation when the window is located a distance W_d  from thev projection of the point on to the plane of the window.

Calculation of Illuminance on a Wall
Skylight which enters a window also falls on the interior wall surface directly. This forms a direct flux on the wall which eventually contributes to the illuminance on the work plane through internal reflection. Figure 3.4.9 illustrates the configuration.

Figure 3.4.9 The configuration of skylight illumination on a wall.

The illuminance on the an internal plane can be obtained as

                                      (3.4.11)

where   h is the angle between the normal of the plane at P  and the line da  from P.  In the case of a vertical plane the relationship of h to q and f is

                   cosh    =   sinq cosa

The illuminance on a vertical wall is given as

                               (3.4.12)

The limits of the integration are

          (3.4.13)

Geometrical approximation method can also be applied in this vertical wall case with identical expression for dE_kv  as in (3.3.8), but with the following limits of integration

      (3.4.14)

                      (3.4.15)

where   q_w    =   tan^-1(W_d  /D  ),

       W_d   =  distance from the projection of P  on the plane of the window to the middle of  the window.

To simplify the evaluation of E_kv  in (3.4.12) assume H_d  = 0 here.  Integrating (3.4.12) with the limits in (3.4.14) and (3.4.15) gives, for the uniform sky model

(3.4.16)

Calculation of Illuminance from an External Surface
The diffuse flux from reflected light on an external surface can reach a point P on a horizontal or a vertical plane through a window. If the illuminance on the surface is known, then the luminance of such surface is obtained as

L_ext  =   r_s E /p                                                                                    (3.4.17)

where   r_s    =  reflectance of the surface,
            E     =  illuminance on the surface.

The same equation (3.3.7) can be used to calculate the illuminance on a horizontal plane, or (3.3.12) or (3.3.8) can be used to calculate the illuminance for a titled plane or a vertical plane respectively. But the limits of integration now are related to the dimensions of the surface within the frame of the window(s).

Calculation of Illuminance due to Ground-Reflected Light
If the total sunlight and skylight illuminance on a ground is E_kH  , and the reflectance of the ground is r_s , then the luminance of the ground is

L_g     =   r_g  E_kH  / p                                                                               (3.4.18)

Generally, the window sill is at a higher level than ground, ground light cannot reach the work plane directly, but can reach the ceiling and the upper part of walls.  Equations (3.4.5)and (3.4.12) can be used for calculation of illuminance on the ceiling and walls.  The luminance from the ground is assumed uniform.

Calculation of Illuminance from Reflected Fluxes
Flux transfer method can be used to calculate the total illuminance on a surface in an enclosed space.  Depending on the level of accuracy required, surfaces in the space can be further broken into smaller segments to be able to distinguish the different illuminance levels between adjacent segments.  But the calculation requires the value of the view factor between each surface.  If there are n segments of surface in an enclosure, up to n ² view factors are required to be evaluated.  In practice, inter reflection will render relatively uniform illuminance in a space, thus it is not necessary to strive to improve accuracy in calculation of inter-reflection fluxes by breaking surfaces into small segments.

Example 3.4.11 Illuminance from Uniform and Clear Sky

Consider a square room of 3m height in Figure 3.4.10 .

Figure 3.4.10 The room configuration and details

A window of height 1.75 m., width 2 m. is situated in the middle of a wall.  All other walls are opaque.

This example will illustrate the procedure and the results of calculation of daylight illuminance in the interior of this room for the cases of uniform and clear sky luminance models, all for the solar altitude angle of 60^o, and transmittance of window of 1.0

Sky Component
In order to see the difference of the illuminance level on the different points on the work plane the work plane is divided into 15 equal segments as shown in Figure 3.4.11. Each segment measures 1.2 x 2 m.

Figure 3.4.11 Points of calculation for direct illuminance.

Illuminance due to skylight through the window will be calculated for each point at the center of each segment on the work plane. The point P_w1  to P_w4  are on the walls, and are at 20% and 70% of distance from the window. These points are at the same level as that of the work plane.  The illuminance due to skylight calculated at P_w1  is taken to represent the illuminance of the half of the left wall near the window, and similarly for other points.

Ground-reflected daylight can reach the upper parts of the walls and the ceiling.  Here it is assumed that it only reaches the ceiling (to simplify calculation) and the illuminance due to ground-reflected daylight at P_c1  at the middle of the ceiling plane represents the illuminance of the whole ceiling.

Internally Reflected Component
The ceiling, wall and work plane are taken to be the surfaces in the room in inter-reflected flux calculation. The average illuminance on the work plane is obtained from summing all the fluxes from skylight and divide the result by the area of the work plane. This is then taken as the direct illuminance for the workplane. Similar procedure is applicable to the wall and ceiling surfaces.

From the given dimensions the view factors between the surfaces are calculated. These, together with the given values of surface reflectances and calculated values of direct illuminance, the total illuminance on each surface is calculated as a solution of the set of flux-balance equations. Subtracting the average direct illuminance from the total illuminance for the work plane, wall and ceiling gives the average reflected illuminance on each surface .

Resultant Illuminance
When the average reflected illuminance for the work plane is added to the sky component calculated for each of the 15 segments the resultant illuminance is the total illuminance in each segment. Similar procedure applies to other surfaces. In this way, the illuminance in each segment is distinguishable from the other. The smaller the segment, the more details are obtained. Even for the diffusive skylight, those segmented areas which do not receive direct skylight will be clearly distinguishable from those which do, and shadow can be discerned.

Calculation of the Sky Component on the Work Plane
This will be illustrated for a point in the middle of a segment on the work plane using 2 methods, numerical integration and geometrical approximation.

First consider the method of numerical integration

Take a point on the left most segment at 30% of distance from the window as an example. Figure 3.4.12 illustrates the configuration. Here the window sill is on the work plane.

Figure 3.4.12 The configuration of the point P and the window.

Here, the width of the window is divided into 5 increments, each at 0.4 m, the height is also divided into 5 increments, each at 0.35 m.  At the given point on the plane of the window the value of w  is 1.8 m., and the value of h  is 1.4 m.  The altitude angle a  and the azimuth angle q of the point with respect to the coordinate at P   are evaluated respectively as

                   q     =   tan^-1(w /D  )  =  p/4  =  45º

                   a    =  tan^-1(h cosq /D  )  =  28.8º.

If the coordinate at P  is taken as the reference, the luminance of the equivalent incremental area da  in the sky dome is given as

  L (q,a,a_s ,g_s )  =   L (45^o,28.8^o,a_s ,g_s )                  (3.4.19)

The value of the luminance depends on the position (a_s , g_s ) of the sun and the sky type.  In this example the solar altitude angle is assumed at 60^o.  The example considers uniform sky and clear sky.  The uniform sky is considered to fall within the class of overcast sky and the horizontal illuminance from skylight is taken to be identical to that of overcast sky.  Table 3.4.2 shows values of the luminance under various conditions.

Table 3.4.2    Luminance at da , a_s = 60^o.

The value of L  (q_i  , a_i  , a_s  , g_s ) sina_i  cosa_i   is the integrand of the integral for the illuminance.  Here, trapezoidal integration can be applied to improve accuracy of the numerical integration.  For the point P  in this example, the value of the illuminance E_khp  evaluated according to trapezoidal integration is given in the table.

Using geometrical approximation , the width of the window as seen from the point P is

Projected width W_wp  =  W_w  cos [tan^-1(W_d  /D  )]

where   W_d  =   2 m.,

            W_w  =  2 m

The projected width is then evaluated as 1.338 m. Figure 3.4.13 illustrates the approximate configuration.

Figure 3.4.13 The configuration of the window.

The size of the window relative to the distance from point P   here is relatively large. For uniform sky, the luminance of the sky is uniform and (3.4.8) applied with the modified window width will give a reasonable result. But for clear sky, it may not be appropriate to use this simplified method. The illuminance at P   for uniform sky using this method is obtained as 0.426 klux .

Direct Illuminance on All Surface
Following the methods described the values of direct illuminance on the point indicated in Figure 3.4.11 for uniform sky calculated based on trapezoidal integration (with 15 incremental areas) and based on the geometrical approximation are shown in Figure 3.4.14. Those for clear sky are also shown. The effect of the azimuthal angle of the sun relative to the orientation of the window is clearly seen from the numerical results. Those areas in the room which is illuminated by the sky patches near the sun are clearly at higher level of illuminance.

a)       Uniform sky-trapezoidal integration.

b)  Uniform sky-geometrical approximation.

c) Clear sky-trapezoidal integration,
a_s = 60º, g_s = 30º

d)      Clear Sky-trapezoidal integration
a_s = 60º, g_s = 180º

Figure 3.4.14 Calculated direct illuminance on various points of the surfaces in the room 1 using different methods, different sky types and different positions of the sun. All values are in klux.

Calculation of Reflected Illuminance

The flux transfer method will be employed. If very detailed calculation is desired, the segmentation used in the calculation of direct illuminance can be employed. But, as has been remarked, the inter-reflection will tend to even out the reflected component and very detailed segmentation is not necessary.

In case c) of Figure 3.4.14 the average illuminance on the work plane, the (combined) wall and the ceiling are obtained respectively as

work plane   0.7815,
wall                  0.2404,
ceiling             0.1589.

These are taken as direct illuminance in the flux transfer calculation.

From the given reflectance values of the surfaces and using the values of view factors obtained in section 3.2 for the room at the same configuration, the total illuminance on each surface is calculated by the flux-balance method as

work plane   0.9102,
wall                 0.3828,
ceiling            0.3213.

Subtracting these values by the corresponding direct illuminance values gives the reflected illuminance values which are

work plane   0.188,
wall                 0.142,
ceiling            0.162.

Adding these reflected illuminance values to the corresponding direct illuminance on each segment gives the total illuminance in that segment. Figure 3.4.15 illustrated the results for case c) of Figure 3.4.14.

Figure 3.4.15 Total illuminance on each surface segment
for case c) of Figure 10.4.29, clear sky.
a_s = 60º, g_s = 30º