Form Insolation Index
Notes on the Relation of Geometric Shape & Solar Irradiation
THANOS N. STASINOPOULOS
National Technical University of Athens, Department of Architecture
Patission 42, GR-106 82 Athens, Greece
This is a paper presented in the recent
PLEA [Passive & Low Energy Architecture] conference in Lisbon, 1-3 June 1998.
The graphs of this page can be seen in larger size using the 'View Image' option of your web browser
ABSTRACT
This paper refers to the effects of geometric shape on the amount of solar irradiation on a solid surface; in order to  explore that relationship, 7 generic convex solids have been studied in 71 variations of proportions & orientation, each one for 3 values of ground albedo (0, 0.2, 1) and in 3 locations (London, Athens, Riyadh). The energy received by the entire exposed surface was estimated as the sum of the irradiation on its facets, acting like flat collectors. 
     The calculation of incident energy was based on the algorithm introduced by Prof. J. K. Page et al [1] for the hourly computation of direct, diffuse & ground reflected irradiance on inclined planes, according to local atmospheric conditions. In each one of the 639 combinations of form, albedo & location, the mean monthly irradiation was calculated for all the radiation components, leading to average irradiance values (e) on the exposed surface. 
     The e values were correlated to the solar energy available on the horizontal plane (e0), in order to compare the performance of all forms as solar receivers. The ratio e/e0 indicates the relative insolation on a given form, and it is called here Form Insolation Index (m); it refers to the capacity of the form to receive more or less solar energy than others under the same conditions due to its geometric characteristics. 
A further analysis revealed a linear relationship of the values of m to the Base-to-Exposed-Surface ratio (B/F), depending on time & location but not on geometric parameters other than that ratio. Based on that, several conclusions were made on the relation between form and insolation, as well as a simple method of estimating solar irradiation on any convex surface using only a factor of the function m=f(B/F). 
 

The generic shapes of the study
SOLAR RADIATION ON SOLID SURFACES
A solid surface of area F exposed to the solar rays can be analysed into a set of planar facets. The three components of global irradiation on each facet (direct, diffuse, ground reflected) depend upon its specific orientation & tilt and also upon parameters which are common for all facets (time, latitude, atmospheric conditions & ground reflectivity). 
     The solar energy R incident upon the entire surface is the sum of the irradiation on its segments. The ratio R/F is the mean irradiance e on the surface and it is independent of the form’s size, as proven in [3]. 
     The value of e indicates the potential of a form to receive more or less radiation than others under the same conditions. In the present work, the relationship between geometric shape and solar irradiation is investigated by comparing the mean irradiance e on several cases of geometric forms corresponding mainly to simple building shapes. Conclusions can equally apply to other man-made structures as well as natural forms, like typical vegetation shapes or topography formations.
FORM INSOLATION INDEX
A comparison between two forms based only on their mean irradiance e is not adequate for general conclusions independent of time & location, since the value of e varies with those two factors. In order to apply a universal correlation basis, it is appropriate to link the mean irradiance e on a particular form to the available solar energy e0, which is usually indicated by the horizontal irradiance. 
     This study introduces the notion of relative mean irradiance or ‘form insolation index’ (m), defined as the ratio e/e0 of the mean solar irradiance received by a solid surface (e) to that received by a horizontal plane (e0) under the same conditions. 
     The m index represents the change of the incident energy if a horizontal flat surface is converted to a solid one of the same exposed area; obviously as a form is transformed into flat & horizontal, its m index approaches 1 (100%). 
     Considering a solid surface as a ‘mechanism’ which receives a fraction of  the generally available solar energy, the m index refers to the ‘efficiency’ (i.e. the ratio of actually received to available energy) of the mechanism in that function, as a direct result of its geometric properties.
ANALYSIS PROCEDURE
The main task of this research was to compare various geometric forms by their m indexes, exploring the relationship between geometric properties and insolation levels. This was done in four steps: 
  • Analysis of the surface in n facets of area fn and calculation of the irradiance in on each one.
  • Calculation of the total energy R on the entire surface by adding the irradiation on each facet multiplied by its area:
      •   
      R = S(in.fn)
      (1)
  • Calculation of the mean irradiance e on the surface by dividing the total energy R to its total area F:

    •   
    e = R / F
    (2)
  •  Calculation of m index by dividing the mean irradiance e to the corresponding horizontal one e0.

    •   
    m = e / e0
     (3)
The irradiance calculation on the planar facets was performed through spreadsheets based on the algorithm introduced by Professor J. K. Page et al [1]. The algorithm allows for hourly computation of direct, diffuse & ground reflected radiation on a plane of any orientation & slope based on the following data: 
  • latitude & altitude of the location;
  • monthly average sunshine duration;
  • monthly Linke atmospheric turbidity factors;
  • annual a+b Angstrom coefficients.

    Figure 1: The 71 form variations of the study
Climatic data from [1], [2] & [4] was used to compute hourly horizontal irradiance (direct & diffuse) for each month. The output was normalised using global radiation data available from [2] & [4]. Hourly irradiance -including ground reflected- on the facets of each form was calculated next, based on their orientation & slope. 
     The algorithm was repeatedly applied for 7 generic types of convex solids in 71 variations of proportions & orientation (Figure 1). Since this study refers to received radiation only, not absorpted or transmitted, all forms were considered as opaque and of zero reflectivity. 
     The procedure was repeated for 3 albedo values (0, 0.2 & 1) and for 3 locations (London, Athens & Riyadh) at steps of 15° latitude, giving a total of 71 x 3 x 3 = 639 cases. The irradiation output in each case was subsequently used to compute the monthly & annual m index for all types of radiation, global, direct, diffuse & ground reflected. 
OUTPUT SURVEY
m indexes
The extensive numerical output of the above procedure was used to search for potential links between the m index and geometric properties of the forms. Figure 2 shows an m index synopsis of all the forms under study as they change during the year. It is clear that m values vary widely from shape to shape, especially during summer. Monthly fluctuations of the global radiation indexes are generally small for low forms (i.e. those with high B/F ratio), but wider for tall ones. This is due to the variable seasonal contribution of the direct component, as the diffuse one remains rather constant in all cases. 
 
LONDON
ATHENS
RIYAD
Global radiation (albedo=0.2)
 
 
Direct radiation
 
 
Diffuse radiation
 
 
Figure 2: Monthly variations of m indexes


In order to correlate m indexes between locations, their values in Athens & Riyadh were compared to those in London. This is illustrated in Figure 3, where it is evident that all forms perform better as solar receivers in higher rather than lower latitudes, contrasting the opposite trend of the available solar energy.
  

ATHENS RIYADH
Figure 3: Monthly m indexes in Athens & Riyadh as % of those in London (global radiation, albedo=0.2)

m index & B/F ratio
The key finding of the study has been a clear linear relation between the m index and the ‘Base-to-Exposed-Surface’ ratio B/F of  all forms. Figure 4 depicts that relation for the direct & diffuse components in the three locations of the study during the year. Figure 5 shows the effects of ground albedo values on the annual m indexes for global radiation, while Figure 6 illustrates those indexes in December, March & June for albedo=0.2. 
     The linear relation applies in all locations, months and radiation types, notably in the case of the diffuse component. Noticable deviations from linearity occur mainly during winter (Figure 6), particularly in London, due to wider variations of the direct component in tall forms (i.e. those with lower B/F ratio).

  

LONDON ATHENS RIYADH
Direct radiation
Diffuse radiation
Figure 4: Mean annual m indexes compared to B/F ratio (direct & diffuse radiation)

LONDON ATHENS RIYADH
albedo=0
albedo=0.2
albedo=1
Figure 5: Mean annual m indexes compared to B/F ratio (global radiation, albedo=0, 0.2 & 1)

LONDON ATHENS RIYADH
December
March
June
Figure 6: Monthly m indexes compared to B/F ratio (global radiation, albedo=0.2)

u factors
In all cases, the relationship between m index &B/F ratio can be expressed by the linear function 

      m = u . (B/F) + v
      (4)
 For B/F=1 (flat shape) it is m=100%, hence u+v = 1; therefore 
      m = u . (B/F - 1) + 1
      (5)
 and 
      u = (m - 1) / (B/F - 1)
      (6)
The monthly u coefficients for each radiation component, as calculated using eqn. (6), vary slightly from shape to shape. Their mean value has been averaged here over all the 71 cases in each location. Table 1 exemplifies average u coefficients for global radiation & albedo=0.2. 


Table 1: Average u coefficients (global radiation, albedo=0.2)

Month
LONDON
ATHENS
RIYADH
1
0.319
0.382
0.412
2
0.356
0.466
0.416
3
0.381
0.512
0.534
4
0.501
0.584
0.640
5
0.519
0.624
0.637
6
0.549
0.624
0.653
7
0.539
0.620
0.654
8
0.538
0.610
0.660
9
0.459
0.560
0.636
10
0.385
0.459
0.493
11
0.223
0.403
0.374
12
0.158
0.307
0.391
Annual average
0.482
0.557
0.559
 
Irradiation calculation using u factors
From eqns. (3) & (5) we can approximate mean irradiance e on a convex form of ‘Base-to-Exposed Surface’ ratio B/F as 
      e = [ u . (B/F - 1) + 1 ] . e0
      (7)
Given that by definition  e = R/F, we find that 
       
      R = [ u . (B/F - 1) + 1 ] . e0 . F
      (8)
Eqn. (8) can be applied to estimate the total irradiation R on any convex solid surface of exposed surface area F and base B, using only horizontal irradiance e0 data and the correspondind u factors. 
     Detailed data on u coefficients for London, Athens & Riyadh, along with relevant applications, can be found in [3]. Since the u coefficients are site specific, the use of eqn. (8) for locations other than the above requires the repetition of  the described procedure, entering the appropriate climatic data into the Excel spreadsheets formulated for this work. 

m indexes & temperature compatibility
The actual m index of a form may differ from the typical value that matches its B/F ratio. Based on the difference ±Dm between the actual & typical values, a form is ‘warmer’ or ‘colder’ than others of the same B/F ratio, i.e. it receives more or less radiation than average. That feature can be positive or negative, according to its timing with seasonal temperatures. 
     Under that prism, the forms of the study, considered as building shapes, have been sorted by their ‘climatic propriety index’ mC, which associates their ‘warm’ or ‘cold’ insolation character to the span between the mean outdoor temperature TO and the limits of the thermal comfort range during each month: 

      mC = Dm . (TO - TC )
      (9)
where TC is the comfort temperature limit closer to TO (if TO is within comfort limits then TC=TO). 
     The annual mC index is the average of the monthly values. Figure 7 shows the forms with the 5 highest & lowest annual mC indexes at each location, assuming a comfort range between 18-25°C (global radiation, albedo=0.2). Due to the diversity of heat transfer and thermal comfort issues, mC index can be viewed as just a rule of thumb on the climatic compatibility of a form, with reference to insolation only. 

  LONDON
Highest
Lowest
 

ATHENS
Highest
Lowest
 

RIYADH
Highest
Lowest
Figure 7: Forms with highest & lowest ‘climatic propriety index’
 
CONCLUSIONS
  • This study has introduced ‘Form Insolation Index’ m as an indicator of the performance of a form as ‘solar receiver’. 
  • It has also displayed that the solar irradiance on a solid surface is a linear function of its ‘Base-to-Exposed Surface’ ratio, with low forms receiving more energy than tall ones. 
  • Using a set of u coefficients, it is easy to estimate global, direct, diffuse, or ground reflected irradiation on any convex form, from only the horizontal irradiance.
REFERENCES
[1] J. K. Page (edit.), Prediction Of Solar Radiation On Inclined Surfaces vol. 3 (Reidel, for the Commission of the European Communities, 1986). 

[2] W. Palz (edit.), European Solar Radiation Atlas  vol. I (Verlang TUV Rheinland, for the Commission of the European Communi-ties, 1984). 

[3] T. N. Stasinopoulos, Geometrikes Morfes & Iliasmos (‘Geometric Forms & Insolation’) (Ph.D. dissertation submitted to the National Technical University of Athens Department of Architecture, in Greek, 1999). 

[4] A. Bowen, Riyadh data from a 1983-86 research programme for the Saudi Arabian National Centre for Science & Technology (private correspondence, 1986).
<c> Thanos N. Stasinopoulos
e-mail
home page 
 Page edited 6.6.98
Counter added 03.03.00