This study refers to the quantitative effects induced by the geometric shape of a body onto the solar radiation received by its surface.
A detailed analysis of solar irradiation on various typical shapes, in connection with several other parameters (proportions, orientation, ground reflectivity & geographic location), has revealed a linear relationship between certain geometric properties of each form and its 'solar efficiency'.
Following that discovery, a simple algorithm has been devised for the estimation of solar irradiation on any convex form, and an appraisal of various forms has been made relating insolation and the seasonal temperature fluctuations at the locations under study (London, Athens, Riyadh).
The study has been based on a computerized version of a special algorithm for the computation of hourly solar irradiation on tilted planes by Page et al.
The thesis was submitted to the Department of Architecture of the National Technical University of Athens in March 1999 and was subsequently approved on 14.6.99. Members of the examining panel were N. Kalogeras, G. Kontoroupis, J. Polyzos, S. Yannas, S. Xenopoulos, E. Evangelinos & M. Santamouris.

The 7 generic shapes of the study

A SUMMARY

Research topic

The amount of solar energy R (global, direct, diffuse, ground reflected) incident upon a solid surface of area F exposed to the solar rays is the sum of the irradiation on the surface segments. It depends on various factors like the size, orientation & slope of each segment, and also on parameters common to the entire surface such as time, latitude, atmospheric conditions & ground albedo (Chapters 2 & 3). 

The ratio e = R/F is the mean solar irradiance on the surface and is independent of the surface size. That ratio can be used to indicate the potential of a form to receive more or less solar radiation than others at the same time & place.

The present analysis of the relationship between geometric shape & solar irradiation is based on the mean solar irradiance e on several shapes that correspond to simple building types (Chapters 7 & 10). Conclusions can equally apply to other artificial & natural forms, like tanks, typical vegetation shapes, topography formations, etc.

The mean solar irradiance e varies over time & place, therefore is not a steady base for assessing forms in terms of insolation. For that purpose it is more appropriate to correlate the mean irradiance e on a form with the solar energy e₀ that is available at the given period & location, typically indicated by the horizontal irradiance (Chapter 8).

The present study introduces the notion of relative irradiance or 'insolation index' m, defined as the ratio e/e₀ of the mean solar irradiance on a surface e to that on the horizontal plane e₀: m = e / e₀.

Considering a solid surface as a solar receiver, the m-index can be taken as an 'efficiency coefficient' in that function, i.e. the ratio of received to available energy. The m-index also indicates the change of incident energy if a horizontal flat surface is converted into a another having the same area. Obviously as a form is transformed into flat & horizontal, its m-index tends to 1 (100%). 

The relationship between geometric properties & insolation levels is explored here by comparing the m-indices of various geometric forms. In each case the corresponding m-index is computed in 4 steps:

• The exposed surface F is partitioned into n facets of area f_n ; solar irradiance i_n on each facet is computed according to its orientation & tilt.
• The total energy R on the entire surface is calculated as the sum of the irradiance on each facet multiplied by its area: R = Σ(i_n^.f_n).
• The mean solar irradiance e on the surface is calculated by dividing the energy sum R to the total area F: e = R/F.
• The m-index is established by dividing the mean solar irradiance e of the surface to the corresponding horizontal one e₀: m = e / e₀.

The irradiance calculation & all the subsequent computations were performed on computer spreadsheets based on the algorithm developed by Professor J. K. Page and other European scientists (Chapter 4 & Appendix A). The algorithm is applied for the calculation of direct, diffuse & ground reflected hourly irradiance on a plane at any orientation & slope, according to the following parameters:

• latitude & elevation of the location
• monthly average sunshine duration
• monthly Linke atmospheric turbidity factors
• annual a+b Angstrom coefficients.

Climatic data from various sources were employed to compute the hourly horizontal irradiance (direct, diffuse & global) for the typical day of each month, assuming unobstructed solar access. All the resultant hourly values h were normalized to the final h', to match the ratio H'/H between measured (H') & calculated (H) global irradiance daily sums, according to the relationship: h'=h^.(H'/H).

Hourly & daily irradiation values (global, direct, diffuse & ground reflected) on each facet of every form was calculated next, according to the normalized horizontal data. This was repeated for 7 generic types of convex solids in 71 variations of proportions & orientation. Since this study refers to incident energy only but not absorbed or transmitted, all forms were considered as opaque and of zero reflectivity.

The procedure was applied for 3 albedo values (0, 0.2 & 1) in 3 locations (London, Athens & Riyadh, located at steps of 15° latitude), giving a total of 71x3x3 = 639 cases. The irradiation output in each case was subsequently used to compute the monthly & annual m-index for all 4 types of radiation.

For each location, ground albedo & radiation type, the monthly output includes:

• Hourly & daily solar irradiance on a horizontal plane (Chapter 11 & Appendix B).
• Daily solar irradiance on the facets of a polyhedral dome (called here 'hemispherical irradiation distribution') (Chapter 12 & Appendix C).
• Total daily solar irradiation on 71 forms (Appendix E).
• m-index of each form (Chapter 13 & Appendix D).

Further processing of the above data has led to various conclusions, like:

• Insolation indices m: The m-index varies from shape to shape, especially during summer. Monthly fluctuations of the global radiation indices are generally narrow for low forms & wider for tall ones. This is due to the higher seasonal variations of the direct component on tall forms, while the diffuse one remains rather constant in all cases (Chapter 13).

• Irradiation distribution: The distribution of the total solar irradiation on the individual facets of a form depends not only on their size & orientation, but also on the time of the year, thus altering the most efficient solar collection section as well as the need for solar control (Chapter 14). 

• Orientation effects: Changes of orientation affect the monthly m-index of a form, but the annual values remain practically constant (Chapter 14).

• m-index & B/F ratio: The m-index & the 'Base-To-Exposed-Surface' ratio B/F of a form feature a direct linear relation. That applies generally to all forms, locations, months & radiation types, notably in the diffuse component. Some deviations from linearity occur mainly during winter -particularly in London- due to wider variations of the direct component in tall forms, i.e. those with lower B/F ratio (Chapter 15).

• u coefficients: The linear relationship between the m-index & B/F ratio can be expressed by a simple function like m= u^.(B/F)+v. For B/F=1 (i.e. completely flat shape) it is m=100%, so u+v=1; hence m=u^.(B/F-1)+1 and u=(m-1)/(B/F-1).
  If we apply computed m & B/F values on the latter formula, the resulting monthly u coefficients differ slightly from form to form. These u coefficients have been averaged over all 71 forms for each location, radiation type & albedo, giving a monthly mean u coefficient, which can be used to estimate the typical m-index of any form that is not included in this study according to its B/F ratio (Chapter 16). 

• Irradiation calculation algorithm applying u coefficients: According to the above equations, the mean solar irradiance e on a convex form of ratio B/F is e=i^.e₀, or e=[u^.(B/F-1)+1]^.e₀. By definition e=R/F, therefore R=[u^.(B/F-1)+1]^.e₀^.F.
  The latter formula can be applied to approximate the total irradiation R on any convex form of exposed surface F & base B, using just horizontal irradiance e₀ data & u coefficients. The u coefficients found in this study refer to only three albedo values, but a simple method has been devised to convert irradiation R data from a given albedo to any other (Chapter 16).

• Climatic insolation index m': The insolation level of a building form can be beneficial or detrimental depending on the ambient temperature T_o, in connection with the comfort temperature T_i. The monthly difference Δt=T_i-T_o is considered as a positive or negative factor that converts m-indexes into 'climatic insolation indexes' m' according to the relation m'=m^.Δt where T_i is the comfort zone limit closest to T_o (if T_o is within the comfort zone then T_i=T_o). 
  Based on the mean monthly ambient temperature T_o in each location & assuming comfort zone between 18-25°C, the forms were sorted according to their annual average climatic index m' for global radiation & albedo 0.2 (Chapter 17).

• Relative climatic insolation indexes i_c: The actual m-index of a form may differ from the typical value m₀ that corresponds to its B/F ratio. Based on the discrepancy between the actual & typical values Δm=(m-m₀)/m₀, a form is considered 'warmer' or 'colder' than others having the same B/F ratio, i.e. it receives more or less radiation than average. That feature can be of positive or negative significance, according to temperature T_o.
  Under that prism, the forms were sorted by their 'relative climatic insolation index' m_c, which integrates Δi & Δt: m_c=Δm^.Δt
  Due to the complexity of heat transfer & thermal comfort issues, m' & m_c indices can be considered no more than general clues for the climatic compatibility of a form with respect to insolation (Chapter 17).

• Differences between locations: The comparison of m-indices in each location against those in the other two has shown that all forms, especially the tall ones, perform better as solar receivers in higher than lower latitudes, in contrast to the opposite tendency of the available radiation (Chapter 18).

As major contributions of this study to the issue of insolation analysis, one can highlight the following:

• Introduction & application of the 'insolation performance' notion, including computation of the associated 'insolation index m' of various typical forms.
• Discovery of a linear relation between insolation level & the proportion of base to exposed surface B/F.
• Formulation of a simple method for calculating solar irradiation on any convex geometric form based on horizontal irradiance data.
• Appraisal & classification of geometric forms in relation to local climatic conditions.

The study includes not only the analysis of quantitative data but also their creation, utilizing methods proposed by other researchers & climatic statistics. In addition to their use in the present work, several of those data (e.g. solar angles or irradiance values) can be useful for other applications, therefore they are included in the Appendices.

The dissertation is divided in 4 parts:
A - INTRODUCTION: Review of solar radiation factors and related thermal properties of the building envelope, based mainly on selected references (Chapters 1-6).
B - THEORETICAL CONTEXT: Introduction to the conceptual framework of the study & research method (Chapters 7-10).
C - OUTPUT PRESENTATION: Presentation, assessment & interpretation of the findings, along with conclusions (Chapters 11-19).
D - APPENDICES: Additional details & assorted data of the research.

The 71 forms of the study