1. Estimation of solar radiation for buildings with complex architectural layouts Arch. Stoyanka Ivanova University of Architecture, Civil Engineering and Geodesy Sofia, Bulgaria

2.  If we want the “solar friendly” architectural thinking to be accepted widely, then we have to think how it could be applied still in the beginning of the architectural creative process – when the idea of the building is shaped and especially in the process of creating of the architectural layout.  It’s possible to use a computer program not only to draw a desired layout, but also to create it. It’s hard to simulate the architectural creative process, but with the increase of the present computer might this is not already a “mission impossible”. 1 Introduction

3.  The program ArchiPlan is created to assist the architect in the process of searching of initial architectural idea – architectural layout.  The program generates hundreds of 2D orthogonal architectural layouts for just few seconds. 1 Introduction

4. In the beginning the architect describes the elements (rooms) of the desired layout:  names of rooms  square surface of each room  functional relations between rooms  requirements for each element for exposure  desired grid, etc. 1 Introduction

5. Example (15 rooms) 1 Introduction

6. Matrix of functional relations between rooms 1 Introduction

7. With these data the program ArchiPlan generates 602 architectural layouts for less than 4 seconds… 1 Introduction

9. Exemplary architectural layouts, generated by the program ArchiPlan: 1 Introduction

10. These layouts have to be evaluated, so the program can offer the architect only the best of them. The program uses the following criteria to rank all generated layouts:  compactness (as a planning quality)  construction’s requirements  energy effectiveness (energy gains and losses) The weight of these three criteria could vary, regarding the architect’s final goal. 1 Introduction

11. The program ArchiPlan sorts the generated architectural layouts on compactness… Let’s see worst 5 and top 10 layouts… 1 Introduction

13. The energy effectiveness of an architectural layout has two sides – energy losses and energy gains. In this too early moment of the architectural process the energy losses could be evaluated with the compactness of the building - the surface-area-to-volume ratio - a proportion between the total exterior surface and internal volume. As less is this number, as more compact is the building and lower the energy losses. 2 Methodology

14.  Since June 30 th a new calculation of supposed energy losses was added to the program. It uses a difference between inside and outside air temperature and supposed heat transmission coefficient.  The energy effectiveness is calculated as follows: 2 Methodology

15. To evaluate energy gains, the program ArchiPlan estimates the quantity of incoming solar energy [Wh] to the building outside surface (exterior walls and roof) for the whole heating season. In this so early stage is convenient to use 2.5D model – with only horizontal and vertical surfaces. 2 Methodology

16. Two methods of solar energy estimation are created. The final goal of the first method is to rank the hundreds of generated architectural layouts considering their energy effectiveness. In this case is important to receive quick results, even with some little inexactness. This is why we need to simplify computation of the summary solar radiation, received by all exterior surfaces. 2 Methodology

17. The second method is created to estimate the solar radiation, received by each particular vertical wall of a layout. The goal is to allow the architect to study how a specific architectural layout can utilize the incoming solar energy in best way. Such information could help him to find the best positions for the wall apertures - windows and glazed doors on the walls of a complex architectural layout. 2 Methodology

18. The exterior surface of our 2.5D model is projected on a plane, normal to solar beam. 3.1 Quick analysis of summary beam radiation

19. Area of roof’s projection: Area of walls’ projection: df – front of projection: Beam radiation: 3.1 Quick analysis of summary solar beam radiation

20. For the diffuse radiation R D [W] is valid the formula:  D ic [W.m -2 ] - the diffuse irradiance on each external surface (walls or roof) of the building  A i - area of this surface [m 2 ] Hofierka and Šúri in their work “The solar radiation model for Open source GIS” described how to estimate the diffuse irradiance on horizontal and inclined surfaces, when they are sunlit, potentially sunlit or shadowed. But this model has to be extended for the cases, when one or more near objects with limited size hide parts of the sky, as it is for the buildings with complex architectural layouts. 3.2 Quick analysis of summary diffuse radiation

21. According Hofierka for inclined surfaces in shadow:  D hc is diffuse irradiance on a horizontal surface  F(γ N ) is a function, which convert the diffuse irradiance on a horizontal surface into diffuse irradiance on an inclined surface under anisotropic clear sky:  r i (γ N )=(1+cos γ N )/ 2 is a fraction of the sky dome, viewed by an inclined surface. For unobstructed surfaces in shadow of our 2.5D model, N=0.25227 according Muneer and Hofierka: a) when γ N =0 (for horizontal surface): F(γ N )=1 b) when γ N =π/2 (for vertical surface): F(γ N )=0.356 3.2 Quick analysis of summary diffuse radiation

22. Let’s consider this wall configuration : 3.2 Quick analysis of summary diffuse radiation

23. For the wall A 0 a part of the sky is hidden by both other walls. I propose such equation to the mentioned model:  DF i is a function, which transforms the value of the diffuse irradiance on a horizontal surface to a value of diffuse irradiance, incoming from partially shaded anisotropic sky, onto an inclined surface (in our example it’s vertical): 3.2 Quick analysis of summary diffuse radiation

24.  α 1 and α 2 are azimuths of the furthest ends of the neighbor walls A 1 and A 2  C 1,2 is a correction coefficient for the visible part of the sky in horizontal direction, between azimuths α 1 and α 2  VT 1 and VT 2 are the correction values due of the spherical triangles T 1 and T 2 above the walls A 1 and A 2 on the stereographic projection. 3.2 Quick analysis of summary diffuse radiation

25. For estimation of C 1,2 this equation is proposed: where α 1 =arctg(dx 1 /dy 1 ) and α 2 =arctg(dx 2 /dy 2 ). If dy 1 =0 then α 1 =-π/2 and VT 1 =0. If dy 2 =0 then α 2 =π/2 and VT 2 =0. If both dy 1 and dy 2 are 0, then C 1,2 =1, VT 1 and VT 2 are 0. In all other cases the values of VT 1 or VT 2 are greater than 0. 3.2 Quick analysis of summary diffuse radiation

26. In the figure are illustrated the variable x 1a and angle β 1, which are necessary to estimate the value of VT 1. A T1 is the area of segment T 1 in the same figure. 3.2 Quick analysis of summary diffuse radiation

27. I propose the following approximate equation of VT 1 to take into consideration at least partially the anisotropic nature of the diffuse radiation: The transforming function DF i (corrected F(γ N )) has to be used to calculate diffuse radiation also on sunlit or partially shaded surfaces. Within this quick method I estimate D ic only once for the center of each examined vertical wall. 3.2 Quick analysis of summary diffuse radiation

28.  Finally the computation of overcast radiation for inclined surfaces is analogous to the procedure described for clear-sky model (Hofierka and Šúri), using the modified values of beam and diffuse irradiance because of the concrete values of beam and diffuse components of the clear-sky index (K b c, K d c ).  The layouts with best energy effectiveness are compact with southerly exposure of the larger dimension of the building and flat south vertical wall. As higher is the percent of diffuse radiation under the real sky, as more important is the compactness. 3.3 Quick analysis of solar radiation under real sky

29. The program ArchiPlan sorts the generated architectural layouts on energy effectiveness… Let’s see worst 5 and top 10 layouts… 3.3 Quick analysis of solar radiation under real sky

31. When the rank list of all generated architectural layouts is ready, the architect could examine the best of them, with the help of a detailed analysis of received solar radiation by each vertical wall to find best place for windows and glazed doors on each floor. 4.1 Detailed analysis of beam and diffuse radiation

32. According detailed methodology in “Solar radiation and shadow modeling with adaptive triangular meshes” by Montero et al:  B 0c is the beam irradiance normal to the solar beam  δ exp is the solar incidence angle between the sun and an inclined surface  Lf is calculated lighting factor for the concrete wall, day and time ( 0 – completely shaded wall; 1 - sunlit wall; between 0 and 1 – partially shadowed wall) 4.1.1 Detailed analysis of incoming beam radiation

33. The quantity of diffuse radiation for each fragment of each wall is different. So for a detailed analysis each wall is divided in horizontal and vertical directions and the program applies to these small fragments the estimations, described in the previous section. 4.1.2 Detailed analysis of incoming diffuse radiation

34. For this wall configuration: 4.1.2 Detailed analysis of incoming diffuse radiation DF i – the value of the diffuse transforming function for the wall with length b and height h, is:  M is number of fragments in horizontal direction  N is number of fragments in vertical direction  dx is size of a fragment in horizontal direction  dz is size of a fragment in vertical direction  (x j, z k ) – center of a fragment of this vertical wall

35. 4.1.2 Detailed analysis of incoming diffuse radiation Instead with this double sum, the exact value of diffuse transforming function for the same vertical wall can be estimated with this double definite integral: The value of the integral is:

36. 4.1.2 Detailed analysis of incoming diffuse radiation For different wall configurations the double integral and its result looks also different. There are 4 basic double integrals and many combinations between them for different wall configurations. I still work on some combinations.

37. 4.2 Balance principles of incoming and received diffuse radiation Balance principle 1 : The quantity of diffuse radiation, crossed an opening (with area A opening ), is equal to sum of the quantities of diffuse radiation, received by the surfaces (with areas A i ), which are behind this opening. From this is easy to come to:  DF opening is value of the diffuse transforming function for the planar surface of the opening  DF i is the value for each receiving surface behind the opening  DF j and A j are value of the transforming function and area of a small fragment j of the surface i

38. Wall ABFE Wall BCGF Wall CDHG Wall ADHE Base ABCD Opening EFGH 4.2.1 Balance principle 1 - example

39. For isotropic sky and AB=2, BC=1, AE=3: All values of DF are calculated with different combinations of mentioned double integral. 4.2.1 Balance principle 1 - example 2 = 2

40. Balance principle 2 : The quantity of diffuse radiation, crossed two or more openings (with area A k ), is equal to sum of the quantities of diffuse radiation, received by the surfaces (with areas A i ), which are behind each of the openings. From this is easy to come to:  DF k is value of the diffuse transforming function for opening k  DF ik is the value for the surface i, generated by the incoming diffuse radiation through opening k.  DF has to be defined twice for the planes of vertical openings - for inside and outside face of examined volume.  The value of the transforming function of the surface i with fragments j, for the incoming diffuse radiation from all openings, is: 4.2.2 Balance principle 2 of incoming and received diffuse radiation

41. Surface ABFE Wall BCGF Wall CDHG Wall ADHE Base ABCD Opening EFGH Opening ABFE 4.2.2 Balance principle 2 - example

42. For isotropic sky and AB=2, BC=1, AE=3: 4.2.2 Balance principle 2 - example 5 = 5

43. Wall BCGF Wall CDHG Wall ADHE Opening EFGH Opening ABFE The summary product of DF and Area for these 3 vertical walls is 3.646776 4.2.2 Balance principle 2 - example Here we have 2 interesting equalities : 3 = 3 5 = 5 Here is another viewpoint to the same example:

44. This is the main practical advantage from both balance principles. It helps to calculate very easy the summary diffuse radiation on a complex wall configuration. The main part of it (pink values) comes across an endless high vertical opening, so for these values we can simplify the outline of the layout in this way: 4.2.2 Balance principle 2 - example

45.  Both equations from the balance principles can be proved for isotropic sky analytically with already mentioned double integrals. It’s really beautiful mathematics, just “music of the projected spheres”.  I wasn’t able to prove both balance rules for anisotropic sky numerically. The proof is not obvious and will be more difficult. It could be possible with better knowledge of sky’s anisotropy.  In spite of this the consequences of the second balance rule can be implemented into calculations for obstructed anisotropic sky and to receive values with good approximation. Under real sky this inexactness will be even more unimportant.  Both balance principles and corresponding equations could also be applied to beam and total radiation, to sunlit or partially shadowed surfaces, and for different sky types. 5 Discussion

46.  It’s proposed a numerical model for estimation of the solar radiation on the external vertical walls of a building with complex architectural layout. This could help the architect to think more about the solar gains and to choose the best solar friendly architectural layout.  Further research is needed to precise the calculation of the diffuse radiation from anisotropic sky for spherical triangles above the vertical walls.  The balance principles and corresponding equations are a good start base for further improvement of this model for both isotropic and anisotropic sky.  Calculations of reflected radiation and analysis of solar radiation in summer will be added in the future. 6 Conclusion and future work

47. 1.Muneer, T., Solar radiation model for Europe. Building services engineering research and technology, 1990, vol. (11) 2.Hofierka, J., Šúri M., The solar radiation model for Open source GIS: implementation and applications, Open source GIS - GRASS users conference, Trento, Italy, 2002 3.Montero, G. et al, Solar radiation and shadow modeling with adaptive triangular meshes, Sol. Energy, 2009, doi:10.1016/j.solenar.2009.01.004 Thank you! 7 References