a solar intensity and shading tool
how to use and methods used in the sheet
Effect of the basic building parameters on the solar intake.
Parameters:
1. length, width, and height ratios (the building volume remains constant = 1m3)
2. cornice/floor height ratio
3. latitude, building azimuth (angle from South)
4. roof slope (then the entire building turns into a roof).
The basic methodology of calculation is as follows:
1. The polar representation of the solar intensity at a particular elevation is a sphere touching the ground.
This stems from the fact that the ratio of the distances from a point to 2 circles (spheres) is constant (a generalisation of the ancient Tangent-Secant Theorem). The point is either inside both circles (spheres) or outside both circles (spheres). If the point is on one circle (sphere) then the other circle (sphere) has the centre at infinity and is a line (a plane).
Therefore, assuming the limit of the atmosphere to be spherical (assuming all the atmosphere is concentrated below this limit and nothing is above it), and the intensity of sunlight being inversely proportional to the thickness of atmosphere traversed (=distance between the observer and the limit of the atmosphere), and the observer being inside the atmosphere, the intensity of sunlight is represented by a sphere around the observer.
Since the observer is extremely close to one side of the atmospherical sphere (the other side being on the other side of the earth), the solar intensity sphere is also very close to the observer. We can assume this sphere is just touching the ground, because the atmospherical sphere is so large that seen from the ground, it is almost a horizontal plane above our heads.
2. The solar intensity (distance from the bottom of the sphere to any point on the sphere) is equal to the scalar product of a vertical unit vector with the sun unit vector, or in other words, to the cosinus of the incidence angle.
3. The solar intensity when the sun is vertical with a clear sky is almost 1kW. We can assign a length of 1 to the vertical vector, and the result will be in kW.
4. The solar incidence per m2 on a particular surface is the scalar product of of the unit perpendicular vector and the solar intensity vector.
5. So we have 2 successive scalar products: 1. sun vector with vertical vector, and 2. sun vector with surface perpendicular vector. The product of these 2 products gives the solar intensity of this surface. Multiplying by the area of the surface, we get the solar intensity at a particular time on this surface.
6. The sun moves every day in a infinite circle in a plane perpendicular to the axis of the earth rotation (roughly pointing towards the Pole Star). This plane (and the centre of the circle) moves during the year. At the equinoxes (21st March and 21st October) this plane is at the centre of the earth, so the sun rays are contained within this plane, called the equinoctial plane. In summer, the plane gradually shifts towards the South (by 23.5 degrees on 21st June) so the sun rays are contained in a cone. In winter, the plane shifts towards the North (by 23.5 degrees on 21st December) and the cone is reversed. The limit between these 2 cones is the equinoctial plane.
7. So finally we have a synthetic representation of the solar intensity at any time, any day of the year at a particular latitude, by intersecting the solar intensity sphere (this sphere is touching the ground and has a diameter of 1kW and is the same everywhere on the earth) with the series of daily solar cones, which can be simplified for convenience with one cone per month, and since the same cones are repeated in November and January, October and February, etc. we get finally only the 3 winter cones (December, January, February), the equinoctial plane (March) and the 3 summer cones (April, May, June).
8. The final shape gives at a glance the type of shape, orientation and shading that we would wish to have for our buildings, depending on the place where we live and the season: It looks a bit like concentric slices of a sphere. What is particularly interesting is that this shape can be approximated in practice with concentric slices, not parallel slices, despite the solar plane moving in a translation and not a rotation. But thinking in terms of rotation reflects the intuitive perception we have of the effect of sunlight. This is because the solar intensity before 8am is very weak, and before 7am, negligible.
9. We can get an instant idea of the intensity of solar irradiation on a particular building surface, by rotating the solar intensity sphere around the point where it touches the ground, by the angle of the surface, and intersecting the two spheres. The exact calculation is just a little bit different, a little bit smaller, and it won't affect the architectural design if we approximate the exact surface by the intersection of the two spheres. It is obvious that the resulting surface has to be at any moment inside both the spheres. So it is also easy to remember the method. See the worksheet "product of 2 circles/spheres" in the same folder.