Direct solar radiation and shading
Duration of direct solar illumination of a bed over a day, and the share of the daily direct-beam flux that reaches it in the presence of shading objects.
Quantities
| Symbol | Quantity | Unit |
|---|---|---|
φ, λ | latitude and longitude of the plot | degrees |
h | solar elevation above the horizon | degrees |
z | zenith distance, z = 90° − h | degrees |
m | relative optical air mass | dimensionless |
I | share of the direct beam reaching a horizontal surface | dimensionless |
s(t) | share of the bed's area not shaded at time t | 0 to 1 |
Δt | integration step | 0.25 h |
T | duration of direct illumination | h/day |
E | share of the daily flux reaching the bed | dimensionless |
Formulas
Optical air mass (Kasten & Young, 1989)
m = 1 / (cos z + 0.50572 · (96.07995 − z)^(−1.6364))
Attenuation of the direct beam (Meinel & Meinel, 1976)
I = 0.7^(m^0.678) · cos z
The cos z factor accounts for the inclination of the beam on a horizontal surface. I is normalised to the flux at z = 0 outside the atmosphere and is not expressed in W/m².
Duration of illumination
T = Σ Δt · s(t), summed over the moments t for which h > 0
Share of the daily flux
E = Σ I(t) · Δt · s(t) ÷ Σ I(t) · Δt
The denominator is evaluated for the same bed on the same day with s(t) ≡ 1. E = 1 denotes the absence of shading and is attainable on any day of the year.
Solar position
Elevation and azimuth follow the NOAA Solar Calculator algorithm: solar declination, equation of time, hour angle. The equation of time shifts true noon relative to mean noon by up to ±16 min depending on the date; omitting it displaces the shadow by the corresponding amount.
Shadow geometry
A tree crown is approximated by a sphere centred at height H − R, where H is the tree height and R the crown radius; its shadow on a horizontal plane is an ellipse elongated by 1/sin h. A building is approximated by a prism up to the eaves; when a ridge height is given, a gable roof is added above it. The building's shadow is its own footprint displaced away from the Sun, together with the roof's shadow: each footprint edge yields a quadrilateral joining that edge's shadow cast from eaves height to the shadow of its projection onto the ridge cast from ridge height. For a rectangular footprint with the ridge along the long axis this gives two slopes and two vertical gable ends. The transmittance of crown and wall is taken as zero.
Illuminated share of the area
The bed polygon is covered by a regular 5 × 5 grid of points. At each point the transmittances of all covering shadows are multiplied; s(t) is the mean over the points. Shading is evaluated for h ≥ 5°: at lower elevations shadow length grows without bound and the direct component becomes negligible.
Applicability and limitations
- The calculation assumes a clear sky and covers the direct component only; diffuse radiation and cloud are not represented.
- The transmissivity of a foliated crown has been measured at 1.3–5.3 % of direct radiation (Konarska et al., 2014; five trees, cherry among them); the midpoint of that range is used. A defoliated crown in the same study transmits 40–52 %, but seasonal leaf development is not modelled: no source states leaf-out dates. In spring and autumn a tree's shade is therefore overstated roughly tenfold.
- The ridge direction is not stored and is taken along the long side of the footprint: the length of the shadow does not depend on it, its shape in the morning and evening does. Roofs more complex than a gable — hipped, mansard, mono-pitch — are not represented; with no ridge height given the top is taken as flat.
- The shadow of the trunk is not represented.
- Objects outside the plot plan, including neighbouring buildings and plantings, are not included.
- Terrain and surface slope are not represented.
Sources
- NOAA Global Monitoring Laboratory. Solar Calculator — calculation details. NOAA Earth System Research Laboratories — full text open Open
- Kasten, F. & Young, A.T., 1989. Revised optical air mass tables and approximation formula. Applied Optics 28(22): 4735–4738 — abstract open, full text paywalled Open
- Meinel, A.B. & Meinel, M.P., 1976. Applied Solar Energy: An Introduction. Addison-Wesley — print publication, no link
- Konarska, J., Lindberg, F., Larsson, A., Thorsson, S. & Holmer, B., 2014. Transmissivity of solar radiation through crowns of single urban trees — application for outdoor thermal comfort modelling. Theoretical and Applied Climatology 117: 363–376 — abstract open, full text paywalled Open