The sun as source of short wave radiation
The source of the short wave radiation arriving at the top of our atmosphere is the nuclear fusion process taking place in the sun. In the center of the sun and under high pressure and high temperature hydrogen is transmuted to helium. The exogenous reaction is a permanent source of energy that will last for the next five billion years; the half of the estimated live cycle of the sun.
A fractional amount of the sun energy is transmitted to the surface of the earth by radiation. The radiation flux density at top of the atmosphere is called solar constant. The solar constant is a function of time (season) that depends on solar-system and terrestrial influencing factors.
The radiation that passes through the atmosphere undergoes an extinction process (dispersion, reflection and adsorption) and only a certain percentage of it arrives at the surface of the earth. The normal radiation is the direct radiation flux density measured on a surface perpendicularly directed to the sun (one imagine a screen that moves with the sun). Dispersed and reflected radiation reappears as diffuse radiation which is still a (non-directed) kind of short wave radiation.
Both, direct and diffuse short wave radiation are balanced together at the building surface. The reflectivity of the surrounding is taken into account, too. Since the radiation is measured usually on a horizontal surface, an angle transformation from the horizontal component to the component normal to the wall surface is required for simulations.
Solar system-related and terrestrial influencing factors on solar constant
One of the influencing factors on the solar constant S is the velocity of circulation of the earth in the orbit. It changes with the season by about 1 hour. The graphics below show time shift against time for the location Tampa in Florida.
A second influencing factor on the solar constant S is the declination of the sun. The declination of the sun is the angle between the equatorial plane of the earth and the earth orbit plane. It changes by more than 40° over the year.
The third influencing factor on the solar constant S is the ecliptical length, the angle between current position of the earth and that at spring begin. With ecliptical length the distance between sun and earth changes. The three influencing factors together cause an annual variation of the solar constant by about 7% or 100 W/m2. The solar constant S reaches a maximum in summer time (mid of year) and a minimum in winter time (end of year).
Solar and local reference systems
For calculation of the sun radiation a s function of position (on the earth) and time the introduction of a reference system is necessary. There are two reference systems: the equatorial system that introduces the solar angles hour angle and declination and the horizontal system that introduces the terrestrial angles azimuth and sun height. The hour angle and the azimuth describe the angle related to the rotation of the earth. The declination and the sun height are a measure for the current vertical position of the sun.
The equatorial system uses a plane parallel to the equator. The horizontal system relates to a horizontal plane from the viewpoint of the respective location on the earth. Therefore, the relation between the angles of both reference systems depend on the respective location.
For radiation calculations the angles of the horizontal system are relevant. The definition of the local angles azimuth and sun height at the building surface coincides with that of the horizontal reference system (see sketches below). Knowing the influencing factors one can develop a set of formulas to calculate the direct sun radiation as function of location and time (still without influences of the atmosphere).
Atmospheric influencing factors
Another influencing factor on the sun radiation is the atmosphere itself or better, its thickness, humidity and degree of pollution.
First, the elevation or altitude of the respective location matters. Since locations can vary by 8 km in altitude and a simple estimation of the thickness of the atmosphere (by knowledge of the total air mass and application of the barometric formula) gives a value of about 10 km, the conclusion is justified that it may have a major influence. The graphics below shows the percentage of radiation arriving at the surface of the earth. In the morning (sun rise) and in the evening (sun set) the traveling distance of the sun beam elongates due to low values of sun height. Elongated traveling distance means higher extinction.
The previous consideration relates to a dry and clean atmosphere. To account for water vapor and dust one introduced the so-called opaqueness factor. Its definition is given by the quotient of logarithms of the radiation of a real and a clear/dry atmosphere. It can be expressed as the number of clean/dry atmospheres needed to have the same extinction effect as the real atmosphere.
Short wave radiation modeling versus measurements
Calculation of the short wave radiation from the considerations above requires estimation of the opaqueness factor. The graphics below show a comparison of the measured and calculated radiation for Tampa, FL with an estimated opaqueness factor of T=8. The estimation is based on the equality of the annual energy amount transmitted by short wave radiation.
The calculated radiation can reproduce the mean course of measured radiation quite well but it doesn't account for short term influences and local fluctuations.
Direct sun radiation on building surfaces
The horizontal component qdir,hor of the normal radiation is measured on a horizontal surface. Its vertical component qdir,ver can be calculated either from the horizontal component (in case of measured radiation) or both components are calculated from the normal radiation qdir,nor (in case of calculated radiation).The direct sun radiation on building surfaces is calculated from both, the horizontal and the vertical components of the normal radiation. Parameters are the slope of the building element and its orientation. The angle between azimuth and facade normal vector is called relative azimuth.
Knowing these parameters the radiation flux to the building surface is the sum of the contributions perpendicular to the building element from the the horizontal and the vertical components of the normal radiation. The radiation flux normal to a vertical wall (East, South and West) and to a flat roof are shown in the graphics below for the location Tampa, FL.
Diffuse sun radiation on building surfaces
On crossing the atmosphere the direct sun radiation is reflected, adsorbed and dispersed. This process is called extinction. The direct sun radiation is partially transmuted into diffuse radiation; another part is reflected and doesn't reach the surface of the earth.
The "missing" radiation can be calculated from the difference between the solar constant and the direct radiation arriving at the surface of the earth. About one third of that "missing" radiation is transmuted into diffuse radiation. The rest is reflected back in the space.
A comparison of calculated and measured diffuse radiation is shown below. The calculated radiation can reproduce the mean course of measured radiation quite well but it doesn't account for short term influences and local fluctuations.
