The geometric relationship of various angles of solar radiation

In the use of solar energy, we need to have a clear spatial concept and mutual relationship with respect to a series of mathematical and physical characteristics related to the geometric angle of solar radiation, relative spatial position, and factors affecting solar radiation intensity. A preliminary understanding of the interrelationships.

We need to refer to the following (Figure 1) and (Figure 2) to further understand the geometric relationship between the following angles:
Geographical latitude angle (φ)-in the use of solar energy, refers to the geographic latitude (angle) where the solar energy application device is located;
The solar declination angle (δ) has been defined before, which refers to the angle (δ) between the sunlight and the earth’s equatorial plane. The “solar declination angle” (δ) for each day of the year can be obtained from the following formula Find:


Where: n is the calculation date, the number of calendar days in a year;
Solar hour angle (ω): the positive sun is 0 degrees at 12 noon local solar hour angle: the morning hour angle is positive, the afternoon hour angle is negative; ω=15° per hour; ω=15x(12-Ts) ; Where Ts = true solar time (i.e. 24-hour clock value);
Sun incident angle (θ): the angle (θ) between the sun’s incident light and the normal to the surface of the solar collector, also known as the direct angle of sunlight.


Azimuth of the sun (ys): The angle (ys) between the projection line of the sun’s incident light on the horizontal plane and the direction of the ground (ie meridian); the azimuth of the sun starts and ends at 0°, and it is westward Positive, negative to the east, the range is ±180°;


Solar altitude angle (a): The angle between the incident sun’s rays and the horizontal line of the ground (a) is called the solar altitude angle; the solar altitude angle (a) of any point on the earth at a certain moment can be obtained by the following formula:


In the formula: a is the solar altitude angle; φ is the local geographic latitude angle (the northern hemisphere takes a positive value, and the southern hemisphere takes a negative value); δ is the solar declination angle; ω is the solar time angle (local 12 noon: ω=0°; It is negative in the morning and positive in the afternoon; the hour angle is 15° per hour; for example: 9 am ω = -45°; 2 pm ω = +30°; and so on).

Solar zenith angle (θz): The angle (θz) between the incident sun’s rays and the vertical line (Z axis) of the ground, which is called the solar zenith angle.

From the two schematic diagrams of the geometric relationship in Figure 1 and Figure 2, we can see that the sun’s altitude angle and the sun’s zenith angle are complementary to each other: that is: ∠α+∠θz = 90°; for the ground plane, because ∠θ=∠θz: Therefore, it can be concluded that the sun’s direct (incident) angle (θ) and the sun’s altitude angle (a) also have complementary angles.

In addition, it can be seen from Fig. 2 that the radiation intensity (intensity) of the sun shining on any inclined surface on the earth is proportional to the sun’s incident angle (θ), which is the cosine of the sun’s direct angle, which is also called “The Law of Cosines”

Figure 1 The geometric relationship between solar declination (δ), geographic latitude (φ), solar altitude (a), solar incident angle (θ), and solar zenith angle (θz)
Figure 2 The geometric relationship between the sun’s direct (incident) angle (θ), the sun’s zenith angle (θz), the sun’s altitude angle (a), and the sun’s azimuth angle (ys)

In fact, when solar radiant energy travels through the earth’s atmosphere, it is generally decomposed into direct radiation that directly reaches the ground and environmental diffuse radiation that is formed by reflection, absorption, and scattering attenuation by the dense atmosphere. Obviously, the amount of solar radiation that can reach the ground is closely related to the length of the way the sunlight travels through the atmosphere, that is, the original degree of the atmosphere.

On the ground plane, the sun’s high skin angle (a) and the Taiyong people’s angle of fire (θz) are complementary angles: that is, ∠α+∠θz = 90° with 0°<∠a<90°. θz=90°, the larger [cos90°=0], which means that the sun’s altitude angle (∠a=0°) is smaller; this is equivalent to early and late hours, the path of sunlight penetrating the atmosphere is relatively long, and the sun Radiation is also greatly affected by atmospheric absorption and scattering attenuation, and the radiant energy that can finally reach the ground plane is also very small. In order to scientifically grasp the relevant influence relationship of the change law, we will pass the sunlight through the atmosphere. The ratio of the actual optical path length to the actual normal thickness of the atmosphere is defined as “atmospheric mass” (m); and set when the solar altitude angle (a) is 90°, that is, when the direct sunlight angle is 0°, [ cos0°=1], air quality (dimensionless): m=1.
Air quality:

Figure 3 The path of the sun’s incidence in the atmosphere

From the above figure and formula (5), we can know by calculation:

Sun altitude angle (α)90°60°45°30°10°
Air quality (1/sinα)1.0001.1551.4142.0005.75811.480
the smaller the solar altitude angle (a), the greater the atmospheric mass (m)

Obviously, the smaller the solar altitude angle (a), the greater the atmospheric mass (m), and the less solar radiation energy the ground can receive; when the solar altitude angle α=0°, it is equivalent to the sun has not yet risen ( (Or has fallen) The possible radiation intensity on the ground is zero.

The main factors affecting the solar altitude angle (a) are:
(1) Geographical factors: the lower the latitude, the closer the tropics to the equator, the richer the solar radiation resources. For residents living on the Tropic of Cancer in the Northern Hemisphere, at 12 noon local time on the summer solstice, the sun is just at the zenith. At this time, the sun’s altitude angle α=90° and the atmospheric mass m=1, which is the sun during the year. When the radiation intensity is strongest. The strongest solar radiation at corresponding locations in the southern hemisphere appears on the winter solstice. The higher the latitude, the smaller the annual average solar altitude angle, and the relatively fewer solar radiation resources. From the above analysis, we can see that solar radiant energy is mainly distributed in low latitude and high altitude areas. Because the average solar altitude angle (α) in this area is relatively high throughout the year, and the atmospheric mass (m) is low, the solar radiation intensity that can be received is relatively high.
(2) Seasonal factors: In the same latitude area, due to the difference in solar altitude and duration of sunshine in different seasons, the intensity of solar radiation in the same area in different seasons varies greatly with seasons. For example, in the northern hemisphere, the solar radiation resources in summer are much higher than the solar radiation intensity in winter. Because the sun is south of the equator in winter, the solar altitude angle to the same area in the northern hemisphere is obviously much lower. The mass of the atmosphere that sunlight needs to penetrate is greater, so the radiation intensity loss is also greater. Based on this, we can know that the solar radiation energy available for collection and utilization in the high latitudes of the earth in winter is relatively less, so the possibility of development and utilization is low, surpassing the areas above 50° north-south latitude, even in summer solar resource utilization. It is also very limited.