How can you calculate solar irradiance on a surface?

1 Solar geometry

To calculate solar irradiance on a surface, it is important to understand the solar geometry, which describes the position and angle of the sun relative to the earth and surface. This geometry depends on latitude and longitude, date and time of year, and tilt and orientation of the surface. Key terms and parameters include solar declination, which varies from -23.45° to +23.45° throughout the year; solar hour angle, indicating how far from noon the sun has moved; solar altitude, measuring how high in the sky the sun is; solar azimuth, indicating direction of the sun from the observer's point of view; surface tilt, affecting how much irradiance is received; and surface azimuth, affecting how well the surface faces the sun.

2 Solar irradiance components

The second step to calculate solar irradiance on a surface is to identify its three components: direct or beam irradiance, which reaches the surface directly from the sun; diffuse or sky irradiance, which reaches the surface after being scattered by the atmosphere; and reflected or ground irradiance, which arrives after being reflected by the ground or other nearby objects. The total solar irradiance on a surface is the sum of these three components, which depend on the solar geometry and the surface characteristics, as well as atmospheric conditions and the albedo (reflectivity) of the ground.

3 Solar irradiance formulas

The third step to calculate solar irradiance on a surface is to apply some formulas that estimate each component based on empirical or theoretical models. One of the most widely used and simple models is the Liu and Jordan model, which assumes a clear sky and uniform diffuse irradiance. The formulas for direct irradiance is I_d = I_0 * cos(\theta) , where I_d is the direct irradiance, I_0 is the extraterrestrial irradiance, and \theta is the angle of incidence. For diffuse irradiance, the formula is I_s = I_dh * (0.5 + 0.5 * sin(\beta)) * (1 + k * cos(\beta)) , where I_s is the diffuse irradiance, I_dh is the direct irradiance on a horizontal surface, \beta is the surface tilt, and k is a constant that depends on the solar altitude and atmospheric turbidity. The formula for reflected irradiance is I_r = I_th * \rho * (0.5 + 0.5 * sin(\beta)) , where I_r is the reflected irradiance, I_th is the total irradiance on a horizontal surface, and \rho is the albedo of the ground. The total irradiance on a surface can be calculated with: I_t = I_d + I_s + I_r , where

I_t</
###### Solar irradiance examples
To calculate solar irradiance on a surface, the fourth step is to apply formulas to examples and compare the results. For instance, let's consider the following conditions: Location 40° N, 0° E; Date and time June 21, 12:00; Extraterrestrial irradiance 1361 W/m2; Atmospheric turbidity 2; Albedo of the ground 0.2. With some online tools or calculators, we can find the values for the solar geometry: Solar declination 23.44°; Solar hour angle 0°; Solar altitude 73.44°; Solar azimuth 180°. Now, let's calculate the solar irradiance on three different surfaces: A horizontal surface 
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\beta        

= 0°, \theta = 16.56°; A south-facing tilted surface \beta = 30°, \theta = 13.74°; An east-facing tilted surface \beta = 30°, \theta = 90°. Using the Liu and Jordan model, we can find the values for each component and total irradiance: Horizontal surface I_d = 1288.12 W/m2, I_s = 834.22 W/m2, I_r = 212.47 W/m2, I_t = 2334.81 W/m2; South-facing tilted surface I_d = 1317.31 W/m2, I_s = 1004.99 W/m2, I_r = 212.47 W/m2, I_t = 2534.77 W/m2; East-facing tilted surface <codeI_d

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>= 0 W/m2, <codeI_s</c>>= 1004.99 W/m2, <c>>= 212.47 W/m2, <c>>= 1217.46 W/m2. As we can see, the south-facing tilted surface receives the highest solar irradiance with a total of 2534.77