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
<code>
\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
>= 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