Druyd:Knowledge

User:

Work

Knowledge

Palos Verdes

Theory and data

Insolation

Sun position

Insolation

Direct light

Indirect light

Indirect light

Storyboard

In addition to direct incident light, a significant portion comes from diffuse light generated both in the atmosphere and on the Earth's surface. Diffuse light generated in the atmosphere results from scattering and refraction. Diffuse light generated by the ground is primarily direct light that is reflected off the ground. In this case, some of the intensity is lost due to absorption, and it's the ground's albedo that determines the amount reflected. In both cases, the generated light is isotropic, meaning that, in a first approximation, it doesn't have a preferred direction.

>Model

ID:(89, 0)

Indirect radiation

Image

>Top

An object receives solar radiation from three sources:
• Direct sunlight from the sun.
• Radiation scattered and refracted in the atmosphere.
• Reflection of light from objects and the Earth's surface.

These sources of solar radiation strongly depend on the direction:

• Direct solar light only occurs in the direction where the sun is located at that moment. This means that if the sun is not visible from a certain point, there will be no direct solar radiation received at that location.

• In contrast, radiation scattered and refracted in the atmosphere, as well as radiation reflected by objects and the surface, is diffuse. This means it doesn't have a preferred direction and can be received from various directions, regardless of the sun's position.

ID:(624, 0)

Indirect captured radiation

Image

>Top

Since diffuse light does not have a preferred direction, we simply estimate its intensity as if it were arriving perpendicularly to the surface, eliminating the need to weight it by the cosine of an angle. The total contribution is equal to the sum of all contributions:

The only difference between these contributions is that those originating from the atmosphere are integrals, while those from reflections only contain the light that was not absorbed, corresponding to the albedo.

Typical albedo values are as follows:

Surface	Albedo [-]
Snow	0.50 - 0.80
Desert	0.30 - 0.40
Vegetation	0.10 - 0.25
Sea and Lakes	0.06

This process can be modeled in such a way that the fraction of the visual field coming from the sky is considered to have an albedo of unity, while other areas contribute proportionally to the occupied visual field. This can be important in situations where:

• In the vicinity of a hill, the visual field of the sky is reduced, resulting in a significant reduction in intensity.

• Coastal areas, where the visual field includes the sea, experience reduced intensity due to the low albedo of water.

• Dry or snowy areas have higher diffuse light due to the high albedo of the ground.

ID:(625, 0)

Multiple broadcast problem

Image

>Top

The main challenge with diffuse light is that it is generated through multiple reflections and diffractions in objects on the Earth's surface and in the atmosphere. Not only is it complex due to geometry, but it also depends on the state of the atmosphere and the degree of diffusion that occurs.

To address this issue, it is assumed that the radiation is isotropic, meaning there is no preferred direction. Under this premise, modeling begins by including coverage and albedos in proportion to the visual field they occupy. Based on this foundation, the transparency of the atmosphere is calculated, and models are employed to calculate a diffusion factor. Finally, this factor is used to modulate the final radiation that arrives isotropically at all surfaces.

ID:(621, 0)

Reflected intensity

Equation

>Top, >Model

At first, we can model the generation of diffuse light ($I_e$, represented by a light yellow LEGO® piece), assuming that the intensity of sunlight ($I_s$, represented by a bright orange LEGO® piece) is modulated by the coverage factor ($\gamma_v$, represented by a light gray LEGO® piece) and the albedo ($a_v$, represented by a sky-blue LEGO® piece).

This relationship can be mathematically expressed considering that the coverage factor allows a factor of $1-\gamma_v$ to pass through and the reflection on the surface is given by the albedo $a_v$:

$ I_e = a_v (1- \gamma_v) I_s $

To simplify, it is assumed that the atmosphere acts as if it reflects the entirety, so it can be modeled with an albedo value of one.

ID:(620, 0)

Diffuse intensity

Equation

>Top, >Model

When light is reflected, it scatters in all directions, which is modeled using a diffusion factor ($k_d$, represented by a LEGO® piece in dark green). The diffuse intensity ($I_e$, represented by a LEGO® piece in light yellow) is multiplied by this factor to create the scattered intensity ($I_i$, represented by a LEGO® piece in light nougat):

This can be described mathematically using the following expression:

$ I_i = k_d I_e $

ID:(619, 0)

Clarity factor

Equation

>Top, >Model

The scattering factor ($k_d$, represented by a LEGO® piece in dark green) is calculated based on the atmospheric clarity factor ($k_t$, represented by a LEGO® piece in light green). The latter depends on the coverage ($\gamma_v$, represented by a LEGO® piece in light gray) and the albedo ($a_v$, represented by a LEGO® piece in light blue).

The relationship has been modeled based on measurements, and there are different levels of sophistication. One of particular interest for situations where there are variations in humidity ($RH$, represented by a LEGO® piece in light blue) in the environment, temperature ($T$, represented by a LEGO® piece in lavender), and the sun's angle ($\theta$, represented by a LEGO® piece in violet) includes these values in the calculation:

The mathematical expression for calculating the atmospheric clarity factor is as follows:

$ k_t = a_v (1- \gamma_v )$

ID:(683, 0)

Relationship between diffusion fraction and transparency index

Image

>Top

Diffuse radiation is generated by the interaction of incident light with the atmosphere and the surface. Its exact calculation is highly complex, so semi-empirical models are used, meaning mathematical models are constructed based on measurements taken in different locations and under various conditions.

These models establish relationships between the diffusion factor $k_d$ and the atmosphere's transparency index $k_t$, which take the following form:

Scarpa, Federico & Marchitto, A. & Tagliafico, Luca A.. (2017). Splitting the solar radiation in direct and diffuse components; Insights and constrains on the clearness-diffuse fraction representation. International Journal of Heat and Technology. 35. 325-329. 10.18280/ijht.350213.

Some models that describe the fraction of diffusion as a function of the clarity index include

• the simple model by Lam and Li (1996 - developed in Hong Kong),

• the Erbs model (1982 - based on data from 5 stations in the USA), and

• the Reindl model (1990 - incorporating data from stations in the USA and Europe), which also accounts for the effects of temperature and humidity.

ID:(618, 0)

Lam and Li (LL) model

Equation

>Top, >Model

In the case of the Lam and Li model, measurements were conducted in Hong Kong, and the results were published in 1996 in the article titled "Lam J.C., Li D.H.W. Correlation between global solar radiation and its direct and diffuse components. Building and Environment. 1996;31(6):527535.. The relationship proposed by this model is summarized by the diffuse radiation fraction $k_d$ as a function of the clarity index $k_t$:

$k_d = \begin{cases} 0.977 & k_t \leq 0.15,\\ 1.237-1.361 k_t & 0.15 \leq k_t \leq 0.7,\\ 0.273 & k_t > 0.7\\ \end{cases}$

ID:(622, 0)

Erbs model (ER)

Equation

>Top, >Model

In the case of the Erbs model, measurements were conducted at 5 stations in the United States, and the results were published in 1982 in the article titled Erbs D.G., Klein S.A., Duffie J.A. Estimation of the diffuse radiation fraction for hourly, daily and monthly-average global radiation. Solar Energy. 1982;28(4):293302.. The relationship proposed by this model is summarized by the diffuse radiation fraction $k_d$ as a function of the clarity index $k_t$:

$k_d = \begin{cases} 1-0.09 k_t & k_t \leq 0.22,\\ 0.9511-0.1604 k_t+4.49 k_t^2 & \\ -16.64 k_t^3+12.34 k_t^4 & 0.22 \leq k_t \leq 0.8,\\ 0.165 & k_t > 0.8\\ \end{cases}$

ID:(617, 0)

Reindl model (RE)

Equation

>Top, >Model

In the case of the Reindl model, measurements were conducted in the USA and Europe, and the effects of solar altitude, ambient temperature, and relative humidity were considered. The results of these measurements were published in 1990 in the article titled Reindl D.T., Beckman W.A., Duffie J.A. Diffuse fraction corrections. Solar Energy. 1990;45(1):17.. The relationship proposed by this model is summarized by the diffuse radiation fraction $k_d$ as a function of the clarity index $k_t$, solar altitude $\alpha$, ambient temperature $T_a$, and relative humidity $RH$:

$k_d = \begin{cases} 1.0-0.232k_t+0.0239\sin\alpha-0.000682 T_a +0.019 RH & k_t \leq 0.3,\\ 1.329-1.716 k_t +0.267\sin\alpha -0.00357 T_a + 0.106 RH & 0.3 \leq k_t \leq 0.78,\\ 0.426 k_t +0.256\sin\alpha - 0.00349 T_a + 0.0734 RH & k_t > 0.78\\ \end{cases}$

ID:(623, 0)

Effect of a window in diffused light

Equation

>Top, >Model

In the case of windows, the glass allows most of the light and energy to penetrate into the interior of the house. For diffuse light, there is no preferred direction, so the incident intensity ($I_i$, represented by the light nougat-colored LEGO® piece) is equal to the intensity that passes through the glass ($I_g$, represented by the dark blue LEGO® piece):

The mathematical expression is simply:

$ I_g = I_i $

ID:(684, 0)

Diffuse radiation captured by a surface

Equation

>Top, >Model

In the case of a wall, diffuse light ($I_i$, represented by a light nougat LEGO® piece), is absorbed without any preference for one direction over another. The absorbed fraction can be estimated based on the material's absorption coefficient and the wall's color ($a_b$, represented by a water-colored LEGO® piece). In this way, the solar intensity contributing energy to the surface ($I_w$, represented by a coffee-colored LEGO® piece) is associated as shown in the diagram:

The mathematical expression is:

$ I_w = a_b I_i $

ID:(685, 0)

Instructions for diffuse radiation

Image

>Top

If we consider all the processes related to diffuse irradiation in the house, we can create an instructional guide as follows:

In this case, the following calculations are identified:

1. Diffuse intensity

2. Atmospheric clarity factor

3. Diffusion factor

4. Scattered intensity

5. Radiation entering through windows.

6. Radiation providing energy through walls and ceilings.

ID:(712, 0)