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Lightning
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Once natural lighting has been assessed, it's possible to determine the amount of additional lighting that may be needed during the day and design the lighting system accordingly. Additionally, it's important to calculate the amount of lighting required for times when natural light is not available.
ID:(91, 0)
Lightning
Description
Once natural lighting has been assessed, it's possible to determine the amount of additional lighting that may be needed during the day and design the lighting system accordingly. Additionally, it's important to calculate the amount of lighting required for times when natural light is not available.
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(ID 3341)
Given that the photon frequency ($\nu$) is the inverse of the period ($T$):
$\nu=\displaystyle\frac{1}{T}$
this means that the speed of Light ($c$) is equal to the distance traveled in one oscillation, which is ERROR:8439, divided by the elapsed time, which corresponds to the period:
$c=\displaystyle\frac{\lambda}{T}$
In other words, the following relationship holds:
| $ c = \nu \lambda $ |
(ID 3953)
The photon flux in a room is described by the photon concentration ($n$) as a function of the time ($t$), using the variables ERROR:10426, the intensity ($I$), ERROR:10424, the albedo ($a$), the speed of Light ($c$), and the photon energy ($\epsilon$), according to the following equation:
| $\displaystyle\frac{ dn }{ dt } = -\displaystyle\frac{1}{6} c \displaystyle\frac{ S }{ V }(1-a) n + \displaystyle\frac{ S_w I }{ \epsilon V }$ |
In the steady-state case, the derivative is zero, and by solving the equation for the photon concentration ($n$), we can define the asymptotic photon concentration ($n_a$) using the following relation:
| $ n_a = \displaystyle\frac{6 I }{ c (1- a )\epsilon}\displaystyle\frac{ S_w }{ S }$ |
(ID 15868)
The variation of the photon concentration ($n$) with respect to the time ($t$),
$\displaystyle\frac{dn}{dt}$
will be equal to the incoming flux:
$\displaystyle\frac{S_w I}{\epsilon V}$
which involves the variables ERROR:10426, the intensity ($I$), the photon energy ($\epsilon$), and ERROR:10425, minus the loss due to absorption by the walls:
$\displaystyle\frac{1}{6}\displaystyle\frac{S (1-a) c n}{V}$
using the variables the speed of Light ($c$), the albedo ($a$), and ERROR:10424, resulting in the following equation:
| $\displaystyle\frac{ dn }{ dt } = -\displaystyle\frac{1}{6} c \displaystyle\frac{ S }{ V }(1-a) n + \displaystyle\frac{ S_w I }{ \epsilon V }$ |
(ID 15870)
Since the variation of the photon concentration ($n$) as a function of the time ($t$) is due to the incoming flux minus the absorbed fraction, the equation can be expressed using the variables ERROR:10426, the intensity ($I$), ERROR:10424, the albedo ($a$), ERROR:10425, the speed of Light ($c$), and the photon energy ($\epsilon$), leading to the following relationship:
| $\displaystyle\frac{ dn }{ dt } = -\displaystyle\frac{1}{6} c \displaystyle\frac{ S }{ V }(1-a) n + \displaystyle\frac{ S_w I }{ \epsilon V }$ |
With the relation for the asymptotic photon concentration ($n_a$) given by:
| $ n_a = \displaystyle\frac{6 I }{ c (1- a )\epsilon}\displaystyle\frac{ S_w }{ S }$ |
and with the relaxation time ($\tau$):
| $ \tau = \displaystyle\frac{ 6 V }{ c (1- a ) S }$ |
the equation can be rewritten as:
$\displaystyle\frac{dn}{dt} = \displaystyle\frac{1}{\tau}(n_0-n)$
whose solution is:
| $ n = n_a + (n_0 - n_a) e^{- t / \tau }$ |
with the initial concentration ($n_0$).
(ID 15871)
Since the variation of the photon concentration ($n$) as a function of the time ($t$) is due to the incoming flux minus the fraction that is absorbed, the equation can be expressed using the variables ERROR:10426, the intensity ($I$), ERROR:10424, the albedo ($a$), ERROR:10425, the speed of Light ($c$), and the photon energy ($\epsilon$), resulting in the following equation:
| $\displaystyle\frac{ dn }{ dt } = -\displaystyle\frac{1}{6} c \displaystyle\frac{ S }{ V }(1-a) n + \displaystyle\frac{ S_w I }{ \epsilon V }$ |
With the relation for the asymptotic photon concentration ($n_a$) given by:
| $ n_a = \displaystyle\frac{6 I }{ c (1- a )\epsilon}\displaystyle\frac{ S_w }{ S }$ |
the equation can be rewritten as:
$\displaystyle\frac{dn}{dt} = \displaystyle\frac{1}{\tau}(n_0-n)$
where the relaxation time ($\tau$) is:
| $ \tau = \displaystyle\frac{ 6 V }{ c (1- a ) S }$ |
(ID 15872)
Examples
(ID 15873)
A simple model to study the required lighting is that of a photon gas occupying the volume of the room. These particles enter the room through the windows from outside and/or from lamps inside the room:
None
The photons move at the speed of Light ($c$) across the space, hitting the walls, where only a fraction corresponding to the albedo ($a$) is reflected. The fraction $1-a$ is absorbed by the walls, and thus the photons exit the system:
Since the walls are not perfectly smooth, the light reflects isotropically, meaning without favoring any particular direction. In the end, there is an incoming photon flux through the windows and/or lamps, and a dominant absorption flux on the walls, which, in a steady-state situation, will be equal to the incoming flux:
(ID 137)
The amount of light, represented by the number of photons entering the room per unit of time, whether through windows or lamps, can be estimated using the variables the intensity ($I$) and ERROR:10426, considering that each photon possesses an energy of the photon energy ($\epsilon$). This relationship is given by the formula:
$\displaystyle\frac{I S_w}{\epsilon}$
Which is illustrated in the following graph:
The photons that enter the space are lost due to absorption by the surfaces of the walls, ceiling, and floor, according to the value ERROR:10424. The number of photons impacting these surfaces is proportional to the photon concentration ($n$), and the fraction absorbed is the complement of the albedo ($a$). Additionally, if the distribution of photons is anisotropic, only 1/6 of the photons near the surface will travel in the direction toward it. Therefore, the flux of absorbed photons can be expressed as:
$\displaystyle\frac{1}{6} n S (1-a)$
This relationship is also represented in the following graph:
In general, the second flux is smaller than the first, implying that the incoming flux is absorbed through multiple reflections on the walls. However, this process is so fast that the human eye cannot perceive it, so the interruption of a light source results in an apparent instantaneous darkening.
(ID 139)
Considering the photon flux entering and the one being absorbed, it is possible to calculate how the photon concentration ($n$) varies as a function of the time ($t$) in ERROR:10425. This is represented in the following graph:
indicating that the variation of the photon concentration ($n$) with respect to the time ($t$),
$\displaystyle\frac{dn}{dt}$
will be equal to the incoming flux:
$\displaystyle\frac{S_w I}{\epsilon V}$
involving the variables ERROR:10426, the intensity ($I$), the photon energy ($\epsilon$), and ERROR:10425, minus the loss due to absorption by the walls:
$\displaystyle\frac{1}{6}\displaystyle\frac{S (1-a) c n}{V}$
with the variables the speed of Light ($c$), the albedo ($a$), and ERROR:10424, resulting in the following equation:
| $\displaystyle\frac{ dn }{ dt } = -\displaystyle\frac{1}{6} c \displaystyle\frac{ S }{ V }(1-a) n + \displaystyle\frac{ S_w I }{ \epsilon V }$ |
(ID 15869)
(ID 15874)
ID:(2093, 0)