The The azimuthal angle and the angle of

The
solar incidence angle ? (in degrees) is a function of the angle of solar
declination ? (in degrees), the latitude of the location of the photobioreactor
? (in degrees), the inclination of the photobioreactor with respect to the
ground ? (in degrees), the azimuthal angle ? (in degrees) as well as the solar
hour angle ? (in degrees) 18. The solar
incidence angle is given by the following equation

For
two sides of the photobioreactor, the angle of incidence is calculated based on
the solar azimuthal and inclination angle for the front and back sides of the
reactor. The azimuthal angle and the angle of inclination of the reactor, for
the front and back side, are computed using the following equations:

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The
angle of solar declination ? is computed by the following
equation:

where,
N is the number of the day in the year.

The solar
hour angle ?, is given by the following equation:

where, 

 is
a function of the actual time t (h), longitude of the location of the reactor ?
(in degrees), meridian of the location of the reactor ? (in degrees) and the
equation of time e. The solar time (

 is
computed by the following equations:

The
solar zenith angle

 (in degrees) and the angle of elevation of the
Sun

 (in degrees) are complementary to each other
and the calculated by the following equations:

The
azimuthal angle ? (in degrees) is given by the following
equation:

In
order to model the solar irradiance on flat panel photobioreactor the direct
and diffuse solar irradiance are taken into account. As solar irradiance data
are measured perpendicular to the surface of the earth, geometric factors are
introduced to obtain solar irradiance on the front and back sides of the
photobioreactor based on its inclination with respect to the surface of the
earth. The front side and back side geometric factors for the reactor for
direct solar irradiance are computed by the following set of equations:

The
geometric factors for the diffuse solar irradiance are a function of the angle
of inclination of the reactor with respect to the ground, ? (in degrees). The
geometric factors for the front side and back side of the reactor for diffuse
solar irradiance are computed by the following equations:

The
geometric factors for the ground reflected diffuse solar irradiance for the
photobioreactor is a function of the reflectivity of the ground surface ?.
Following the same approach as above the geometric factors are computed by the
following equations:

The
total solar irradiance on the front and back side of the photobioreactor (

) (W/m2) is computed by the
following equations:

 

Light distribution in parallel flatpanel
photobioreactors

In
large scale microalgae cultivation in photobioreactors often a series of
reactors are placed parallel to each other. Such a configuration results in
shading and significant part of the reactor surface is unable to receive direct
solar irradiance. The height of shadow on vertical photobioreactor panels is
given by the following equation:

where,
h (m) is the height of the reactor,

 (m) is the distance between the parallel
reactor panels and

 is
the solar zenith angle (in degrees). In order to perform the simulation the
panel is divided into two parts. The upper part of the panel receives both the
direct and diffuse solar irradiance whereas the lower part receives only
diffuse solar irradiance. The separation between these two parts depends on the
solar zenith angle and it is computed for every time step during the day time. Parallel
positions of the photobioreactors also influence the penetration of diffuse
solar irradiance in the space between the panels, where the intensity decreases
from the top to the bottom. Thus, the geometric factors of diffuse solar
irradiance for the front and back side of the reactor panels become a function
of height and are given by the following equations: