Kipp&Zonen CM 121 B/C Shadow Ring - Manual (Page 24)

6. APPENDIX
Instruction manual CM 121 23
6. APPENDIX
6.1 Theoretical derivation of the correction factor, for uniform sky radiation
The relation between the correction factor C and the intercepted part S of the downward
component of the sky radiation is
C = 1 / (1 -S) (6.1)
S can be expressed in the view angle V of the ring, the sun's declination D and the latitude B of the
observation site.
Let U
0
be the angle between the sun at sunrise (or at sunset) and the sun at true noon in the plane
of the ring.
U
0
is computed with the formula:
cosU
0
= - tanB . tan D (6.2)
Let us consider a part of the ring subtending a solid angle V.dU as seen from point M, the center of
the ring, and as seen from the pyranometer of
V . cosD . dU (6.3)
V is assumed to be constant within the range of D, due to the special U-profile shadow ring.
Not every part of the ring equally affects the total downward component of the sky. This is because
this component is proportional with the cosine of its zenith angle Z. Radiation with a zenith angle Z
within this solid angle causes a downward component
L . V . cosD . cosZ . dU (6.4)
L is the radiance (= brightness) of the sky in W/m2 sr. L is assumed to be uniform over the
complete sky. So the complete ring part above the horizon intercepts a downward corn-
ponent
2L . V . cosD .
0
I
U0
cosZ . dU (6.5)
Z can be expressed in the declination D, latitude B and time (by the hour angle U) with the
formula
cosZ = sinB . sin D + cosB . cosD . cosU (6.6)
After integrating we find the total intercepted downward radiation to be
2L . V . cosD . (U
0
. sinB . sinD + sinU
0
. cosB . cosD) (6.7)
The total irradiance of a horizontal surface by a sky with radiance L (W/m2 sr) is L (W/m2).
The intercepted part S of the sky radiation therefore is
S = 2V . cosD . (U
0
. sinB . sinD + sinU
0
. cosB . cosD) / π (6.8)
From this formula the list of correction factors is computed taking into account formula (6.2)
and (6.1) and V = 0.185 rad. Mind that U
0
is in radians.
Actually V varies within 2% in dependence of the declination D but this gives only rise to an error
less than ± 0.5% in the calculated correction factors.