Factoring in the Actual Length of Day for Solar Energy Calculations
It can be shown
that if TD is the length of day - i.e., the time in hours
from sunrise to sunset (not taking into account the refraction of the
earth's atmosphere) - then TD satisfies
cos(TD
/24) = -(tan )(tan
L)
where
is the sun's position north of the equator (the solar declination) and
L is the observer's latitude. (This equation can be obtained by a careful
application of spherical trigonometry that will not be given here.)
Solving for TD gives
TD
= (24/) arccos {(-tan
)(tan L)} hr.
In our previous
article on collecting energy from the sun, the assumption was made that
energy could be collected over a 12-hour period. In fact, this assumption
is not adequate since solar energy can be collected only over the period
of actual daylight. Thus, in the expression
dt
= (12/) dA hr
we should replace
the factor 12 with TD to give
dt
= (TD /)
dA hr.
Making this substitution
gives for the energy E collected over an actual daylight period TD
dE
= D
(sin A) (TD /)
(3,600 sec/hr) dA
= D
(3,600 TD /)
[(sin A) dA]
and
E = D
(3,600 TD /).
Now, let's complete
the midsummer and midwinter calculations again and compare the results
with those obtained previously.
1. As before,
L = 42°. At midsummer,
= 23.5°; at midwinter, =
-23.5°.
2. In midsummer, D
= (137mW/cm2) cos (42° - 23.5°) = 130 mW/cm2.
3. In midsummer,
TD = (24/)
arccos {(-tan 23.5°)(tan 42°)} Hr = 15 hr.
4. The energy collected by our 1 cm2 collector is
E = (130 mW/cm2)(3,600sec/hr)(15hr/)
= 2.2 x 106 mJ.
5. In midwinter, D
= (137mW/cm2) cos (42° + 23.5°) = 56.8 mW/cm2.
6. In mid winter,
TD = (24/)
arccos {(-tan (-23.5°))(tan 42°)} Hr = 8.9 Hr.
7. The energy collected by our 1 cm2 collector is
E = (56.8 mW/cm2)(3,600 sec/hr)( 8.9 hr/)
= 5.8 x 105 mJ.
The ratio between
midsummer and midwinter is
(5.8
x 105 mJ)/(2.2 x 106 mJ) = 0.26
which means that
over an actual daylight period my solar collector actually collects
26% of the energy d in midwinter that it would in midsummer. (No wonder
our winters are soooooooo cold!!!)
Physicists often
make the kinds of approximations that you have seen in these two articles.
The comparison of results between the 12-hour day and the actual length
of day approximations shows the level of care that always must be exercised.