Factoring in the Actual Length of Day for Solar Energy Calculations

It can be shown that if T_D 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 T_D satisfies

cos(T_D /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 T_D gives

T_D = (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 T_D to give

dt = (T_D /) dA hr.

Making this substitution gives for the energy E collected over an actual daylight period T_D

dE = _D (sin A) (T_D /) (3,600 sec/hr) dA
= _D (3,600 T_D /) [(sin A) dA]
and
E = _D (3,600 T_D /).

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,
T_D = (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,
T_D = (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.