The question as it appears in the 05/22 edition of Specs & Techs from GlobalSpec:
You've built two solar hot water panels. Each panel is virtually identical  a 1 m square (approximately) aluminum sheet painted black, with a copper tube (also painted black) zigzagging across the surface, and contained in a hardwood frame with glass over the face. However, you misjudged the length of copper tubing available; one panel (Panel A) has nine zigzags, while the other panel (Panel B) has 13. Therefore, the panels will not perform exactly the same. The panels are plumbed in a series. Should the water be pumped to flow through Panel A and then Panel B, or the reverse?
Thanks to Del the cat who submitted the original question (which we revised a bit).
(Update: May 29, 9:21 AM EST) And the Answer is...
Let average temperature of water in the 1^{st} panel = T
Let the average temperature of water in the 2^{nd} panel = T+4
(The pump starts at an 8 degree differential between hot water cylinder and solar panel outlet pipe)
Let the mean surface temperature of panels=k
Efficiency is proportional to number of copper tube zigzags
(9 in A 13 in B) so,
Let the efficiency of panel A = E
Therefore the efficiency of panel B = 13E/9
The heat gained will be proportional to the temperature difference between panel and water multiplied by the efficiency.
Running water through panel A then B
Heat H(AB) gained will be:
H(AB) =(kT)*E + (k(T+4))*13E/9
= E(kT+ 13k/9 13T/9  52/9)
= E( 22k/9 22T/9  52/9 )
= E/9 (22k22T52)
= E/9 ( 22(kT) – 52)
Heat gained running water through B then A
H(BA) = (kT)* 13E/9 + (k(T+4))E
= E ( 13k/9 – 13T/9 + k – T  4)
= E (22k/9 – 22T/9  4)
= E/9 ( 22k22T  36)
= E/9 ( 22(kT) 36)
E/9(22(kt)… is common to both expressions and thus it can be seen that H(BA) is greater than H(AB)
Let's put some figures in to get a feel for the effect.
Ignoring the constant E/9
If we take k as 90 degrees and T as 55 degrees (I think these are reasonable, certainly in England in May)
H(AB)= 77052 = 718
H(BA)= 77036 = 734
This represents about a 2.5% difference.

Re: Solar Hot Water Panels: Newsletter Challenge (05/22/07)