# What is the equilibrium temperature of a collector plate?

## Homework Statement

You would like to put a solar hot water system on your roof, but you're not sure it's feasible. A reference book on solar energy shows that the ground-level solar intensity in your city is 750 W/m^2 for at least 5 hours a day throughout most of the year.
Assuming that a completely black collector plate loses energy only by radiation, and that the air temperature is 20 ∘C, what is the equilibrium temperature of a collector plate directly facing the sun? Note that while a plate has two sides, only the side facing the sun will radiate because the opposite side will be well insulated.

## Homework Equations

dQ/dt=eσAT^4
T=(T_h^4-T_c^4)
e=5.67E-8 W/m^2*K[/B]

## The Attempt at a Solution

[/B]
Okay, so I know that Power is equivalent to dQ/dt. The emissivity(e) is equal to 1 because it is an ideal black collector plate. However, the question does not have the dimensions of the plate, so I am assuming it is 1. I know that I have to find the Hot temperature.

P=eσA(T_h^4-T_c^4)
P=eσAT_h^4-eσAT_c^4
P+eσAT_c^4=eσAT_h^4
P+eσAT_c^4/(eσA)=eσAT_h^4

(P+eσAT_c^4/(eσA))^1/4=T_h
OR
T=((P/Aeσ)+T_c^4)

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CWatters
Homework Helper
Gold Member
However, the question does not have the dimensions of the plate, so I am assuming it is 1.
You don't need to know the area because the solar intensity is given per square meter.

Otherwise all looks ok.

gneill
Mentor
P=eσA(T_h^4-T_c^4)
P=eσAT_h^4-eσAT_c^4
P+eσAT_c^4=eσAT_h^4

P+eσAT_c^4/(eσA)=eσAT_h^4
##~~~~~##
(P+eσAT_c^4/(eσA))^1/4=T_h
OR
T=((P/Aeσ)+T_c^4)
Can you explain what you've done on the indicated line? To me it looks like you've divided one term on the LHS of the equation by eσA and done nothing to the RHS. Always be sure to use enough parentheses to disambiguate the order of operations and grouping of terms. On the last line you've left out taking the quartic root of the RHS.

Other than the above probably typographical issues, it looks like you're on the right path. What value do you calculate for the temperature of the plate?