# What is the equilibrium temperature of a collector plate?

• Galactium
In summary, to calculate the equilibrium temperature of a collector plate directly facing the sun, we use the equation P=eσAT_h^4-eσAT_c^4, where P is the solar intensity, e is the emissivity (equal to 1 for an ideal black collector plate), σ is the Stefan-Boltzmann constant, A is the area of the plate, and T_h and T_c are the hot and cold temperatures, respectively. We can rearrange the equation to solve for T_h, which gives us T_h=((P/Aeσ)+T_c^4)^1/4. Since the solar intensity is given as 750 W/m^2 and the air temperature is 20 ∘C, we
Galactium

## 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)

Galactium said:
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.

Galactium said:
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?

## 1. What is the definition of equilibrium temperature?

Equilibrium temperature is the temperature at which a system reaches thermal equilibrium, meaning there is no net flow of heat between the system and its surroundings.

## 2. How is equilibrium temperature determined for a collector plate?

The equilibrium temperature of a collector plate is determined by considering the energy balance of the system, taking into account factors such as incoming solar radiation, heat loss to the surroundings, and the properties of the collector plate itself.

## 3. What factors can affect the equilibrium temperature of a collector plate?

The equilibrium temperature of a collector plate can be affected by various factors, including the intensity of incoming solar radiation, the material and design of the collector plate, and the presence of any insulation or shading.

## 4. Why is the equilibrium temperature important for a collector plate?

The equilibrium temperature is important for a collector plate because it affects the efficiency of the system in converting solar energy into heat. By optimizing the equilibrium temperature, the collector plate can operate at its maximum efficiency and produce the most energy.

## 5. How can the equilibrium temperature of a collector plate be controlled?

The equilibrium temperature of a collector plate can be controlled by adjusting factors such as the angle of the collector plate, the material and design of the plate, and the use of insulation or shading. Additionally, the use of a tracking system to follow the movement of the sun can also help control the equilibrium temperature.

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