Converting sunlight to electricity with solar cells has an efficiency of 15%. It's possible to achieve a higher efficiency (though currently at higher cost) by using concentrated sunlight as the hot reservoir of a heat engine. Each dish in the figure concentrates sunlight on one side of a heat engine, producing a hot-reservoir temperature of 500 degrees Celsius. The cold reservoir, ambient air, is approximately 30 degrees Celsius. The actual working efficiency of this device is 30%. What is the theoretical maximum efficiency?
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Well, let's do some calculations, shall we? So, if the hot reservoir is at 500 degrees Celsius and the cold reservoir is at 30 degrees Celsius, we can use the Carnot efficiency formula to find the theoretical maximum efficiency.
Carnot efficiency = 1 - (Tc/Th)
Tc = temperature of the cold reservoir
Th = temperature of the hot reservoir
So, plugging in the values:
Carnot efficiency = 1 - (30/500)
And that gives us...
Carnot efficiency = 1 - 0.06
Wait for it...
Carnot efficiency = 0.94
So, the theoretical maximum efficiency of this device would be 94%. But hey, I guess it's just trying its best at 30%, you know? It's like the underachiever at the efficiency party.
To calculate the theoretical maximum efficiency of the device, we can use the concept of the Carnot efficiency. The Carnot efficiency is the maximum efficiency that any heat engine can achieve given a specific temperature difference between the hot and cold reservoirs.
The formula for Carnot efficiency (η_carnot) is:
η_carnot = 1 - (T_cold / T_hot)
Where T_cold is the temperature of the cold reservoir in Kelvin and T_hot is the temperature of the hot reservoir in Kelvin.
Given that the cold reservoir (ambient air) is approximately 30 degrees Celsius, we first need to convert it to Kelvin by adding 273.
T_cold = 30 + 273 = 303 K
The hot-reservoir temperature produced by the concentrated sunlight is 500 degrees Celsius.
T_hot = 500 + 273 = 773 K
Substituting the values into the Carnot efficiency formula:
η_carnot = 1 - (303 / 773)
= 1 - 0.392
= 0.608
The theoretical maximum efficiency for this device is approximately 60.8% (or 0.608 when expressed as a decimal).
efficiency = 1- Tc/Th= 1- (273+30)/(27+500)
= 1- 0.39 = 0.608
=0.608 x 100 = 60.8 %