• nineeyes
In summary, the problem discusses an engine in outer space that operates on the Carnot cycle and transfers heat through radiation. The rate of heat transfer is proportional to the fourth power of the absolute temperature and the area of the radiating surface. It is shown that for a given power output and a given high temperature, the area of the radiator will be at a minimum when the low temperature is three-fourths of the high temperature. The Stefan's Law is used to find the net energy radiated by the engine and then differentiated to find the minimum. The differentiation can be done with respect to either the low temperature or high temperature.
nineeyes
Problem :
Consider an engine in outer space which operates on the Carnot cycle. The only way in which heat can be transferred from the engine is by radiation. The rate at which heat is radiated is proportional to the fourth power of the absolute temperature and the area of the radiating surface ($$Q_L$$ is proportional to $$A(T_L)^4$$). Show that for a given power output and a given $$T_H$$ the area of the radiator will be an minimum when $$T_L/T_H=3/4$$ .

I was guessing I need to try to show Q_L is a minimum using the given ratio. I can find the efficiency but after fooling around with it a few times in some equations I haven't come up with much, I generally have problems when few numbers are provided.

Any hints that can be provided would be great, Thanks!

OK This is simple.

First of all I would like to tell you that , if a body is at temperature T ,it radiates heat energy(E) given by:

$E=esAT_L^4$

where T_L is the temperature of the engine.

Now outside temperature is T_H

Now amount of energy radiated by the engine reduces because Outside region also supplies some energy into the engine.Therefore now the net Energy radiated becomeS:

$E=esA ( T_L^4 - T_H^4)$
s in above equation is the stefan's constant.And the above equation is the Stefan's Law.

Now differentiate it to get the minima...You will get the answer.Easy isn't it?

Last edited:
Hi,
Sorry, but we have not yet encountered this equation in my class. I was wondering do I differentiate with respect to A? If I do, doesn't that just eliminate the A from the equation? I was thinking I would need to somehow solve for A , in terms of T_H and the Power Output.
Sorry if I misunderstood what you meant.
Thanks for the help.

Differentiate it w.r.t $T_H$ or $T_L$.

Do you know we can find the maxima or minima of an expression by simply differentiating it ?...The same concept we apply to the above problem.

1. What is a Carnot Engine?

A Carnot Engine is a theoretical heat engine that operates on the principle of reversible adiabatic and isothermal processes. It serves as a model for a perfect engine and is used to determine the maximum efficiency of any heat engine.

2. How does a Carnot Engine work?

A Carnot Engine works by taking in heat from a high temperature reservoir, converting some of it into work, and then releasing the remaining heat into a low temperature reservoir. This process is repeated in a continuous cycle to produce work.

3. What is the efficiency of a Carnot Engine?

The efficiency of a Carnot Engine is given by the Carnot efficiency formula: Efficiency = (Th - Tl) / Th, where Th is the temperature of the high temperature reservoir and Tl is the temperature of the low temperature reservoir. This means that the efficiency of a Carnot Engine is always less than 100% and is dependent on the temperature difference between the two reservoirs.

4. What are some real-life applications of the Carnot Engine?

The Carnot Engine is a theoretical model, but its principles are applied in various real-life applications such as refrigerators, air conditioners, and heat pumps. These devices use the Carnot Cycle to transfer heat from a colder environment to a hotter one, thereby cooling the colder environment.

5. What are the limitations of the Carnot Engine?

The Carnot Engine is a theoretical model and cannot be built in reality. It also assumes the absence of any friction or heat loss, which is not possible in real-world systems. Additionally, the efficiency of a Carnot Engine decreases as the temperature difference between the two reservoirs decreases.

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