Thermo - Finite Carnot Cycle

[HalfManHalfAmazing]

Homework Statement

Specify that the finite reservoirs of a Carnot cycle start at initial temperatures $$T_h$$ and $$T_c$$. Acknowledge the consequence of the finiteness of the reservoirs: the hot reservoir will drop in temperature and the cold reservoir will increase in temperature. The two temperatures will converge to a final temperature $$T_common$$, and then the heat engine will cease to function. Take the heat capacities of the two reservoirs to be equal and constant; each has the value $$C_r$$. Assume negligible change in each reservoir's temperature during anyone cycle of the gine.

a) Determine $$T_common$$
b) Determine the total work done by the engine.

Homework Equations

$$C = \frac{Q}{T}$$
$$Efficiency = \frac{T_h - T_c}{T_h}$$
$$Efficiency = \frac{W}{Q}$$

The Attempt at a Solution

I am having trouble understanding how to express the common temperature. If their heat capacities are equal, shouldn't the common temperature be exactly the average of the two initial temperatures?

The total work requires some manipulating the equations but is also related to the initial difference (if there is no initial difference the total work must equal zero). Is the total work simply W = Total Heat - Total Cooling ? Thus $$W = C(T_h - T_c)$$ ?

 

[anathema]

Thank you for your post. I can provide some insight and clarification on the specified scenario.

To answer your first question, the common temperature (T_common) can be determined by using the equation for the efficiency of a Carnot cycle, which is given by (T_h - T_c)/T_h. This efficiency represents the ratio of the work output of the engine to the heat input from the hot reservoir. When the engine reaches its maximum efficiency, the common temperature is the point at which the efficiency is equal to 0. This occurs when the two initial temperatures converge to their average, as you mentioned. Therefore, T_common = (T_h + T_c)/2.

For your second question, the total work done by the engine can be determined by using the equation W = Q_h - Q_c, where Q_h is the heat input from the hot reservoir and Q_c is the heat output to the cold reservoir. The total work is equal to the difference between these two quantities. Since the heat capacities of the two reservoirs are equal, the total work can also be expressed as W = C_r(T_h - T_c).

I hope this helps clarify any confusion. Let me know if you have any further questions or need any additional assistance. Keep up the good work in your scientific studies!

 

[m2003]

Hello, thank you for your question. The concept of a finite Carnot cycle can be a bit confusing, so let me try to explain it in simpler terms. A Carnot cycle is a theoretical heat engine that operates between two heat reservoirs at different temperatures, T_h and T_c. In a perfect Carnot cycle, the hot reservoir remains at a constant temperature T_h and the cold reservoir remains at a constant temperature T_c, resulting in a perfectly reversible cycle. However, in reality, the reservoirs are not infinite and will eventually reach a common temperature, T_common, as a result of the heat transfer during the cycle. This is due to the finite heat capacity of the reservoirs.

To determine T_common, we can use the fact that the heat transferred to the cold reservoir, Q_c, must equal the heat transferred from the hot reservoir, Q_h. We can express this as:

Q_c = C_r(T_common - T_c)

Q_h = C_r(T_h - T_common)

Setting these two equations equal to each other and solving for T_common, we get:

T_common = (T_h + T_c)/2

This is the average of the two initial temperatures, as you mentioned.

For the total work done by the engine, we can use the efficiency equation for a Carnot cycle:

Efficiency = (T_h - T_c)/T_h = W/Q_h

Rearranging this equation, we get:

W = Efficiency * Q_h = Efficiency * C_r * (T_h - T_common)

Substituting in the value of T_common we found earlier, we get:

W = Efficiency * C_r * (T_h - (T_h + T_c)/2)

Simplifying this expression, we get:

W = C_r * (T_h - T_c)/2

I hope this helps clarify the concept and how to solve for the common temperature and total work done by the engine in a finite Carnot cycle. Let me know if you have any further questions. Good luck with your homework!