# Thermo - Finite Carnot Cycle

1. Jan 31, 2007

### HalfManHalfAmazing

1. The problem statement, all variables and given/known data
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 any one cycle of the gine.

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

2. Relevant equations

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

3. 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 $$W = C(T_h - T_c)$$ ?

Last edited: Jan 31, 2007