Efficiency of reversible carnot engine must be 0?

Question

In order for heat to transfer in a reversible process, the temperature difference must be approximately zero. Therefore at the isotherms in a carnot engine, there is negligible temperature difference. How does the temperature of the (surroundings) from source to sink then suddenly change after a sudden adiabatic expansion/compression?

If the temperature of the source and sink are constant, then the efficiency of the ideal carnot engine comes to 0. Then how come this is better than the otto cycle? Additionally, I'm slightly confused as to what the temperature of source and sink is in an otto cycle, since the temperature of the surroundings continuously change during the isochoric parts.

To my understanding, the carnot cycle consists of isothermal expansion, where work is done by adding heat to the system, this is done at negligible temperature difference ideally. (rapid) Adiabatic expansion, reducing temperature. Isothermal compression, where heat is taken away and negative work is done, followed by adiabatic compression. It is possible to derive efficiency in terms of temperature by mathematically computing the expression $\frac{W}{Q_h}$ to get $1-\frac{T_c}{T_h}$. But how can $T_c$ be different from $T_h$?

This could be solved by rapidly transferring the engine into another reservoir?

Answer 1

First of all, the adiabatic expansions are not rapid in the ideal Carnot cycle. They are slow and quasi static. An expansion does not have to be rapid in order for it to be adiabatic.

So the way to carry out the cycle is

1. Do the isothermal expansion reversibly and isothermally in contact with the hot reservoir at temperature Th.

2. Remove the cylinder from contact with the hot reservoir and insulate the cylinder. Carry out the adiabatic reversible expansion slowly until the gas temperature reaches Tc.

3. Remove the insulation from the cylinder and put it into contact with a cold reservoir at Tc. Do the isothermal compression reversibly and isothermally in contact with this cold reservoir.

4. Remove the cylinder from contact with the cold reservoir and insulate the cylinder. Carry out the adiabatic reversible compression slowly until the gas temperature reaches Th.

Answer 2

How does the temperature of the (surroundings) from source to sink then suddenly change after a sudden adiabatic expansion/compression?

The short answer is you have two separate surroundings. The temperature of each does not change. The temperature of the system gradually changes during the reversible adiabatic (isentropic) expansion and compression in order to match the temperature of the new surroundings prior to the isothermal compression and expansion, respectively.

Hope this helps.