Understanding the Carnot cycle

Question

My textbook explains the Carnot cycle as follows:

1. Heat is added to the boiler, where the steam inside expands isothermally at high temperature (T(High))--(both valves closed).
2. The intake valve is opened (exhaust valve closed) and the steam expands adiabatically. The expanding steam does work by pushing the piston out.
3. As a result of the expansion, the gas cools to the lower temperature (T(Low)).
4. The exhaust valve opens (intake valve closed). The piston compresses the cool gas and pushes it out to the exhaust (isothermal compression). The condenser removes heat from the steam to the air outside, and steam condenses to lower-temperature liquid.
5. A pump moves water to the boiler (adiabatic compression).

I understand step 1, 2, and 3; however, step 4 and 5 I don't grasp. I know that The first steps entail expanding a gas while keeping it at the same temperature, opening the intake valve in order to use the expanding gas to move a piston and do work, and I understand that, according to the first law of thermodynamics, the internal energy of the gas decreases as a result of this work. However, what is the point of step 4 and 5? Why does the gas need to be compressed? Why can't it just go back to the boiler as is? In addition to answering, could someone, in plain language, explain what steps 4 and 5 are for? For example, I am confused about the point that step 4 says that the gas is already cool because it lost energy from doing work. If so, then why is there a lower-temperature reservoir at all. Also, what does "compress" mean in the contexts of step 4 and 5? What is the purpose of this "compression?"

Answer 1

Here's a hands-wavy explanation, which might help you're intuition.

The alternative to step 4 and 5 would be to vent the steam to the atmosphere. But the steam is still much hotter than the atmosphere, so a lot of energy will be wasted. So you want to recapture that energy, and the logical place to put it is back into the boiler.

However, taking it out of the piston is much easier if there's no back pressure, which is achieved by condensing the exhaust gas. This ensures that the gas on the exhaust side has very low pressure and saves energy by allowing the piston to push out the steam more easily.

Once you've recondensed the steam, you've got hot water, which should be returned to the boiler. However, the boiler is at a higher pressure, so you need a pump to push it back in.