Why is there a cold reservoir in a Carnot engine?

I'm trying to wrap my head around the idea behind a Carnot engine.

The main question I'm trying to figure out is how a Carnot engine actually works in producing work.

I'll provide a picture for reference.

Specific questions I need addressing are:

• How gas doing work on its environment benefits us. For instance, how does this power a steam train or something?

• Why an adiabatic compression is used instead of just doing an isothermal compression the whole time so as to keep a high temperature.

• How the working substance delivers $$Q_{cold}$$ to the cold reservoir yet its temperature remains the same. It sounds like its losing heat from the reservoir, but its temperature stays the same. Is this due to the compression? It sounds also like the cold reservoir just prevents some of the heat being converted to work. It seems like it's purpose is only to prevent the engine from being perfectly efficient.

• Why a path of $$cd$$ then $$da$$ is necessary, hypothetically, instead of just drawing a slightly curved line from $$c$$ to $$a$$ (so that it's not just doing $$ac$$ in reverse), calling it path $$ca$$, and drawing it such that the positive work is as small as possible. It may not be possible on physical grounds, but I can't really explain to myself why we couldn't just do that. Or after $$c$$, drawing an isobar until the gas is at the same volume as it was at $$a$$, and then drawing an isochor back up to $$a$$, which looks like it would minimize positive work. Is this just the only physically possible path?

• If net work is produced by the process, why can it not be used to continue the process indefinitely and produce effectively an infinite amount of work?

I just find the whole process extremely vague to me, especially considering how I'm meant to wrap my head around a bunch of processes with a finite work integral as an engine. If possible, I'd appreciate a response to each bullet.

The short answer to the title question is a cold reservoir is needed to get you back to the original state and complete the cycle while performing net work.

Taking each of your questions (all my comments on processes assume the processes are reversible), and breaking up some of the bullets because of multiple points covered:

• How gas doing work on its environment benefits us. For instance, how does this power a steam train or something?

The cycle can perform work as part of a steam power cycle, such as to operate a turbine, or operate as a piston/cylinder in a reciprocating engine, etc. There is no limitation, other than the fact that it is very impractical cycle. That is because for the two isothermal and adiabatic processes to be reversible, the processes must be carried out infinitely slowly. So while the cycle is the most thermally efficient possible heat engine cycle (work out divided by heat in), the rate of work (power) is too slow.

Bottom line, the Carnot Cycle serves to set an upper limit on thermal efficiency but is too impractical to actually use. Someone once said, if you put a Carnot engine in your car you would get fantastic gas mileage, but pedestrians would be passing you by!

Why an adiabatic compression is used instead of just doing an isothermal compression the whole time so as to keep a high temperature.

The first process is an isothermal expansion. That process takes heat in and does work out. Now if you wanted to reverse the process and do an isothermal process to get back to the original state, you would need to do an isothermal compression at the same temperature as the expansion. For that you need to do work on the gas and transfer heat back out to the high temperature reservoir. The compression work equals minus the expansion work so the net work done would be zero. The only way to do net work is to follow the isothermal expansion with an adiabatic expansion to get the temperature down to the cold temperature reservoir.

How the working substance delivers $$Q_{cold}$$ to the cold reservoir yet its temperature remains the same.

The temperature of the gas approaches that of the cold reservoir in the limit (that’s what makes it reversible) but for any real process there must be a finite temperature difference in order for heat transfer to occur.

It sounds like its losing heat from the reservoir, but its temperature stays the same. Is this due to the compression?

Heat is transferred TO the cold reservoir, not the other way around. And yes it is transferred to the cold temperature reservoir by isothermal compression.

It sounds also like the cold reservoir just prevents some of the heat being converted to work. It seems like it's purpose is only to prevent the engine from being perfectly efficient.

Exactly. The necessity of the cold reservoir to complete the cycle necessitates that some of the original heat to the system must be rejected by the system to the cold reservoir. But it’s not its purpose to prevent 100% efficiency. The second law of thermodynamics prohibits 100 % efficiency in a thermodynamic cycle. That is, it prohibits taking heat from a single reservoir and entirely converting it to work in a thermodynamic cycle.

Why a path of …..

Without trying to address all of the if, and, ors, the short answer to this question is that any paths other $$cd$$ followed by $$da$$ will result in a lower thermal efficiency. To convince yourself, try other paths and calculate the efficiency and you will see it is always less.

If net work is produced by the process, why can it not be used to continue the process indefinitely and produce effectively an infinite amount of work?

Not sure what you are getting at, but the cycle can be performed an infinite amount of times and you would get an infinite amount of work. But for each cycle you will need to get heat in and reject heat out, so the efficiency will always be the Carnot efficiency.