Well this is a very odd question. Is it a trick question or extra credit for a class?
Keep in mind I’m more than a decade out of my ME classes but from what I recall…
The first problem I see is in the wording of the question. A Carnot refrigerator is a refrigeration cycle using a Carnot cycle. Well if you understand a refrigeration cycle, it is essentially a heat pump; it transfers heat from inside the box to outside. So keeping that in mind, you are asking if a heat pump can power another heat pump. That really doesn’t make sense because a heat pump cycle does not produce energy, it merely transfers heat from one “location” to another. In theory a Carnot cycle states that the energy put into the cycle can be 100% recovered so that in essence there is no net gain or loss in enthalpy or energy. So back to your question, a Carnot refrigerator doesn’t produce energy so it can’t necessarily power anything. Consequently, a true Carnot heat pump doesn’t even require power or energy input so it wouldn’t need a power source to begin with. You can toss the whole Carnot refrigerator part out of the question because in reality it doesn’t apply. Or if you want to look at it macroscopically, two true Carnot systems is essentially one big Carnot system. Depending on how you want to look at, you could either view it as a system that transfers heat to one location which is then transferred to another location. But since there is no loss in energy then the first system could have transferred the heat; there was really no need to break it up into two cycles even though there is no penalty to do so. The other way to look at it is as two perfect heat pumps, one that removes heat from a system while the other puts it back in. In the end, it’s just a big circle that goes nowhere.
The second part of the question regarding the 2nd law of thermodynamics is pretty much answered in your question. The 2nd law states that entropy must never decrease; it must increase or in an ideal situation stay the same. A realist’s take on the 2nd law would say that in nature, there is no such thing as a perfect system so that entropy must always increase. A Carnot cycle, almost by definition, is a completely reversible cycle, enthalpy and entropy stay constant (doesn’t change). So in “theory” any Carnot cycle or any combination of Carnot cycles obey the 2nd law because the entropy does not decrease. Now a realistic take is that it does not realistically obey a real world take on the 2nd law because no system exists in nature that is a perfect Carnot cycle. Hence, all this is essentially covered when someone says, hypothetically say you had a completely frictionless and 100% efficient “insert object, like engine or heat pump here”.
So in summary, the first part about a Carnot refrigerator powering a heat pump doesn’t make any sense because a Carnot cycle doesn’t produce, nor require power. In terms of heat transferring, you are either going to move heat in a circle or to a location where a single Carnot cycle could have done the same thing, so the splitting up of the two cycles offers nothing but to convolute the second part of the question. The second part of the question is answered simply by understanding what the 2nd law of thermo states and what a Carnot cycle is in definition. In a hypothetical and theoretical sense, a Carnot cycle is the ideal 1st and 2nd law cycle. It’s basically a 100% efficient cycle as defined by the 1st and 2nd laws of thermo. Meaning if I were to obey these two laws, this cycle is the theoretical best I could do. So in definition, it must obey these laws. However, remember, in reality there is no such thing as a Carnot cycle, so this cycle complies with the 2nd law in theory only.
