Heat pumps: can $COP\leq 1$ and when does $COP=\frac{1}{\eta}$ hold?

Question

Heat pump COP is defined as $$\xi=\frac{|Q_{HOT}|}{|W|}$$

Where $Q_{HOT}$ is the heat given to the hot source. I have two linked doubts about it

1. Under what conditions I can say that $\xi=\frac{1}{\eta}$ where $\eta$ is the efficiency of the system if it works as an heat engine (instead of heat pump)?
2. Can $\xi\leq1$?

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$1)$ We have $$\eta=\frac{W}{Q_{HOT}}\implies \xi=\frac{1}{\eta}$$

But is it always possible to talk about an hypotetical heat engine that works between the same sources and exchanges same heats and work?

My guess would be that the conditions is that the heat engine must be reversible. So in that case, $(2)$ would be valid iff the engine is reversible. Is that correct or is the condition different?

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$2)$ Looking at point $1$ it seems impossible that $\xi=1$ because that would mean $\eta =1$. More in general it should be $\xi \geq 1$ (since $\eta \leq 1$).

But consider the heat pump in the diagram. I listed the trasformations below. $$A \rightarrow B= \mathrm{reversible \, isothermic \, compression \, from \, V_A \, to \, V_B}$$ $$B \rightarrow C= \mathrm{adiabatic \, expansion \, from \, V_B \, to \, V_C}$$ $$C \rightarrow A= \mathrm{ (irreversible) \,adiabatic \, compression \, from \, V_C \, to \, V_A}$$

(The blue curve is an irreversible process and thus should not be represented as I did)

We have $Q_{cycle}=Q_{A->B}<0$ so $W_{cycle}=Q_{A->B}<0$ so in this case $\xi=1$. And this heat pump seems possible to me because we have $\Delta S_{universe}=-\Delta S_{A->B,system}>0$.

In the end this is just a system that takes work and trasform it entirely in heat, which is (as far as I understood) allowed by the laws of thermodynamics.

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I would really appreciate any suggestion on these two points on heat pumps which are not clear to me.

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Just to clarify the sign convention I use is the one in picture

Answer 1

Say there are two heat reservoirs, one at higher temperature than the other. Heat pump extracts heat $Q_{cold}$ from colder reservoir, adds energy due to work $W$ to it and dumps the total $Q_{hot}=Q_{cold}+W$ into the hotter reservoir. An engine works in the opposite sequence: It extracts $Q_{hot}$ from hotter reservoir, converts part of it into work $W$, and rejects the rest to colder reservoir $Q_{cold}=Q_{hot}-W$, as is required by second law.

Now for heat pump $COP=\frac{Q_{hot}}{W}$ and for engine $\eta=\frac{W}{Q_{hot}}$, and for given $Q_{hot}$, only when both of them run on reversible cycle will they be equal. Otherwise, for a given $Q_{hot}$, $W$ for heat pump is always more than that for engine.

Now $COP=1$ is possible, in fact it is the worst case scenario (the most inefficient heat pump). But $COP<1$ is not possible because it violates energy conservation. You have extracted $Q_{cold}$ and added $W$ to it, so what you dump into hot reservoir must necessarily be equal to sum of the two and therefore always $\geq W$.