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
- 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)?
- Can $\xi\leq1$?
$1)$ 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?
$2)$ 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.
I would really appreciate any suggestion on these two points on heat pumps which are not clear to me.
Just to clarify the sign convention I use is the one in picture