Ok, so the question is about the concept of increasing entropy. We obtain the result (by utilizing the Clausius inequality theorem) $dS = \frac{dQ_{rev}}{T} \geq \frac{dQ_{irr}}{T}$.
Then it's stated that for a closed system, $dQ_{irr}$ is zero and therefore $dS \geq 0$. Fair enough, the total energy in a closed system is constant and therefore no heat (thermal energy in transit) can flow in or out. The thing that bothers me though, is that I cannot imagine any process where an amount $dQ_{rev}$ can transferred to or from the system if the system is closed. And that leads me to $dQ_{rev}=0$ as well which results in $dS=0$ .
Now I know that there is a problem, as entropy is in fact generated when heat flows between subsystems in the isolated system, which one could calculate. The problem originates from the statement that $dQ_{irr}$ is zero. In the isolated system an irreversible amount of heat can (and will be) transferred between subsystems of different temperatures. Even though the net heat transfer is zero, $\frac{dQ_{irr}}{T_1} + \frac{-dQ_{irr}}{T_2}$ should also be zero in an isolated system with two subsystems of temperatures $T_1$ and $T_2(>T_1)$ for this to work out, which is incorrect. The explanation is much appreciated.
Kind regards!
