When I studied thermodynamics for the first time I didn't really get much the conceptual understanding on reversibility, but nonetheless I've got a rough understanding and a mathematical criterion for it.
The rough understanding I got was the following: If we consider a process connecting two equilibrium states we might ask whether the inverse process could occur naturally or not. If it can the process would be reversible and otherwise it would be irreversible.
The criterion for a reversible process would be $\Delta S =0$. The whole point is that the entropy maximum postulate states that the entropy must be maximized. If when a constrant is removed a system goes from $A$ to $B$ with $\Delta S > 0$, then certainly $S_B>S_A$, hence naturally the system could never return from $B$ to $A$, because it wouldn't maximize the entropy.
On the other hand if $\Delta S = 0$, we would have $S_A = S_B$ and nothing would prevent the return.
This is a mathematical criteria and more than that, requires the idea of entropy.
My question here is: suppose we are simply given the description of a process (for example: "a piece of hot metal is thrown into cold water" or "a pendulum with a frictionless support swings back and forth"), in that case we don't know the fundamental relation, and there's no mathematics whatsoever here.
In that case, just from a conceptual description of a process how can one judge whether the process is reversible or irreversible? What is a criterion that can be used when we are discussing the process just conceptually without math involved and without any knowledge about entropy?