Is this a spontaneous or a non-spontaneous process? An insulated tank initially contains a liquid-vapor mixture of water. Then, an electric heater is turned on and kept inside the tank until all the liquid becomes saturated vapor.

According to the 2nd Law of Thermodynamics, a spontaneous process occurs when the change in entropy of the universe is greater than zero.

In this case, since the tank is insulated, $Q=0$ and the change in entropy of the surroundings is also equal to $0$; hence, $\Delta S_\mathrm{universe}$ = $\Delta S_\mathrm{system}$. Since adding work to the system (via the electric heater) results in an increase of the thermal energy of water and its evaporation, $\Delta S_\mathrm{system}&gt;0$. Therefore, $\Delta S_\mathrm{universe} &gt; 0$, and the process is spontaneous.

But wouldn't the necessity of adding work to the system (in order for the water to evaporate) mean the process is non-spontaneous?
Think like this, if the process occurs it is spontaneous.

Without the heating the process is not spontaneous because it does not occur.

With the warm up, you've modified the system. It is this system that is spontaneous (with heating).

Your last statement is right for the unheated system.

Notice, there are two distinct situations.