I'm kind lost about the principles behind a thermosiphon steam reboiler. I'm mean: when it's necessaty to increase the heat duty, normally, in a steam flow/reboiler level cascade control strategy, the set point of the reboiler level is decreased. The resulting increase of heat duty is consequece of the an larger heat exchange surface, or is consequence of a bigger temperature difference between the steam's temperature (lower as consequence of the smaller pressure - steam expasion due to increase of "free" volume) and wall/boiling liquid's temperature, or is a sum of both effects?

The most common method to vary the heat flux in a thermosyphon reboiler is to use a valve in the steam supply to the reboiler.  Adjusting this valve varies the pressure in the steam chest and since the condensation pressure fixes the condensation temperature you are able to set the temperature driving force.  In this case a steam trap would be used to discharge the condensate while maintaining the back pressure.

Another method, which is less common, is to put the control valve in the condensate outlet.  This works by varying the back-up of condensate in the steam chest, thus effectively making the reboiler a variable-area device since the heat transfer coefficient from the liquid condensate is negligible compared with the HTC for the condensing steam.

Considering the classic heat transfer equation of
Q = U x A x