Hi Willi,
You're right, I missed the part of 'on-demand'.
But you're free to calculate for your own situation; instead of a batch-process you have a continuous process. Should throw in flow-rate, etc. Shouldn't be too hard to calculate, but I'll leave that up to you.
As TomW says, batch or continous process, it's much alike in this case. If you assume a zero thermal leak (i.e., your water wouldn't cool while in storage, or in this case, in transit from your heater to the shower head), you're already almost there, I think.
When you say 'it's obvious that I need more power for on-demand [...] but it wouldn't be continuous' you've got me puzzled...
So you're saying 12V@3A (36W) isn't enough? Okay, let's make it 360W (30A); it would of course only run 1/10th of the time (6.5 hours in my batch example). Total energy required would still be the same: 6.5hrs*30A=195Ahr... Still need the same size battery.
My figures (50kg; 50deg.C) were only for a quick calculation; I have no idea how much water an 'average' shower uses; I don't know what temperature; or how long (I shower 10 minutes; a friend of mine 2 (!!) minutes); etc.
If you really want to size your system correctly you will need to do these kind of calculations; otherwise the risk is very high that your system is too small (cold water) or too large (=too expensive).
(BTW, I realize that this is a mostly American board, but having grown up with metrics & used them through my entire educational career, if I start using feets and gallons I would make too many errors to get at useful results)
Good luck,
Peter,
The Netherlands.