What boils faster? At work we got a new kettle recently. It's a fancy one with a water filter in it. You fill up the top half of the jug and it goes through a Brita filter into the bottom half of the jug where the heating element is.
It got me wondering what would be faster to boil the water.
Given a fixed volume of water and a kettle that provides a fixed amount of energy over time...
Would the water boil quicker (I.e. Reach 100 degrees and turn the kettle off) if you were to wait for all the water to filter through before turning the kettle on. Or if you were to turn the kettle on as soon as the water started filtering.
By quicker here I mean the time between turning the kettle on and it turning itself off. Not bothered about the time before turning the kettle on.
 A: I haven't done a thorough analysis, I am basing this more on a qualitative analysis (and honestly, in this case I don't think that's going to be the best, any differences are going to be minimal compared to the time scale of the heating, so in reality any small factor I forget to consider could completely change the answer).
Hopefully this will address some of the complexity of this.
On paper obviously you need to add the same amount of energy to heat the same mass of water up to the same temperature. Since we have a constant energy source, they should take the same amount of time. 
That said, the reality is it will not evenly heat. Adding the water slowly would potentially force mixing as you go. It also may interrupt the natural convection of the fluid. I'm not sure if the forced mixing would help or hurt the convective currents.
My thought is that slowly pouring the water will help.  Mainly because the natural convective currents are trying to force the new cool water downwards, the same as the pouring water would be doing.  This helps to maximize the temperature difference at the heating element for better heat transfer.
The other slight advantage of pouring water is you would be constantly generating heat from the fluid friction.
For those reasons I'd say it's marginally better to heat while pouring.
A: Kettle's fuse goes off in the very beginning, where is not enough water and the kettle heats up too much.
Since this is not a response you expected, I have a second one. In principle - boiling should take a little more time with the water slowly pouring through the filter, because of a higher heat loss
Here is a small 
simulation that can be actually found at http://nbviewer.jupyter.org/urls/dl.dropbox.com/s/306bc6dwa59ea6d/konvice.ipynb (if you find an error, we can correct it)

The x-axis is time, y-axis is temperature, red line is with slowly pouring water, blue line is a full kettle from the beginning.
Check the (red) stable plateau of the temperature for 50 sec.  - it is a balance until all water gets inside the kettle (1 litre here). This part can influence the boiling time, because just here there are higher losses.
For curiosity, I have added Stefan-Boltzman law for heat irradiation losses, but no real difference is visible. There could be also losses by air-cooling, I dont think they would be also important, it is just a limited time at 45 degrees Celsius. 
Rule of thumb - mixing is so good, that temperature inhomogenities I dont think they play any role.
