[GpsFrontier]

Do I need any special thermometer for it? I am not sure what the impact of the nutrient solution is on a "normal" thermometer.

No special thermometers needed. I use a fish tank thermometer myself, I got it at a local store (WalMart) for under $2 US dollars. A glass one, not the kind that stick to the side of the tank, I don't think those are accurate enough (especially over time). It does have a suction cup on it to stick it to the glass, but it's a regular glass thermometer. I don't leave it in all the time, it only takes about 5 min to get a good reading. I only check it when I have a concern. I just tied a string on it so I can drop it in and retrieve it easily. P.S. I store it in a empty plastic spice container to protect it from breaking.

About the bucket system: if I understand correct, in the lower bucket I place a piece of tube that will determain how much water will stay in the bottom acting as reservoir. However the top bucket will be above that level, so I assume I fill up the bottom bucket with hydroton pebbles so it reaches the top bucket? If not: please explain how the plant could wick up the solution.

Well there are really two ways to run it, either a flood and drain (ebb & flow) or a drip system. I am suggesting running it as a drip system, because it would be less complicated, and use a lot less nutrient solution than it would take to flood a flood and drain system. You are correct about the bottom bucket reserving nutrient solution at the level you decide. However running it as a drip system the pump feeds the growing medium from the top, making it's way down through the growing medium, and draining into the bottom bucket. The tube height in the bottom bucket is determined by the overflow tube. Once it overflows the tube it drains down into the PVC tubing that takes it back to the reservoir where (one pump) recycles it back. Each bucket will have it's own reserve, but all of the plants would be feed by one pump.

The size of the buckets, the placement of the bottom overflow tube are both key to the design. And are reliant on what materials you have to work with. Once the root system becomes large enough it will grow down into the bottom bucket where the reserve nutrient solution is. but if the top bucket is smaller than the top one, and there is enough space between both buckets, you can raise the overflow tube high enough so the reserve water level is soaking the bottom inch or two of the growing medium. This will wick up throughout the growing medium. It's all really about what you have to work with, and how you ultimately can design it. I can maybe do a quick image of what I mean tomorrow to give you a better idea. It is basically a DWC setup, but using much more growing medium to hold moisture. I don't see much difference between a dutch bucket system and a DWC system other than that. I believe a typical dutch bucket system is feed by a drip system, but like I said it can be run like a flood and drain also, just more complicated (fittings, running lines etc.).

How to determine the reservoir size? It seems to depend on the
Number of plants x
Bucket size x
Average uptake per day x
Ammount that keeps the pump from running dry

You are right here, all factors you need to consider, but I might add that as the plants get bigger they will suck up more water daily. Also they will suck up more as it becomes hotter. I don't have any magic formula (or even know where to find one). But as a comparison I was using a 18 gallon reservoir for my broccoli plants (by the way that is the bucket system in the earlier pictures, reservoir is under the table), I started out using just 8 gallons. But as the plants got bigger they were drinking much more water daily, I was replacing between 2-4 gallons a day. Many times it got to low and the pump was going dry, so I filled it to the full 18 gallons.

This gave me a buffer so I wasn't checking water level, and pH etc. twice a day. But as the water level fluctuates, so does the concentration of the nutrients in it (PPM/TDS), in the long run the bigger the reservoir the better. But also you don't want to waist nutrients either. This is part of the learning curve, and why I suggest starting small first so you can experience these things first hand.