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Is possible estimate the needed size of an geodesic dome (like in the Eden project) for creating an real hydrosphere - especially clouds (and rain)?

With other words, under what circumstances can happen clouds formation in an closed space -isolated from the outer atmosphere?

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The most important physics with respect to cloud formation happens in what is called the Atmospheric Boundary Layer (ABL). A lot of research is done in this field, since the effects of clouds is the major source of incertainty in all climate prediction models.

To get some sort of cloud formation, in my opinion you would need to have some kind of ABL inside of your dome. The typical height of this ABL is ~800m (in the morning) up till 2km (in the evening). So if your dome is that large, you will be able to form clouds. But it is not that easy: heat fluxes (representing radiation), temperature gradient, humidity, pressure, condensation, everything should be controlled in the dome to spontaneously form these clouds.

Another aspect to be aware of is the size of the base of the dome. This has to be larger than the largest turbulent structures in this domain, so to be sure to let these structures develop, it should be at least a few times the height of the dome.

If this dome is too large for you, it is probably possible to make clouds in a smaller dome, but then you still need to capture the ABL. To achieve this, you need to change the conditions: increase temperatures and temperature gradients, replace the air by some other fluid, etc etc. It will in no case be pleasant for people. Actually, some experimental physicists have been able to model this ABL in a water tank (~1m³). However, cloud formation was off course impossible to achieve in such a experimental set-up.

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