So I created a DC power supply of 15 V (special transformator connected to the net) and I have a graphite anode and stainless steel cathode. When I put them in a beaker next to each other in water, the electrolysis takes place as I see gas bubbles. However, when i put them in another container that consists of two beakers connected with a tube to allow liquid to flow through but to separate the gases, the reaction doesn't run while it is the same setup more or less. How come? It is true that the tube connecting the beakers is relatively thin and the distance is also quite large (10 cm or sth), so does this lead to a higher required overpotential? I really don't know why it doesn't work in the larger container. Any help is appreciated!
Solution resistance in a two electrode cell can make a big difference as it is totally uncompensated. You may be applying 15 V between the electrodes, but the redox reaction only occurs at a very close proximity to the surface (<10 nm), so it's the interfacial potential we're concerned with, which can be significantly different than the applied voltage if the solution is highly resistive. A tight constriction in the cell can certainly increase the resistance and you should make sure you're using an electrolyte as plain water is very resistive.
Resistance is proportional to length and reverse proportional to сross section of the conductor. When you increase distance by 20 cm from 2-3 cm (10 times resistance) AND indtroduce ion bridge about 1 or less cm in diameter (probably also 10+ times resistance, likely more), you effectively increase the resistance by several orders of magnitude, proportionally reducing the current and the speed of the reaction.
In industrial-grade applications when speed of reaction is of essence, like eletrolitic cutting, the distance between electrodes may be as small as several millimeters. Strong electrolite flow is also used.