Ian,
Beyond the suggestion of having a larger exit pipe diameter than the inlet pipe, I see from your pix that the exit from the second barrel should have a vent system. All the other pipes are naturally self-venting by nature of the way you plumbed them (air can escape at the same moment the water does), but the exit from the second barrel comes straight up and goes nearly straight back down. This leaves an air trap in the horizontal pipe section where it goes through the barrel wall. I would install a Tee fitting in this pipe in place of the ell fitting on the outside of the barrel with the top of the tee left open. Add a short, vertical extension or riser pipe to the open (top) end of the tee to prevent any overflow there, extend this riser just above the top of the barrel.
The next part of the system that needs to be addressed is the Black Box filter. I do believe that you need to supply water to this filter via a pump. I don't know if gravity supply is going to cut it. Since it has spray bars internally, it will require some measure of positive pressure, probably beyond what your gravity feed system is capable of supplying. The water is just going to "dribble" out of the spray bar/s otherwise and your second barrel will overflow, as you reported.
You might need to install a pump on your second barrel's outlet for this, or, before trying that.... Try installing your exit pipe from the second barrel at the bottom of the barrel, and don't plumb it with any rise. Now, you will have the pressure of the height of the entire water column in that second barrel acting almost as a pump outlet. Possibly, the barrel won't fill up to full capacity because the outlet is going to be flowing right out the bottom. However, since your Black Box filter requires some back pressure to operate its spray bar/s, the barrel may actually fill to the optimum level to satisfy this. This is definitely a "maybe" situation as I don't know what the recommended pressure of the Black Box filter is, but it would be a relatively simple experiment to find out the answer, without purchasing an additional pump.
As it stands right now, you only have the pressure from the weight of a 40mm diameter column of water 75 to 90 cm in height (just the water which is in the vertical section of the pipe leaving the second barrel) feeding your Black Box filter and you also have the problem with the air trap as well. This is obviously not sufficient pressure nor volume to operate the spray bar "nozzles" and prevent your second barrel from overflowing.
As for the sizing of inlet vs outlet plumbing, you don't always have to use larger outlet pipes than inlets. Explanation: If you are gravity feeding a series of three barrels, the same pressure and volume of water entering one barrel and leaving it is virtually the same throughout the chain if there is no filter media in the barrels. If you put filter media in the middle barrel, you might restrict the water flow through it. It might overflow. But, adding a larger diameter outlet pipe won't alter that overflow problem. A larger outlet pipe won't make the water move through the filter media any faster.
However, if you are feeding the first barrel with a pressurized pump, then you change the situation. You might have a 2 inch pipe feeding into the barrel under high pressure and volume, but a 2 inch pipe going out via gravity flow. With NO filter media in the barrel, the volume of water entering may be too much water to exit the 2 inch pipe via non pressurized flow and the barrel will overflow.
Just wanted to make these statements regarding inlet/outlet pipe dimensions and when/why they are important. So if you are using all gravity flow, you don't need to oversize your exit pipes between barrels. The added expense is not necessary in such scenarios, but they ARE necessary in other applications.
Catfishnut
