The maximum flow rate you want in your pipe is probably around 8-10 foot per second. Much less is you are worried about silt eroding your copper pipes.
Typical "rivers" run at 5 - 25 fps. In real life, your inlet and pipe has to handle peak flows, or you need a retarding basin or a flood plain. (Actually, even with retarding basins and flood plains, you always get flood damage somewhere on river systems).
The flow rate in the pipe will depend on slope. At the pipe entrance, the flow rate will be the river rate, so the entrance has to be bigger than the effective pipe diameter by the ratio of the two flow rates. At the exit, your pipe should open out and flatten out, to get the flow rate you want into your settling basin. You can do that in an open trench.
You will also get better flow rates at slightly less-than full (say 95%), and better flow rates in your copper pipe than on a river bed at the same slope and fill ratio, but if your pipe is going to be 5 times too big, to handle normal peak flows, you don't really need to worry about that.