Wind (nano-/pico-)turbines installed on bicycles for regenerative charging of batteries?

Hilariously, such a product does exist. It comes from notorious crapgadget vendor Thanko. Note that it only purports to charge two AA batteries, and it's not an efficient way to do that. They make no estimate of how long it would take to fully charge those batteries.

I did some playing around with this calculator. Based on some guesses and estimates, it looks like the Thanko wind turbine would generate 0.001 kW, or 1 W of power with a wind speed of 20 km/h (this includes default estimates for efficiency, which are probably very optimistic). A single AA LiIon battery has about 3 watt-hours of capacity, so it would take 3 hours to charge (assuming a 100% efficient charger).

If you were trying to power a 250-W e-bike motor with that AA battery (and could magically convert its voltage at 100% efficiency) you'd get 43 seconds at full blast out of it.

Ross's comment about a perpetual motion machine is bang-on. The second law of thermodynamics says that any transfer of energy from one form to another is wasteful.

This is a machine that tries to take the wind generated by forward motion, convert it to rotational energy, convert that to electrical energy, convert that to chemical energy (in the battery) and then convert chemical energy back to electrical, to rotational energy, and finally back to acceleration.

Trying to convert wind energy to electricity will increase the drag on the bicycle by more than the amount of rotational energy you would see in the turbine, and then you'd get more losses when you convert that rotational energy to electricity, and still more losses when you convert that electrical energy to chemical energy in the battery.

Any turbine and generator system will also increase the weight of the bike, meaning it requires more power to accelerate the bike.

The turbine will increase the drag on the bike, and require more power from the drive system (rider and/or motor) to compensate for it - quite a bit more power than the amount of power that would be collected in the battery.

Saying "It's a micro-turbine, the drag is small" is silly. If it generates a very small amount of drag, it generates an even smaller amount of power.

In theory you might be able to only deploy the turbine when the bike is coasting downhill and recover energy that would have been lost to braking. That means, however, that you would only generate electricity on steep downhills where braking would have been required - a small portion of most rides. Plus, any turbine large enough to collect meaningful amounts of energy is going to be both large and massive, and the simple volume of space it takes up is likely going to increase the total drag of the bike even when it is retracted. A mechanism to deploy and retract the turbine would also be relatively large, complex, and heavy, and require some power to operate.

All in all, it seems likely that any such system would reduce the range of the e-bike, not increase it, and it would certainly make it heavier.