What is the effect of air friction to the velocity of satellite? I have heard satellite's speed increases with air friction. But I'm in confusion how is it possible?
I would expect the speed of the satellite to slow down upon reentry; not speed up. Most terrestrial satellites, even ones in low Earth orbit, do not encounter sufficient atmosphere or air resistance to significantly degrade their orbits so that they reenter the atmosphere. It is true that the Iridium satellites in LEO often do this. They have relatively brief service lives (a few years) before reentry occurs owing to their skirting the upper atmosphere, but this is the exception.
In order for atmospheric friction to increase the speed of a satellite, the atmosphere of the planet it is orbiting would need to be one that undergoes super rotation, or in other words, the atmosphere would have a velocity greater than the velocity of the satellite. This would be an unusual situation, to say the least, as the atmosphere would also tend to escape the planet.
Air friction does increase the velocity of the satellite. sort of.
Assume for example, that the satellite is in a circular orbit with the appropriate velocity for the height it is at. Now it hits small patch of atmosphere, that slows the satellite a little (yes, slows).
The satellite is now at its apogee in an new elliptical orbit. As it proceeds in pure vacuum, it drops lower towards its perigee, and must, by conservation of both angular motion and energy, speed up.
At the moment of perigee, the satellite just happens to encounters another small patch of atmosphere, which again slows it a little. It just happens that the slowing is enough to give the satellite a circular orbit, at the new, lower height, and thus at a higher velocity, than it had at the start of this process.
The three steps of frictional slowing, orbital speed-up and frictional slowing combine to give an orbital speed-up
Orbital dynamics are weird...