Why don't valence electrons escape in a superconductor? Here's what I imagine: we cool a material so that there is very little thermal vibration of atoms and the valence electrons can flow without resistance. However, why don't the free electrons fly off when the circuit is energised?
 A: Superconductors are solid lattices composed out of atoms and molecules. The conduction band when temperatures are very low allow the phenomenon of non resistance, but this can occur only while a lattice exists. If at that low temperature the lattice would disintegrate there would be no superconductivity.
A solid expressed by the lattice has a surface, that surface is characterized by a work function which is the amount of energy an electron from the conduction band in this case, would need to get out to vacuum.
The  energy of electrons in superconductivity  is examined in this paper and one would have to to study this, to see whether some electrons can have enough kinetic energy to be able to escape, but I think the argument that escaping electrons would destroy the lattice is overall valid.
If this is not there, as argued above, the lattice would disintegrate.
The work function for the elements is measured and is seen here . At very low temperatures it might diminish but it cannot go to zero and the motion of the electrons in the superconducting direction must  not give enough energy to destroy the lattice, since super-conductivity is stable where it appears.
