Let's forget about the temperature of the metal for a minute, assuming that all of the rigid parts are perfect insulators. Then, the temperature of the gas is our only concern. Allow the expansion, and then allow the gasses in the two vessels to reach thermal equalibrium. I know this cannot be done in practice, but at this point, we are totally into the theoretical. Because this is a free expansion, and therefore 100% irreversible, the overall enthalpy of the gas will remain the same. For an ideal gas, this means the temperature will be the same as when the process started. For a real gas, it will be a little different depending on exactly what the gas is, and the proportion of gas that was originally high pressure, and low pressure in the system. This is the Joule-Thompson effect that JohnN has been obsessing over.
Essentially what has happened here is the potential energy in the bank has been converted first to kinetic energy in pushing the gas through the valve, and then back to thermal energy when the gas stops.
The transient result, however is that the bank gets cooler, and the cylinder being filled gets warmer during the actual expansion.
