"Not quite - it's not a closed system, so it's not perpetual motion. The sun adds energy to the system, so it's not impossible."
The sun adds the initial energy. But on this scheme, having burnt the methane (including the methane produced as such by the digester) they have CO2. And as this is essentially a system for using CO2 and hydrogen (from ???) and more energy (from ???) the only limit to repeating this indefinitely is ??? As I said, the Underpant Gnomes version of perpetual motion, powered by ???