[Christ]

Alright I lied a little bit... your friend heat wrapped his down pipe because keeping the heat in the pipe before the turbine increases the pressure difference across the turbine housing.

The problem with your thought that a turbine is inherently a heat engine isn't that it doesn't function with heat, its just that heat is not the root principle of its operation, which is what I'm trying to explain this whole time. A pressure differential of any kind, with any fluid medium, will drive the turbine. In the case of the exhaust driven turbocharger specifically, you can see that the turbocharger presents a restriction to flow, initiating a post-combustion pressure stage. The pipe between the engine and turbine is a heat sink, and we already known that pressure increases in a closed space as temperature is increased. Its a common fault, and I don't blame you for thinking this way at all.

Wrapping that pipe iaulates it against later heat loss, which keeps a maximum pressure differential between the input and output of the turbine allowing it to operate with greater efficiency. The maths refer to DeltaP here when calculating the output power of the turbine shaft, asking for the pressure differential as an input, and this can be derived with some certainty using a temperature input if one wishes to analyze a freeze frame of the potential output under a given set of circumstances. In order to fully analyze a turbocharger system, you would need the inputs of mass flow and engine exhaust flow, as well as downpipe volume and turbine sizing, pitch, etc.. a much more complicate set of maths that is such less commonly used where simplicity is a factor.

What the latter would tell you is, under at set of conditions in which a turbine might be found to operate, what efficiency and output it wud have. This is part of how flow maps re created for specific designs of turbochargers.

I think I covered everything now...