## Tuesday, December 23, 2014

### Formula 1 turbines and enthalpy

A couple of interesting developments occurred around the exhaust systems on both the Ferrari and Mercedes-engined Formula 1 cars in 2014: the Ferrari-engined vehicles acquired insulation around the exhaust-pipes, and the Mercedes-equipped cars appeared with a so-called log-type exhaust.

The purpose of the insulation was to increase the temperature of the exhaust gases entering the turbine. Similarly, increasing the exhaust gas temperature was a purported beneficial side-effect of the log-type exhaust on the Mercedes.

A couple of general points about the physics of turbines might provide some useful context here. First, the work done by the exhaust gases on the turbine comes from the total enthalpy (aka stagnation enthalpy) of the exhaust gas flow.

This is perhaps a subtle concept. The total energy E in the fluid-flow through any type of turbine consists of:

E = kinetic energy + potential (gravitational) energy + internal energy

However, to understand the change of fluid-energy between the inlet and outlet of a turbine, it is necessary to introduce the enthalpy h, the sum of the internal energy e and the so-called flow-work pv:

h = e + pv ,

where p is the pressure, and v is the specific volume, (the volume occupied by a unit mass of fluid).

One way of looking at the flow-work is that it is part of the energy expended by the fluid maintaining the flow; the fluid performs work upon itself, (in addition to the external work it performs exerting a torque on the turbine), and this work can be divided into that performed by the pressure gradient and the work done in compression/expansion.

Another way of looking at it is that the energy released into the fluid from a combustion process may have been released at a constant pressure as the fluid performed work expanding against its environment. The internal energy e doesn't take that into account, but the enthalpy h = e + pv does. As the diagram above from Daniel Schroeder's Thermal Physics suggests, the enthalpy counts not only the current internal energy of a system, but the internal energy which would be expended creating the volume which the system occupies.

For a system which is flowing, it possesses energy of motion (kinetic energy) in addition to enthalpy. The so-called total enthalpy hT is simply the sum of the enthalpy and kinetic energy:

hT= e + pv + 1/2 ρ v2 ,

where ρ is the mass density and v is the fluid-flow velocity.

This quantity is also called the stagnation enthalpy because if you brought a fluid parcel to a stagnation point, at zero velocity, without allowing any heat transfer to take place to adjacent fluid or solid walls, the kinetic energy component of the total energy in that parcel would be transformed into enthalpy.

In the case of a Formula 1 turbine, there is no difference in the potential energy of the exhaust gas at the inlet and outlet, so this term can be omitted from the expression for the change in energy. What remains entails that the rate at which a turbine develops power is determined by subtracting the enthalpy-flow rate at the outlet from the enthalpy-flow-rate at the inlet. The greater the decrease in total enthalpy, the greater the power generated by the turbine.

As the exhaust gases pass through the turbine, they lose both kinetic energy and static pressure, but gain some internal energy due to friction. As a consequence, the entropy of the exhaust gas increases, and the enthalpy reduction is not quite as large as it would otherwise be (see diagram above from Fluid Mechanics, J.F.Douglas, J.M.Gasiorek and J.A.Swaffield).

However, (and here is the crux of the matter), for a given pressure difference between the turbine inlet and outlet, the reduction in total enthalpy increases with increasing temperature at the inlet. In other words, this is another expression of the fact that the thermal efficiency of a turbine is greater at higher temperatures (a fact which also dominates the design of nuclear reactors).

So, all other things being equal, increasing exhaust gas temperature with insulation or a log-type exhaust geometry will increase the loss of total enthalpy between the inlet and outlet of the turbine, increasing the power generated by the turbine.

However, there is another side to this coin: the required pressure drop between the turbine inlet and outlet for a desired enthalpy-reduction, decreases as the inlet temperature increases. Hence, if there is a required turbine power-level, it can be achieved with a lower pressure drop if the exhaust gases are hotter. This could be important, because the lower the pressure at the inlet side of the turbine, the lower the back-pressure which otherwise potentially inhibits the power generated by the internal combustion engine upstream. So increasing exhaust gas temperatures might be about getting the same turbine power with less detrimental back-pressure on the engine.