Torque on a Turbine, Air Breathing Rocket Engine

[Abraham]

Hi All,

I'm a junior in HS. I have a research project due this semester for Physics B (first year introductory college physics).
Last semester I did some research on boundary layers, friction loss in pipes.
This semester I've decided to do some investigation on air breathing rocket engines.

The particular one I'm looking at (sorry, no idea what its called) uses a cryogenic fuel to absorb heat from the airframe of the craft moving at high speed. The fuel (e.g. Methane, H2) vaporizes and expands, driving a turbine, which is connected to a driveshaft which turns a compressor. This compressor compresses air with the fuel and ignites the mixture in a combustion chamber / rocket engine.

My primary concern is the expanding fluid in the turbine.

The rotor is a centrifugal turbine, I'm not sure what its called exactly, (but it's not an axial flow turbine). The methane enters one stage near the center at radius r1, and exits at the edge at r2.

From the equation of continuity: a1*v1 = a2*v2
where
a=cross sectional area between blades
v=velocity at r1 or r2

And torque on the turbine, T = w/g (r1V1*cos(α1)-r2V2*cos(α2))

Many equations I've looked at, including a1v1 =a2v2, don't mention situations where the fluid expands.

My problem is figuring out the torque on the turbine imparted by the expanding methane fuel. How do I calculate torque if the fluid is expanding?

Thanks for the help!

(And sorry if the answer is all too obvious)

 

[jaap de vries]

I think the equation that you show,
T = w/g (r1V1*cos(α1)-r2V2*cos(α2))
is still valid regardless of expansion,
It is called the Euler turbine equation I know it better as;

T = Qm * (Vt2*R2-Vt1*R1)
T = Torque
Qm = (Rho * A * V) = mass flow at either inlet or outlet
Vt = Tangential velocity
Note that 'slip' is not taken into account here.

when the fluid in compressible then the continuity equation simply becomes.
Rho * A * V = constant
I believe Fred Garvin might know this one best.

Good luck, Jaap