I finally got a handle on the diameter and blade length of the direct drive turbine needed to work well with a rotary. A 19,000 RPM Formula 1 engine would require a smaller turbine. This correlates well with the geared turbine used
on the R3350 TC engine. The 11.3 inch diameter R3350 turbine speed is 16,000 RPM at cruise power and 19,000 RPM at takeoff power. Gear ratio is 6.52:1. Engine RPM is 2900.

A stationary blow down turbine with air flowing over the blades generates maximum torque. As the turbine accelerates the torque reduces in a straight line. As the turbine blades reach the speed of the air flowing over them no torque is generated. This will be the maximum theoretical RPM of the turbine. The maximum HP generated by the turbine will be at a RPM that is roughly half of the peak RPM. Torque at mid RPM will be roughly half of peak torque. Since HP is RPM times torque peak HP is generated at half peak RPM and half peak torque roughly speaking.

If the total projected blade area of the turbine is the same as the cross sectional area of the exhaust pipe feeding it the velocity of the gas flowing over the blades will be the same as the exhaust pipe gas velocity. The projected area is that area between the inner diameter of the blades and the outer diameter. Not the total blade surface area.

The part of the exhaust system closest to the blades is called the nozzle. It is usually possible to neck down the nozzle area to speed the air flowing over the blades at the modest expense of increasing the back pressure on the engine. This is analogous to changing the A/R ratio of a turbo charger. Usually there is an overall increase in net HP of the turbo compound system because gas velocity is more important than back pressure. The blades respond to the gas velocity squared just like lift on a wing responds to an increase in forward speed. The force the blades exert on the hub is the density of the gas times the surface area of the blades times the lift coefficient times the gas velocity squared. Just like a wing. Lift coefficients on the highly cambered blades would be between one and one and a half.

Since you need to keep the projected cross sectional area of the blades more or less the same as the exhaust pipe area the length of the blades must change as the overall diameter of the turbine changes. In general a large diameter turbine with short blades will generate more torque and peak HP and will be generated at a lower RPM. The up side to this is the gear ratio required between turbine and crankshaft is reduced. The draw back is leakage around the ends of the blades. The aspect ratio of the blades in wing terms is low. A smaller diameter turbine with longer blades will operate at a higher RPM but the blades aerodynamic losses may be lower due to a more favorable aspect ratio. What also must be considered is the centrifugal loads on the blades. Longer blades put more stress on the roots of the blades which is the critical area. Another factor of course is metal loses it strength as the temperature of the metal increases. There is an area of operation bounded on one side by the turbine inlet temperature and on the other side by the RPM. these two lines eventually converge so one must keep the turbine operating in the safe area between the two curves. Turbine design is best left up to people with experience. One of those companies is Barber Nichols in the US.

I used QB (below) to help with the calcs. A spread sheet would also work.
The results are in the attached jpg. The theoretical free speed of the 10 inch
turbine is 23,000 RPM. Of course the turbine would not live at that speed. This turbine will be generating max HP around half of that or about 11,500 RPM. As long as it is connected to the e-shaft (crankshaft) it will not see any RPM higher than the engine.  This is a racing application. For an aircraft engine the turbine will need to be a bit bigger.

Check my logic and math.

Paul Lamar

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CLS

PRINT : GasVel = 1000'FPS

PRINT "Gas  Vel  FPS ="; GasVel

PipeR = 1.5: AreaPipe = PipeR ^ 2 * 3.1416

PRINT "Area Pipe sq in ="; AreaPipe
PRINT

R1 = 5: R2 = 4.77

PRINT "Turbine Dia OD in ="; R1 * 2

PRINT "Turbine Dia ID in ="; R2 * 2: PRINT

FlowArea = R1 ^ 2 * 3.1416 - R2 ^ 2 * 3.1416

PRINT "Flow Area sq in ="; FlowArea

Circum = R1 * 2 * 3.1416

PRINT : PRINT "Circum Inches ="; Circum

PRINT : PRINT "Circum Feet ="; Circum / 12

PRINT : PRINT "Rev per second ="; GasVel / (Circum / 12)

PRINT : PRINT "Rev per minute ="; (GasVel / (Circum / 12)) * 60