Sounds like a well thought out thesis and project
I remember from my science fair days ----- documentation is EVERYTHING... you can never document enough.
Keep good records - even record failures or errors.
More to your point....
If you want to approach sonic tip speeds, you would best not do it with a conventional molded prop. You would run the risk of shattering the blade.
11" diameter = 34.54" circumference * 18,000 rpm = 621720 in/min = 51810 ft/min = 9.8125 miles/min * 60 = 588.75 miles/hr
An 11" prop turning about 18,000 rpm equates to a tip speed around 588 mph
In dry air at sea level with a temperature of 21 °C (70 °F) the speed of sound is 344 m/s (1230 km/h, or 770 mph, or 1130 ft/s)
So the mach number at 18,000 rpm is 588.75/770 = .765
That is approaching M0.8, which is considered where transonic begins - so that is a pretty quick tip speed, and no doubt some airflow over the blade is well exceeding M.8 by a good margin. So you would see some transonic effects and a drop in efficiency.
Why 18,000 rpm here? I do not believe you are going to turn any "usual" 11x4 prop much over 18,000 rpm with a SJ-50, even with a full size nitro pipe properly tunned. Perhaps ya might get up to 18,500.
Mach 1, or a tip speed of 1130 ft/s or 813600 in/min , for the 11" prop requires 813600/34.54 = 23,555 rpm
For your experiment, getting to that M.76 number would be probably useful in showing the performance drop on a curve.
What you would want to do, is use the exact same prop - and use the throttle to set your rpm reference points.
Key here, is you will want to use a carbon fiber prop, just to be safe. There are some available in that size.... you may have to look to Europe, or even look toward some of the CL and FF guys.
One suggestion though.......
You may want to try this experiment with an electric motor.
A few reasons.
1) you are not mechanically RPM limited - Getting to 20,000 rpm with an 11x4 is a matter of AMPS and VOLTS
2) You can measure AMPS and Volts. You know Watts, you can equate/chart the POWER (Watts or HP) needed to turn the prop at that RPM.
3) Electric is smoother - no accelerations - less stress on the blades and prop hub - so you may get away with using a standard blade prop.
4) You can perform the demonstration inside at the science fair without pissing off many people
Anyway.... its a good project !! Stick with it.... and stay safe!
Let me know how you make out.....
Bob