Gas Turbine Design Update

        Designing a turbine for a turbopump is critical considering the work the pump will do is ultimately derived from the gas turbine efficiency that is driving the impeller on the same shaft. Being of such importance also results in the turbine being extremely complex to design. Even after weeks of research, we barely scrapped together a turbine design based off a few initial velocity triangles of the stator vanes that direct the gas into the rotor blades or the turbine itself. 

example velocity triangle setup from MIT

The power produced by the turbine has to be equal to or greater than the combined power required of the impeller to produce desired flow rate along with the frictional losses due to off tolerances, bearing, or seal resistance. Thus the equation to derive the power from the turbine is as follows:                                                                                                                                                                                                                                                                Pt = nt mdott cpT1[1-(p2/p1)^(k-1)/k

Where P is power output of turbine, n is efficiency, m dot is mass flow rate(of the turbine), c is specific heat at a constant pressure, T is the absolute temperature,  p1 is initial pressure of system, p2 is the exit pressure, and k is the ratio of specific heats. The equation to derive nt is as follows:

                                                               nt = LtNt/mdot*dh

Where L is torque produced, N is specific speed, and dh is change in enthalpy.

After attempting to plug in some parameters into bladegen, a feature in ANSYS to model turbomachinery, the geometry that resulted for the v1 turbine rotor blade is shown below.

Stress Analysis of Spinning Disks

 As a first order analysis to determine whether or not a plastic disk would shatter spinning at high rotational speeds, a quick rotational stress calculation was done. From The Engineering Toolbox, the stress can be calculated as shown in the two images below. ABS plastic was chosen as the material. 

Method 1
Item	Value	Units	Notes
RPM	5000	rev/min	From Zach's Design
Radius	0.0458	m	From Zach's Design
Density	1060	kg/m^3	Abs Plastic
Stress	202751.49	Pa
	29.41	psi
Ult.Tensile Strength	5500	psi	Source
FOS	187

In Google Sheets the stress was found to be ~29 psi with a 0.0458 m radius disk spinning at 5000 RPM. 

The second method is from a forum where the stress was calculated as shown.  

Method 2
Item	Value	Units	Notes
RPM	5000	rev/min	From Zach's Design
Radius	0.0458	m	From Zach's Design
Density	1060	kg/m^3	Abs Plastic
Poisson's	0.35		Source
Max Stress	254706.56	Pa
	36.94	psi

This method produced a maximum stress of 37 psi, closely matching the previous result. The main difference with this method was ABS plastics' Poisson's ratio was taken into account.