## Turbo Charged – The Thrust part 1

June 7th, 2008 JoshSo… my attempt at a simple blog on how to turn a turbo into a useful engine is spilling into an uber-multi-page super rant. Oh well… time to suck up and crunch the numbers I promised in the last article. For this I will be using this NASA page as a general reference.

First we state the givens/assumptions:

- Thrust = 100 lbs(force)
- operating fluid – air
- ambient conditions – STP (standard temperature and pressure)
- exhaust pressure = ambient
- engine is fixed to ground, no forward velocity
- maximum operating temperature =

Then the variables:

- T = temperature
- p = pressure
- h = specific enthalpy (look this up- it’s an amazing property)
- V = velocity
- cp = specific heat – you used this in high school
- y = specific heat ratio
- n = adiabatic nozzle efficiency (adiabatic = no heat is absorbed or escapes the system)
- NPR = nozzle pressure ratio
- A = nozzle exit area
- R = gas constant
- anything with “i” after it = condition at the front of the nozzle
- anything with “e” after it = condition at the exit of the nozzle
- anything with a “t” after it = “total” which means that there are multiple parts to the variable (pt = total pressure = velocity pressure + static pressure)

Next we have the formulas:

- F = mdot * V
- pte/pti =(Tte/Tti)^(y/(y-1)) = 1
- h = cp * T
- hte =he + Ve^2/(2n)
- Ve = sqrt(2*cp*Tte*n*[1-{1/NPR}^(y-1)/y])
- mdot = (A*pt/sqrt(Tt))*sqrt(y/R)*((y+1)/2)^(-(y+1)/2(y-1)

Ok, so now that we have a bunch of formulas we need to decide what we want them to tell us and then which ones to use to get it. So, looking at the nozzle, we have a state point going in to the nozzle, and a state point coming out of the nozzle. We know that we want our maximum thrust to be 100 lbs-force, and since that is a maximum, we know that the flow is choked at the nozzle exit (mach = 1).

The easiest formula is #1, which can be solved for mdot, with velocity = mach 1:

mdot = F/V = 100 lbf / speed of sound at sea level = 2.88 lb/s

So the total mass flow through the nozzle (and through the engine) is 2.88 lbs of air every second. That’s quite a bit of air! Now that we have the mass flow, we need to figure out the area of the nozzle exit. That requires 4 more variables: total pressure, total temperature, the Mach number, and gamma for air. We already know the Mach number is 1 for choked flow, gamma for air = 1.4 for stp, and total pressure and temperature are limited by material and design properties. Total pressure is limited by the compressor- most turbocharger compressors have pretty good efficiency at 2 bar pressure ratio. Total temperature is limited by the highest temperature that the housing material can withstand. Lets call it 400F.

Using the calculator on this page, the area of the nozzle is going to be 1.21 sq ft. This is huge! There are a number of ways to reduce the size of the nozzle, such as revising the total temperature, or total pressure, which I might look at in the next post. Or, since I have been more interested in making an engine with shaft power output, I think I will probably just try to design that… But who knows.