HW#5 - Jerry M. Seitzman

AE4451/Seitzman
Spring 2015
Due Mon., March 2
Problem Set #5: Turbine Engines
 Homework solutions should be neat and logically presented, see format requirements
at http://seitzman.gatech.edu/classes/ae4451/homeworkformat.html.
 If appropriate, include a sketch of the flow/system, and indicate clearly your choice of
control surface.
 Always indicate any assumptions you make. If you use any results or equations
from the class notes or text in you solutions, please note and reference them (but you
better be sure they are applicable to the problem at hand).
 Try to solve the problem algebraically first. If possible, only use numbers/values in
the final steps of each solution.
1. Polytropic Efficiency
For turbomachinery, the component’s adiabatic efficiency can be modeled using the
concept of the polytropic efficiency p. It turns out that well-designed, modern
compression machines have similar polytropic efficiencies, independent of the
overall pressure ratio. The polytropic efficiency is the efficiency of an adiabatic
device with an infinitesimal (very small) pressure change, and like the adiabatic
efficiency, p<1. Since a finite pressure change can be thought of as many
infinitesimal pressure changes, we can produce a relationship between the adiabatic
efficiency of a real device and the polytropic efficiency.
For a compressor or fan with a stagnation pressure ratio Pr (outlet/inlet) and a
polytropic efficiency p, the adiabatic efficiency is given by

Pr
Pr
 1

 1
 p
1
.
1
assuming a thermally and calorically perfect gas.
Using this definition, find an expression for the exit stagnation temperature (To3) of a
compressor with an inlet temperature To2, a polytropic efficiency p, a stagnation
pressure ratio po3/po2, and a normalized specific heat cp/R.
2. Ram-Turbine (former 2nd midterm problem)
An air-powered turbine (or ram-turbine) is shown below. On an aircraft or missile in
flight, this device directs freestream air (at ambient temperature T a) into an internal
combustor. The exhaust of the combustor runs a power turbine that drives an
electric generator (to power electrical systems on the vehicle).
Electric
Generator
Ta
Combustor
Nozzle
Turbine
To,c To,d
To,b
Te
For this problem, you are to assume that: 1) the gas passing through the ramturbine is thermally & calorically perfect (=1.4); 2) all processes are reversible and
the nozzle is perfectly expanded; 3) there is no heat loss from any device; 4) the fuel
addition does not significantly change the total mass flowrate: (1+f)1; and 5) the
ram-turbine is designed to produce zero specific thrust.
a) For an aircraft flight velocity of 240 m/s, what is the nozzle exhaust velocity of
such a ram-turbine system?
b) Write an expression for Te in terms of the combustor exit temperature (T o,c), the
flight Mach number (M) and whatever other properties of air you need. (It may
help to draw a T-s diagram for this ideal cycle.)
c) Find the power that can be delivered to the electric generator IF 0.5 kg/s of air
enters the ram-turbine, M=0.90, Ta=220 K and the maximum temperature limit
of the ram-turbine is 1200 K.
2