4261: Air-Breathing Engines
Homework #2
Assigned: January 31, 2012
Due: February 14, 2012
Question 1: The purpose of this question is to continue to make the connection between aircraft and propulsion system performance beyond the level that we were able to investigate in one lecture.
Recall from your fluids and aerodynamics courses that the drag of an aircraft may be described as being composed of a component termed the parasitic drag , which increases with the square of the flight velocity, and a term called the induced drag (drag due to lift), which decreases in proportion to the inverse of the flight velocity squared. The total drag coefficient may be expressed as:
C
D

C
D
0

C
D i

C
D
0

2
C
L
 eAR
1.1
Where:
L

1
2

V
2
SC
L
1.2
D

1
2

V
2
SC
D
1.3
C
D0
is the parasitic drag coefficient and is not a function of C
L
, and the induced drag coefficient varies approximately with the square of C
L
. AR is the aspect ratio, e is
Oswald’s efficiency factor, and S is a characteristic area.
Our goal in this is to examine 2 conditions of interest:
1.
Minimum Drag , which maximizes the range (distance traveled) of an aircraft
2.
Minimum Power , which maximizes the endurance (time aloft) of an aircraft
Part 1: For steady, level flight and a given weight, show that the minimum drag occurs at the condition of maximum lift-to-drag ratio. Show that the velocity at which the vehicle should fly for minimum drag is:
2
1.4
V min, drag
 



4
W
S

1
2
1
C
D
0
1

AR







1
4
Part 2: For a given amount of available fuel energy (Joules), the maximum endurance
(time aloft) is obtained at a flight condition corresponding to the minimum rate of energy expenditure (Joules/s), or P req,min
. The propulsion system power requirements to maintain steady, level flight are:
1

T req

D P req

T req
V

DV 1.5
With these assumptions, it may be show that:
P req

1
2

V
3
SC
D
0

W 2 1

AR
1.6
1

VS
2
Find the velocity that the vehicle should fly at for minimum power and also derive an expression for the maximum time aloft assuming constant Isp.
Part 3: Fill in the following table to summarize your results.
Minimum Drag
Maximum Range
Minimum Power
Maximum Endurance
C
L
C
D
C
L
/C
D
Flight Velocity
Compare the expressions that you have obtained for minimum drag (maximum range) and minimum power (maximum endurance). By what percentage should the aircraft be flying faster or slower relative to the minimum drag condition, in order to maximize endurance? What is the corresponding L/D ratio of the maximum endurance condition relative to the maximum L/D condition? Sketch this behavior on a plot.
Question 2: Brayton Cycle Application
Consider a gas turbine engine that operates on an ideal Brayton cycle for a new large turbo-prop aircraft which will serve as a high efficiency regional transport. The gas turbine delivers shaft power output of 50 MW to the propeller blades, which is sufficient to propel the aircraft. The minimum temperature in the cycle is 300 K and the maximum is 1700 K. The minimum pressure in the cycle is 100 kPa and the compressor pressure ratio is 15. a.
What is the thermal efficiency of the cycle? b.
What fraction of the turbine output is required to drive the compressor? c.
What is the power output of the turbine?
2

Question 3: Consider a turbojet powered aircraft during landing with the throttles set to some fraction of that required to achieve take-off thrust. As the aircraft is rolling down the runway at 100 miles per hour, the engine consumes 150 kg/s of air and produces an exhaust velocity of 200 m/s. a.
What is the thrust of this engine? You may assume that the exhaust jet is ideally expanded and that the mass flow rate of fuel is small compared to the mass flow rate of the air. b.
Now suppose that the pilot engages the thrust reversers which deflect the engine exhaust normal to the direction of travel (half up and half down) without affecting the operating condition of the engine, as shown generally in the picture below.
Clam shell thrust reverser on the DC-9 Clam shell thrust reverser schematic
What is the magnitude and direction of the thrust produced by the engine? c.
With the thrust reversers still engaged, the aircraft comes to a stop on the taxiway.
The engine is now consuming 80 kg/s of air and producing an exhaust velocity of
150 m/s. What is the magnitude and direction of the thrust produced by the engine? d.
You are asked to design some vanes used in the thrust reverser. Estimate the vertical forces (± y-direction) on the turning vanes as a function of the vane turning angle, see sketch below. You may assume steady flow and no acceleration of the vehicle. y
 x
The engine exit mass flow is
 e
U e
A e
. The engine exit area is A e
, and assume that the exit pressure, P e
, is equal to the atmospheric pressure, P a
.
3