1. AME 514 Applications of Combustion
Lecture 10: Hypersonic Propulsion I: Motivation, performance parameters

2. Advanced propulsion systems (3 lectures)
Hypersonic propulsion background (Lecture 1)
Why hypersonic propulsion?
What's different at hypersonic conditions?
Real gas effects (non-constant CP, dissociation)
Aircraft range
How to compute thrust?
Idealized compressible flow (Lecture 2)
Isentropic, shock, friction (Fanno)
Heat addition at constant area (Rayleigh), T, P
Hypersonic propulsion applications (Lecture 3)
Ramjet/scramjets
Pulse detonation engines
AME Spring Lecture 10

3. Why use air even if you're going to space?
Carry only fuel, not fuel + O2, while in atmosphere
8x mass savings (H2-O2), 4x (hydrocarbons)
Actually even more than this when the ln( ) term in the Breguet range equation is considered
Use aerodynamic lifting body rather than ballistic trajectory
Ballistic: need Thrust/weight > 1
Lifting body, steady flight: Lift (L) = weight (mg); Thrust (T) = Drag (D), Thrust/weight = L/D > 1 for any decent airfoil, even at hypersonic conditions L mg T D
AME Spring Lecture 10

4. What's different about hypersonic propulsion?
Stagnation temperature Tt - measure of total energy (thermal + kinetic) of flow - is really large even before heat addition - materials problems
T = static temperature - T measured by thermometer moving with flow
Tt = temperature of the gas if it is decelerated adiabatically to M = 0
 = gas specific heat ratio = Cp/Cv; M = Mach number = u/(RT)1/2
Stagnation pressure - measure of usefulness (ability to expand flow) is large even before heat addition - structural problems
P = static pressure - P measured by pressure gauge moving with flow
Pt = pressure of the gas if it is decelerated reversibly and adiabatically to M = 0
Large Pt means no mechanical compressor needed at large M
AME Spring Lecture 10

5. What's different about hypersonic propulsion?
Why are Tt and Pt so important? Isentropic expansion until exit pressure (P9) = ambient pressure (P1) (optimal exit pressure yielding maximum thrust) yields
… but it's difficult to add heat at high M without major loss of stagnation pressure
AME Spring Lecture 10

6. What's different about hypersonic propulsion?
High temperatures:  and molecular mass not constant - dissociation - use GASEQ ( to compute stagnation conditions
Example calculation: standard atmosphere at 100,000 ft
T1 = 227K, P1 = atm, c1 = m/s, h1 = kJ/kg
(atmospheric data from
Pick P2 > P1, compress isentropically, note new T2 and h2
1st Law: h1 + u12/2 = h2 + u22/2; since u2 = 0, h2 = h1 + (M1c1)2/2 or M1 = [2(h2-h1)/c12]1/2
Simple relations ok up to M ≈ 7
Dissociation not as bad as might otherwise be expected at ultra high T, since P increases faster than T
Problems
Ionization not considered
Stagnation temperature relation valid even if shocks, friction, etc. (only depends on 1st law) but stagnation pressure assumes isentropic flow
Calculation assumed adiabatic deceleration - radiative loss (from surfaces and ions in gas) may be important
AME Spring Lecture 10

7. What's different about hypersonic propulsion?
WOW! HOT WARM COLD
5000K K K K
N+O+e- N2+O N2+O N2+O2
AME Spring Lecture 10

8. Thrust computation
In airbreathing and rocket propulsion we need THRUST (force acting on vehicle)
How much push can we get from a given amount of fuel?
We'll start by showing that thrust depends primarily on the difference between the engine inlet and exhaust gas velocity, then compute exhaust velocity for various types of flows (isentropic, with heat addition, with friction, etc.)
AME Spring Lecture 10

9. Thrust computation
Control volume for thrust computation - in frame of reference moving with the engine
AME Spring Lecture 10

10. Thrust computation - steady flight
Newton's 2nd law: Force = rate of change of momentum
At takeoff u1 = 0; for rocket no inlet so u1 = 0 always
For hydrogen or hydrocarbon-air FAR << 1; typically 0.06 at stoichiometric
AME Spring Lecture 10

11. Thrust computation
But how to compute exit velocity (u9) and exit pressure (P9) as a function of ambient pressure (P1), flight velocity (u1)? Need compressible flow analysis, next lecture …
And you can obtain a given thrust with small [(1+FAR)u9 - u1] and large large (P9 – P1)A9 or vice versa - which is better, i.e. for given , u1, P1 and FAR, what P9 will give most thrust? Differentiate thrust equation and set = 0
Momentum balance on exit (see next slide)
Combine
 Optimal performance occurs for exit pressure = ambient pressure
AME Spring Lecture 10

12. 1D momentum balance - constant-area duct
Coefficient of friction (Cf)
AME Spring Lecture 10

13. Thrust computation
But wait - this just says P9 = P1 is an extremum - is it a minimum or a maximum?
but Pe = Pa at the extreme cases so
Maximum thrust if d2(Thrust)/d(P9)2 < 0  dA9/dP9 < 0 - we will show this is true for supersonic exit conditions
Minimum thrust if d2(Thrust)/d(P9)2 > 0  dA9/dP9 > 0 - we will show this is would be true for subsonic exit conditions, but for subsonic, P9 = P1 always since acoustic (pressure) waves can travel up the nozzle, equalizing the pressure to P9, so it's a moot point for subsonic exit velocities
AME Spring Lecture 10

14. Propulsive, thermal, overall efficiency
Thermal efficiency (th)
Propulsive efficiency (p)
Overall efficiency (o)
this is the most important efficiency in determining aircraft performance (see Breguet range equation, coming up…)
AME Spring Lecture 10

15. Propulsive, thermal, overall efficiency
Note on propulsive efficiency for FAR << 1
p  1 as u1/u9  1  u9 is only slightly larger than u1
But then you need large to get required Thrust ~
(u9 - u1); but this is how commercial turbofan engines work!
In other words, the best propulsion system accelerates an infinite mass of air by an infinitesimal u
Fundamentally this is because Thrust ~ (u9 - u1), but energy required to get that thrust ~ (u92 - u12)/2
For hypersonic propulsion systems, u1 is large, u9 - u1 << u1, so propulsive efficiency usually high (i.e. close to 1)
AME Spring Lecture 10

16. Specific Thrust
Specific thrust – thrust per unit mass flow rate, non-dimensionalized by sound speed at ambient conditions (c1)
For any 1D steady propulsion system
For any 1D steady propulsion system if working fluid is an ideal gas with constant CP, 
AME Spring Lecture 10

17. Specific Thrust
Specific thrust (ST) continued… if P9 = P1 and FAR << 1 then
Thrust Specific Fuel Consumption (TSFC) (PDR's definition)
Usual definition of TSFC is just , but this is not dimensionless; use QR to convert to heat input, use c1 to convert the denominator to a quantity with units of power
Specific impulse (Isp) = thrust per weight (on earth) flow rate of fuel (+ oxidant if two reactants, e.g. rocket) (units of seconds)
AME Spring Lecture 10

18. Breguet range equation
Consider aircraft in level flight
(Lift = weight) at constant flight
velocity u1 (thrust = drag)
Combine expressions for lift & drag and integrate from time t = 0 to t = R/u1 (R = range = distance traveled), i.e. time required to reach destination, to obtain Breguet Range Equation
Lift (L)
Thrust
Drag (D)
Weight (W = mvehicleg)
AME Spring Lecture 10

19. Rocket equation
If acceleration (u) rather than range in steady flight is desired [neglecting drag (D) and gravitational pull (W)], Force = mass x acceleration or Thrust = mvehicledu/dt
Since flight velocity u1 is not constant, overall efficiency is not an appropriate performance parameter; instead use specific impulse (Isp) = thrust per unit weight (on earth) flow rate of fuel (+ oxidant if two reactants carried), i.e. Thrust = mdotfuel*gearth*Isp
Integrate to obtain Rocket Equation
Of course gravity and atmospheric drag will increase effective u requirement beyond that required strictly by orbital mechanics
AME Spring Lecture 10

20. Breguet & rocket equations - comments
Range (R) for aircraft depends on
o (propulsion system) - depends on u1 for airbreathing propulsion
QR (fuel)
L/D (lift to drag ratio of airframe)
g (gravity)
Fuel consumption (minitial/mfinal); minitial - mfinal = fuel mass used (or fuel + oxidizer, if not airbreathing)
This range does not consider fuel needed for taxi, takeoff, climb, decent, landing, fuel reserve, etc.
Note (irritating) ln( ) or exp( ) term in both Breguet and Rocket:
because you have to use more fuel at the beginning of the flight, since you're carrying fuel you won't use until the end of the flight - if not for this it would be easy to fly around the world without refueling and the Chinese would have sent skyrockets into orbit thousands of years ago!
AME Spring Lecture 10

21. Breguet & rocket equations - examples
Fly around the world (g = 9.8 m/s2) without refueling
R = 40,000 km
Use hydrocarbon fuel (QR = 4.5 x 107 J/kg),
Good propulsion system (o = 0.25)
Good airframe (L/D = 20),
Need minitial/mfinal ≈ aircraft has to be mostly fuel - mfuel/minitial = (minitial - mfinal)/minitial = 1 - mfinal/minitial = 1 - 1/5.7 = 0.825! - that's why no one flew around with world without refueling until 1986 (solo flight 2005)
To get into orbit from the earth's surface
u = 8000 m/s
Use a good rocket propulsion system (e.g. Space Shuttle main engines, ISP ≈ 400 sec)
Need minitial/mfinal ≈ 7.7 can't get this good a mass ratio in a single vehicle - need staging – that's why no one put an object into earth orbit until 1957
AME Spring Lecture 10