MAE 4262: ROCKETS AND MISSION ANALYSIS

1
MAE 4262 ROCKETS AND MISSION ANALYSIS

• Performance of a 1-D Isentropic Nozzle
• January 31, 2008
• Mechanical and Aerospace Engineering Department
• Florida Institute of Technology
• D. R. Kirk

2
SUMMARY OF KEY EXIT VELOCITY, Ue, EQUATIONS

• For high Ue (high Isp), desire
• Propellants with low molecular weight, M
• Propellant mixtures with large heat release, QR
• High combustion chamber pressure, P02
• NOTE Sometimes subscript 2 is dropped, but still
  conditions in combustion chamber

3
SUMMARY OF KEY THRUST, T, EQUATIONS
Comparison to best theoretical
Measure from actual rocket (parameters that can
be easily measured on a thrust stand)
4
OVERVIEW

• Document covers development of Figure 11.3 of
  Hill and Peterson
• Figure 11.3 is one of most important and common
  curves in rocket propulsion
• Figure 11.3 is plot of thrust ratio vs. area
  ratio
• Figure compares two non-dimensional numbers
• Abscissa is ratio of nozzle exit area to minimum
  area, or nozzle exit area to throat area (minimum
  area always occurs at throat), Ae/A
• Ordinate is ratio of thrust with diverging to
  converging nozzle, T/Tconv
• Curve is plotted for constant ratio of specific
  heats, g cp/cv 1.2
• Curve would shift for g 1.4 or any other value
• Curves correspond to various ratios of Pa/P0
• Pa/P0 ambient (atmospheric) to combustion
  chamber pressure
• P0 is approximately constant for most rockets

5
FIGURE 11.3 HILL AND PETERSON
6
COMPARISON OF CONVERGING vs. DIVERGING NOZZLES

• Examine ratio of thrusts, with and without a
  diverging section
• Examine performance benefit of having diverging
  portion
• Metric of comparison
• Excellent Web Site http//www.engapplets.vt.edu/f
  luids/CDnozzle/cdinfo.html

Chamber, Pa
Chamber P0
Chamber P0
Converging Nozzle
Converging-Diverging Nozzle
7
COMMENTS CONVERGING NOZZLE (CTconv)

• For nozzle with only a converging section ?
  analysis is straightforward
• Pa/P0 is varied in equation

Evaluate at Me 1 Sonic exit condition
For converging nozzle Ae/A 1
8
THRUST COEFFICIENT, CTconv, FOR CONVERGING NOZZLES

• Maximum Thrust Coefficient when Pa 0 (expansion
  into a vacuum)
• Ae/A1

9
COMMENTS DIVERGING NOZZLE (CT)

• Requires more analysis than simple converging
  nozzle
• IMPORTANT POINT We can not vary Pe/P0 and Ae/A
  independently
• Connected through Mach Number, Me

Expression for Pe/P0
Vary Pa/P0 and Ae/A
Given A/A ? 2 Me Solutions Subsonic and
Supersonic
10
MACH NUMBER vs. A/A
Differences in Cp/Cv Amplified as M ?
For Given A/A ? 2 Solutions Subsonic and
Supersonic Mach
Highly Sensitive Region Small Changes in A/A ?
Large Changes in M
11
PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC
NOZZLE

• WHAT DID WE DO HERE?
• Set Pa/P0 0.05, g 1.2
• For any Ae/A determine supersonic Me
• Using this Me calculate P0/Pe
• Calculate CT
• Plot CT/CTconv (or T/Tconv) as function of Ae/A
  (which is equivalent to plotting CT as a function
  of Me (supersonic))

Function is Maximized when Pe Pa
12
PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC
NOZZLE
Maximum Thrust (Pe Pa)
Diverging Portion Increases Thrust
In terms of calculation, we could allow T/Tconv
to become negative, but as we will soon see, we
can deal with this part of the curve more
realistically
Diverging Portion Reduces Thrust
13
PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC
NOZZLE
Nozzle is Ideally Expanded Pe Pa
Curve can also tell us where Pe gt or lt Pa IF Pe
gt Pa Nozzle is Under-Expanded IF Pe lt Pa Nozzle
is Over-Expanded
14
PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC
NOZZLE
Nozzle is Ideally Expanded (Pe Pa)
Nozzle is Under-Expanded (Pe gt Pa)
Nozzle is Over-Expanded (Pe lt Pa)
15
PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC
NOZZLE
Nominal Range of Pa/P0
Decreasing Back Pressure or Increasing Altitude
16
PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC
NOZZLE
Line of Maximum Thrust Connects Locus of Maxima
For each value of Pa/P0 there is an optimum area
ratio for nozzle
17
PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC
NOZZLE
Small Ratios of Pa/P0 Require Very Large Area
Ratios
18
EXAMPLE ROCKET LAUNCH Ae/A 20
Burnout (Under-Expanded)
? Vertical Flight
Max Thrust (Ideally Expanded)
Launch (Over-Expanded)
Notice we are closer to Optimum Thrust
on Under-Expanded Side
19
PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC
NOZZLE
What can physically happen to supersonic flow in
this region? For this combination of pressure
ratios and area ratios, a shock enters nozzle
20
MODEL OF SHOCK IN EXIT PLANE

• We can plot shock line by located a shock at exit
  plane of nozzle
• Requires 1 additional equation
• Flow across a normal shock to connect conditions
• For a given g only one Pa/P0 for which a normal
  shock will locate in plane of a nozzle of given
  area ratio Ae/A

21
PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC
NOZZLE
On this line a normal shock wave located at exit
of nozzle
22
PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC
NOZZLE
If Pe reduced substantially below Pa flow can
separate A rough approximation for this condition
is Pe/Pa lt 0.4 NOTE Axial thrust direction is
not usually altered by separation and CT can
actually be increased over non-separated case
23
THRUST COEFFICIENT PLOTS

• Taken from Rocket Propulsion Elements, 6th
  Edition, by G. P. Sutton
• Notation
• p1 p0
• p2 pe
• p3 pa
• CF CT
• A2/At e Ae/A
• k g cp/cv
• Comments
• Plots are only CF (CT), they are not normalized
  by CTconv as in Figure 11.3
• Large region of separated flow
• Asymptotic behavior as p1/p3 ? 8
• pa/p0 ? 8 in HP

24
THRUST COEFFICIENT VS. NOZZLE AREA RATIO FOR g1.2
25
OPTIMUM EXPANSION SUMMARY
26
KEY POINTS ON PERFORMANCE CURVE

• How does a rocket flying vertically move on
  Performance Curve?
• High Pa/P0 to Low Pa/P0
• P0 usually remains constant during flight
• Pa ? as altitude ?
• As Pa/P0 ? very large Ae/A for maximum thrust
• How does optimal Ae/A vary as rocket flies
  vertically?
• Required Ae/A for maximum thrust increases as
  rocket altitude increases
• If T/Tconv lt 1, diverging portion of rocket is
  hindrance
• Actual rockets never operate in this region
• Best nozzle gives best performance (Isp, range,
  etc.) over flight envelope
• If nozzle operation is still unclear, review HP
  3.2-3.7
• Lecture on operation of C-D nozzles coming soon

27
COMMENTS ON ACTUAL NOZZLES

• Model of thermal rocket thrust chamber
  performance
• Model has many simplifications ? measure of best
  theoretical performance
• Actual rockets benefit from diverging nozzle
  portion, operate above T/Tconv 1
• Actual thrust chambers (non-idealities important
  to consider)
• Pressure losses associated with combustion
  process
• Actual flow in nozzle is not isentropic
• Friction
• Heat losses
• Shocks within nozzle
• Chemistry
• Frozen Flow Propellant composition remains
  constant
• Shifting Equilibrium Composition changes with
  propellant temperature
• Actual shape of nozzle affects performance

28
SUMMARY WHAT HAVE WE DONE?

• Simplified model of thermal rocket thrust chamber
• Model resulted in connection between
  thermodynamics and exit velocity, Ue
• Propellants with low molecular weight to achieve
  high exit velocity (high Isp)
• Desirable to have propellant mixtures with large
  QR/M
• Desirable to have high combustion chamber
  pressure, P0
• For a given thrust, higher P0 leads to lower A
  (smaller rocket)
• Increasing P0 leads to difficulties (stress, heat
  transfer, chemical issues)
• Model resulted in connection between
  thermodynamics, geometry and exit velocity
• Developed Characteristic Velocity, c, and Thrust
  Coefficient, CT
• Compare actual rockets to theoretical predictions
• Developed plot of Performance Characteristics of
  a 1-D isentropic rocket nozzle
• BASICS OF THERMAL (CHEMICAL) ROCKET PROPULSION
  AND PERFORMANCE