MAE 4262: ROCKETS AND MISSION ANALYSIS

Basics of Chemical Rocket Performance

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

Schematic Diagram of a Conventional Liquid Rocket Motor (Figure 11.1 in Sforza)

2

Northrop Grumman: LOX/LH2 Engine

3

Northrop Grumman: LOX/LH2 Engine

4

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

1

02

1

02

1

0202

1

0202

1212

11

2

12

p

p

M

Q

p

pQU

p

pT

M

RU

p

pTCU

eReRe

ee

epe

5

SUMMARY OF KEY THRUST, T, EQUATIONS

T

eaeeT

eaee

CcmT

A

A

p

p

p

p

p

p

pA

TC

M

RT

m

Apc

A

A

p

p

p

p

p

p

pA

T

*

*00

1

0

11

2

0*

01

1*

0*

*00

1

0

11

2

0*

11

2

1

2

2

11

11

2

1

2

Measure from actual rocket(parameters that can be easily measured on a thrust stand)

Comparison to best theoretical

6

OVERVIEW

• Next page shows a 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, = cp/cv = 1.2

– Curve would shift for = 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

• Compare with Figure 11.7 (page 448) in Sforza

7

T/Tconv versus Ae/A*

8

Figure 11.7 (page 448) in Sforza

9

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/fluids/CDnozzle/cdinfo.html

Converging Nozzle Converging-Diverging Nozzle

convT

T

conv C

C

T

T

ChamberP0

ChamberP0

Chamber, Pa

10

COMMENTS: CONVERGING NOZZLE (CTconv)

• For nozzle with only a converging section → analysis is straightforward

• Pa/P0 is varied in equation

0

1

*00

1

0

11

2

0*

1

0

120

120

1

21

11

2

1

2

1

2

2

11or

2

11

p

pC

A

A

p

p

p

p

p

p

pA

TC

p

p

Mp

pM

p

p

aT

eaeeT

e

ee

conv

Evaluate at Me = 1Sonic exit condition

For converging nozzleAe/A* = 1

11

THRUST COEFFICIENT, CTconv, FOR CONVERGING NOZZLES

• Maximum Thrust Coefficient when Pa = 0 (expansion into a vacuum)

• Ae/A*=1

0

1

1

21

p

pC aTconv

12

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

*00

1

0

11

2

0*

120

11

2

1

2

2

11

A

A

p

p

p

p

p

p

pA

TC

Mp

p

eaeeT

ee

Expression for Pe/P0

Vary Pa/P0 and Ae/A*

12

1

2* 2

11

1

21

e

e

e MMA

A Given A/A* → 2 Me SolutionsSubsonic and Supersonic

13

MACH NUMBER vs. A/A*

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3 3.5

Mach Number

A/A

*

Cp/Cv=1.2Cp/Cv=1.4

Differences in Cp/Cv Amplified as M ↑

Highly Sensitive Region:Small Changes in A/A* → Large Changes in M

For Given A/A* → 2 SolutionsSubsonic and Supersonic Mach

14

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 10 100 1000

Ae/A*

T/T

con

v

Pa/Po=0.050

WHAT DID WE DO HERE?1) Set Pa/P0 = 0.05, = 1.22) For any Ae/A* determine

supersonic Me3) Using this Me calculate P0/Pe4) Calculate CT

5) 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

*

11

12

*1

1

2

1

2

A

A

p

p

p

p

p

p

pA

TC e

o

a

o

e

o

e

oT

15

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 10 100 1000

Ae/A*

T/T

con

v

Pa/Po=0.050

Diverging Portion Reduces Thrust

Maximum Thrust (Pe = Pa)

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 Increases Thrust

16

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 10 100 1000

Ae/A*

T/T

con

v

Pa/Po=0.050

Nozzle is Ideally Expanded Pe = Pa

Curve can also tell us where Pe > or < Pa

IF: Pe > Pa Nozzle is Under-ExpandedIF: Pe < Pa Nozzle is Over-Expanded

17

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 10 100 1000

Ae/A*

T/T

con

v

Pa/Po=0.050

Nozzle is Over-Expanded (Pe < Pa)

Noz

zle

is U

nder

-Exp

ande

d (P

e >

Pa) Nozzle is Ideally Expanded (Pe = Pa)

18

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1 10 100 1000Ae/A*

T/T

con

v

Pa/Po=0.05

Pa/Po=0.03

Pa/Po=0.02

Pa/Po=0.01

Pa/Po=0.005

Pa/Po=0.003

Pa/Po=0.002

Pa/Po=0.001

Pa/Po=0.0

Decreasing Back Pressureor

Increasing Altitude

Nominal Range of Pa/P0

19

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1 10 100 1000Ae/A*

T/T

con

v

Pa/Po=0.05

Pa/Po=0.03

Pa/Po=0.02

Pa/Po=0.01

Pa/Po=0.005

Pa/Po=0.003

Pa/Po=0.002

Pa/Po=0.001

Pa/Po=0.0

Line of Maximum Thrust: Connects Locus of Maxima

For each value of Pa/P0 there is an optimum area ratio for nozzle

20

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1 10 100 1000Ae/A*

T/T

con

v

Pa/Po=0.05

Pa/Po=0.03

Pa/Po=0.02

Pa/Po=0.01

Pa/Po=0.005

Pa/Po=0.003

Pa/Po=0.002

Pa/Po=0.001

Pa/Po=0.0

Small Ratios of Pa/P0 Require Very Large Area Ratios

21

EXAMPLE: ROCKET LAUNCH Ae/A* = 20

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1 10 100 1000Ae/A*

T/T

con

v

Pa/Po=0.05

Pa/Po=0.03

Pa/Po=0.02

Pa/Po=0.01

Pa/Po=0.005

Pa/Po=0.003

Pa/Po=0.002

Pa/Po=0.001

Pa/Po=0.0

↑ Vertical Flight

Launch (Over-Expanded)

Max Thrust (Ideally Expanded)

Burnout (Under-Expanded)

Notice we are closer to Optimum Thrust on

Under-Expanded Side

22

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1 10 100 1000Ae/A*

T/T

con

v

Pa/Po=0.05

Pa/Po=0.03

Pa/Po=0.02

Pa/Po=0.01

Pa/Po=0.005

Pa/Po=0.003

Pa/Po=0.002

Pa/Po=0.001

Pa/Po=0.0

What can physically happen to supersonic flow in this region?

For this combination of pressure ratios and area ratios, a shock enters nozzle

23

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 only one Pa/P0 for which a normal shock will locate in plane of a nozzle of given area ratio Ae/A*

12

1

2*

00

120

2

2

11

1

21

2

11

1

1

1

2

ee

e

e

e

aa

ee

ee

a

MMA

A

p

p

p

p

p

p

Mp

p

Mp

p

24

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1 10 100 1000Ae/A*

T/T

con

v

Pa/Po=0.05Pa/Po=0.03Pa/Po=0.02Pa/Po=0.01Pa/Po=0.005Pa/Po=0.003Pa/Po=0.002Pa/Po=0.001Pa/Po=0.0Shock Line

On this line a normal shock wave located at exit of nozzle

25

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1 10 100 1000Ae/A*

T/T

con

v

Pa/Po=0.05

Pa/Po=0.03

Pa/Po=0.02

Pa/Po=0.01

Pa/Po=0.005

Pa/Po=0.003

Pa/Po=0.002

Pa/Po=0.001

Pa/Po=0.0

If Pe reduced substantially below Paflow can separate

A rough approximation for this condition is: Pe/Pa < 0.4

NOTE: Axial thrust direction is not usually altered by separation and CT can actually be

increased over non-separated case

26

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 = = Ae/A*

– k = = 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 → ∞

• pa/p0 → ∞ in H&P 27

THRUST COEFFICIENT VS. NOZZLE AREA RATIO FOR =1.2

28

OPTIMUM EXPANSION SUMMARY

29

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 < 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, lecture on operation of C-D nozzles coming soon

30

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 performance31

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

32