Chemical Rocket Propellant Performance Analysis · REAL NOZZLES Calculated Losses in the Space...

15/01/10Sapienza Activity in ISP-1 Program Pagina 1

PROPULSIONE SPAZIALE

Chemical Rocket Propellant PerformanceAnalysis

REAL NOZZLES

Compared to an ideal nozzle, the real nozzle has energy losses and energy that is unavailablefor conversion into kinetic energy of the exhaust gas. The principal losses are:

1. The divergence of the flow in the nozzle exit sections causes a loss. The losses can bereduced for bell-shaped nozzle contours

2. Small chamber or port area cross sections relative to the throat area or low nozzlecontraction ratios A1/At cause pressure losses in the chamber and reduce the thrust andexhaust velocity slightly

3. Lower flow velocity in the boundary layer or wall friction can reduce the effectiveexhaust velocity by 0.5 to 1.5%

4. Solid particles or liquid droplets in the gas can cause losses up to 5%

5. Unsteady combustion and oscillating flow can account for a small loss

6. Chemical reactions in nozzle flow change gas properties and gas temperatures

7. There is lower performance during transient pressure operation, for example during start,stop, or pulsing

8. For uncooled nozzle materials, such as fiber reinforced plastics or carbon, the gradualerosion of the throat region increases the throat diameter during operation. In turn thiswill reduce the chamber pressure and thrust and cause a reduction in specific impulse

9. Non-uniform gas composition can reduce performance (due to incomplete mixing,turbulence, or incomplete combustion regions)

Chemical Rocket Propellant Performance Analysis

PROPULSIONE SPAZIALEDaniele Bianchi

2 / 23

REAL NOZZLES

Calculated Losses in the Space Shuttle Booster RSRM Nozzle



2 / 23

ENERGY LOSSESTwo types of energy conversion processes occur in any propulsion system: the generation of

energy (conversion of stored energy into available energy) and, subsequently, the conversion

into kinetic energy, which is the form of energy useful for propulsion

Typical energy losses for a chemical rocket

• The combustion efficiency for chemical rockets is a measure of the source efficiency forcreating energy. Its value is typically high (approximately 94 to 99%)

• A large portion of the energy of the exhaust gases is unavailable for conversion intokinetic energy and leaves the nozzle as residual enthalpy



3 / 23

ENERGY LOSSESTwo types of energy conversion processes occur in any propulsion system: the generation of

energy (conversion of stored energy into available energy) and, subsequently, the conversion

into kinetic energy, which is the form of energy useful for propulsion

Propulsive efficiency at varying velocities

• The propulsive efficiency determines how much of the kinetic energy of the exhaust jet isuseful for propelling the vehicle. The propulsive efficiency is a maximum when theforward vehicle velocity is exactly equal to the exhaust velocity (residual kinetic energy ofthe jet is zero)



3 / 23

PERFORMANCE COEFFICIENTS AND PARAMETERS

The thrust of a rocket motor can be expressed as:

F = m CF c∗

we now have split the thrust into two contributing factors:• The thrust coefficient CF = F/(p0At) which is a nozzle quality factor• The characteristic velocity c∗ = (p0At)/m which is a propellant quality factor

Combustion chamber with the regions which are important for the characteristic velocity and thrust coefficient

The specific impulse can be expressed in terms of c∗ and CF:

Ispg0 = c∗CF



4 / 23

CHARACTERISTIC VELOCITY AND SPECIFIC IMPULSE

The characteristic velocity has been used frequently in propulsion literature and is defined as:

c∗ = p0At

m= 1

Γ

√RT0

M

• It is a function of the propellant characteristics and combustion chamber design• It is independent of nozzle characteristics• It is used as a figure of merit in comparing propellants and combustion chamber designs

High values of c∗ can be obtained if the average molecular weight of the reaction products is

low or if the available chemical energy is large (high combustion temperatures)

The thrust and the specific impulse can be expressed in terms of c∗ and CF:

Thrust F = mc∗CF Specific impulse Is = c∗CF

g0

The influence of variations in the specific heat ratio γ on various parameters (such as c∗ or CF)is not as large as the changes in chamber temperature, pressure ratio, or molecular mass. It is:

• 1.67 for monatomic gases such as helium and argon• 1.4 for cold diatomic gases such as hydrogen, oxygen, and nitrogen• for triatomic and beyond it varies from 1.1 to 1.3 (methane is 1.11 and ammonia 1.33)



5 / 23

TYPICAL ROCKET EXHAUST GASES PROPERTIES

Results of calculations for several different liquid and solid propellant combinations are given:

Theoretical Performance of Liquid Rocket Propellant Combinations



6 / 23

THEORETICAL ROCKET PERFORMANCE (TAF AND C∗)

mixture ratio

ad

iab

ati

c t

em

pe

ratu

re, K

0 1 2 3 4 5 6 7 8 9 10 11 121000

1250

1500

1750

2000

2250

2500

2750

3000

3250

3500

3750

4000

4250

UDMH/NTO

MMH/NTO

RP1/LOX

LH2/LOX

CH4/LOX

Pc = 100 bar

mixture ratio

ch

ara

cte

ris

tic

ve

loc

ity

, m

/s

0 1 2 3 4 5 6 7 8 9 10 11 121000

1200

1400

1600

1800

2000

2200

2400

2600

UDMH/NTO

MMH/NTORP1/LOXLH2/LOX

CH4/LOX

Pc = 100 bar

• The peak of Taf is at stoichiometric conditions (O/F)st

• The peak of c∗ is always shifted towards lower O/F than (O/F)st



7 / 23

THEORETICAL ROCKET PERFORMANCE (ISP AND ISPVAC )

mixture ratio

I spg

0, m

/s

0 1 2 3 4 5 6 7 8 9 10 11 121750

2000

2250

2500

2750

3000

3250

3500

3750

4000

4250

4500

4750

5000

UDMH/NTO

MMH/NTO

RP1/LOX

LH2/LOX

CH4/LOX

Pc = 100 bar

pexit

= 0.25 patm

mixture ratio

I sp

,va

cg

0, m

/s

0 1 2 3 4 5 6 7 8 9 10 11 121750

2000

2250

2500

2750

3000

3250

3500

3750

4000

4250

4500

4750

5000

UDMH/NTO

MMH/NTO

RP1/LOX

LH2/LOX

CH4/LOX

Pc = 100 bar

Ae/A

t = 150

• Isp expressed in [m/s] is obtained assuming pa = pe (adapted nozzle)

• Ispvac expressed in [m/s] is obtained assuming pa = 0 (vacuumcondition)



8 / 23

CHEMICAL ROCKET PROPELLANT PERFORMANCE ANALYSIS

• For maximum specificimpulse, the optimum O/Fis ≈ 2.3 for frozenequilibrium and 2.8 forshifting equilibrium

• The maximum values of c∗

are at slightly different O/F

• The optimum O/F is notthe one for highesttemperature, which isusually close to thestoichiometric value(≈ 3.4)

• The temperature and themolecular weight at thenozzle exit increase forshifting equilibrium due torecombination reactions

• Much of the carbon isburned to CO2 and almostall of the hydrogen to H2O

Performance of LOX/RP-1 as a function of O/F



9 / 23








O/F

ch

ara

cte

ris

tic

ve

loc

ity

, m

/s

mo

lec

ula

r w

eig

ht,

km

ol/k

g

ch

am

be

r te

mp

era

ture

, K

1 2 3 4 5 6 7 8 9 101000

1200

1400

1600

1800

2000

2200

2400

2600

2800

0

5

10

15

20

25

30

35

40

500

1000

1500

2000

2500

3000

3500

4000

characteristic velocitymolecular weightchamber temperature

stoichiometriccondition (max. T

f)

max. c*

nozzle expansion ratio 40Pressure 1000 psi (69 bar)Fuel RP1(L)Oxidizer O

2(L)

Performance of LOX/RP-1 as a function of O/F (assuming frozen

equilibrium)



9 / 23








O/F

ch

ara

cte

ris

tic

ve

loc

ity

, m

/s

sp

ec

ific

im

pu

lse

, s

ch

am

be

r te

mp

era

ture

, K

1 2 3 4 5 6 7 8 9 101000

1200

1400

1600

1800

2000

2200

2400

2600

2800

250

275

300

325

350

375

400

425

450

475

500

500

1000

1500

2000

2500

3000

3500

4000

characteristic velocityspecific impulse (vacuum)chamber temperature


f)

max. Ispvac

max. c*


2(L)

Performance of LOX/RP-1 as a function of O/F (assuming frozen

equilibrium)



9 / 23








O/F

ch

ara

cte

ris

tic

ve

loc

ity

, m

/s

sp

ec

ific

im

pu

lse

, s

ch

am

be

r te

mp

era

ture

, K

1 2 3 4 5 6 7 8 9 101000

1200

1400

1600

1800

2000

2200

2400

2600

2800

250

275

300

325

350

375

400

425

450

475

500

500

1000

1500

2000

2500

3000

3500

4000


max. Ispvac

max. c*


2(L)

Performance of LOX/RP-1 as a function of O/F (frozen equilibrium and

shifting equilibrium)



9 / 23








O/F

ch

ara

cte

ris

tic

ve

loc

ity

, m

/s

sp

ec

ific

im

pu

lse

, s

ch

am

be

r te

mp

era

ture

, K

1 2 3 4 5 6 7 8 9 101000

1200

1400

1600

1800

2000

2200

2400

2600

2800

250

275

300

325

350

375

400

425

450

475

500

500

1000

1500

2000

2500

3000

3500

4000



2(L)

Performance of LOX/RP-1 as a function of O/F (frozen equilibrium,

shifting equilibrium and Bray’s throat freezing point)



9 / 23





• The optimum O/F is notthe one for highesttemperature, which isusually close to thestoichiometric value (8.0)


O/F

ch

ara

cte

ris

tic

ve

loc

ity

, m

/s

mo

lec

ula

r w

eig

ht,

km

ol/k

g

ch

am

be

r te

mp

era

ture

, K

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161000

1200

1400

1600

1800

2000

2200

2400

2600

2800

0

5

10

15

20

25

30

35

40

500

1000

1500

2000

2500

3000

3500

4000characteristic velocitymolecular weightChamber temperature


f)

max. c*

nozzle expansion ratio 40Pressure 1000 psi (69 bar)Fuel H

2(L)

Oxidizer O2(L)

Performance of LOX/LH2 as a function of O/F (assuming frozen

equilibrium)



10 / 23







O/F

ch

ara

cte

ris

tic

ve

loc

ity

, m

/s

sp

ec

ific

im

pu

lse

, s

Ch

am

be

r te

mp

era

ture

, K

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161000

1200

1400

1600

1800

2000

2200

2400

2600

2800

250

275

300

325

350

375

400

425

450

475

500

500

1000

1500

2000

2500

3000

3500

4000characteristic velocityspecific impulse (vacuum)Chamber temperature

max. Ispvac


2(L)

Oxidizer O2(L)


f)

max. c*

Performance of LOX/LH2 as a function of O/F (assuming frozen

equilibrium)



10 / 23







O/F

ch

ara

cte

ris

tic

ve

loc

ity

, m

/s

sp

ec

ific

im

pu

lse

, s

Ch

am

be

r te

mp

era

ture

, K

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161000

1200

1400

1600

1800

2000

2200

2400

2600

2800

250

275

300

325

350

375

400

425

450

475

500

500

1000

1500

2000

2500

3000

3500


max. Ispvac


2(L)

Oxidizer O2(L)

max. c*

Performance of LOX/LH2 as a function of O/F (frozen equilibrium and

shifting equilibrium)



10 / 23







O/F

ch

ara

cte

ris

tic

ve

loc

ity

, m

/s

sp

ec

ific

im

pu

lse

, s

Ch

am

be

r te

mp

era

ture

, K

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161000

1200

1400

1600

1800

2000

2200

2400

2600

2800

250

275

300

325

350

375

400

425

450

475

500

500

1000

1500

2000

2500

3000

3500



2(L)

Oxidizer O2(L)

Performance of LOX/LH2 as a function of O/F (frozen equilibrium,

shifting equilibrium and Bray’s throat freezing point)



10 / 23

FROZEN VS SHIFTING EQUILIBRIUM GAS COMPOSITION

Combustion chamber Nozzle exit

• Dissociation of molecules requires considerable energy and causes a decrease in thecombustion temperature, which in turn can reduce the specific impulse

• Atoms or radicals such as O or H and OH are formed. As the gases are cooled in theexpansion, the dissociated species recombine (shifting equilibrium) and release heat intothe flowing gases. Only a small percentage of dissociated species persists at the nozzle exit



11 / 23


O/F

Co

mp

os

itio

n, m

as

s f

rac

tio

n

1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0 H2

O2

H2O

H

O

OH

CO2

CO

solid lines = chamber compositiondashdotted lines = exit composition


2(L)

Mass fraction

O/F

Co

mp

os

itio

n, m

ole

fra

cti

on

1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0 H2

O2

H2O

H

O

OH

CO2

CO



2(L)

Mole fraction





12 / 23


O/F

Co

mp

os

itio

n, m

as

s f

rac

tio

n

2 3 4 5 6 7 8 9 10 11 12 13 14 15 160.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0 H2

O2

H2O

H

O

OH



2(L)

Oxidizer O2(L)

Mass fraction

O/F

Co

mp

os

itio

n, m

ole

fra

cti

on

2 3 4 5 6 7 8 9 10 11 12 13 14 15 160.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

H2

O2

H2O

H

O

OH



2(L)

Oxidizer O2(L)

Mole fraction





13 / 23


O/F

Mo

lec

ula

r m

as

s, k

g/k

mo

le

Ex

it t

em

pe

ratu

re, K

2 3 4 5 6 7 8 9 10 11 12 13 14 15 163

6

9

12

15

18

21

24

250

500

750

1000

1250

1500

1750

2000

2250

2500Molecular mass (chamber)

Molecular mass (exit)

Exit temperature


f)


2(L)

Oxidizer O2(L)

Shifting equilibrium

O/F

Mo

lec

ula

r m

as

s, k

g/k

mo

le

Ex

it t

em

pe

ratu

re, K

2 3 4 5 6 7 8 9 10 11 12 13 14 15 163

6

9

12

15

18

21

24

250

500

750

1000

1250

1500

1750

2000

2250

2500Molecular mass (chamber)

Molecular mass (exit)

Exit temperature


f)


2(L)

Oxidizer O2(L)

Frozen





14 / 23

THRUST AND THRUST COEFFICIENT

If the diverging part of the nozzle did not contribute to the thrust, and if there were no flow in

the combustion chamber, the thrust delivered by the rocket motor in vacuum, F0 would be

F0 = p0At

Combustion chamber without the nozzle. Wall pressure is assumed constant

As the divergent section of the nozzle does contribute to the thrust, and as there is a flow in thecombustion chamber, the thrust F differs from F0.

The thrust coefficient is defined as the ratio of actual thrust and the reference thrust:

CF = Fp0At



15 / 23

THRUST AND THRUST COEFFICIENTThe thrust coefficient is the thrust divided by the chamber pressure and the throat area

CF =F

p0At= Γ

√2γγ − 1

[1 −(

pe

p0

) γ−1γ

]+(

pe

p0−

pa

p0

)Ae

At

For any fixed pressure ratio p0/pa, the thrust coefficient CF has a peak when pe = pa. This peak

is the optimum thrust coefficient and is an important criterion in nozzle design



16 / 23

THRUST AND THRUST COEFFICIENTThe thrust coefficient is the thrust divided by the chamber pressure and the throat area

CFopt =F

p0At= Γ

√2γγ − 1

[1 −(

pe

p0

) γ−1γ

]For any fixed pressure ratio p0/pa, the thrust coefficient CF has a peak when pe = pa. This peak

is the optimum thrust coefficient and is an important criterion in nozzle design

Optimum thrust coefficient CF as a function of p0/pe , ε, and γ



16 / 23


Thrust coefficient as a function of ambient pressure ratio p0/pa and area ratio ε for different γ

Thrust coefficient CF versus nozzle area ratio for γ = 1.20

The CF is a convenient parameter for seeing the effects of chamber pressure or altitude

variations in a given nozzle, or to correct sea-level results for flight altitude conditions



17 / 23


Thrust coefficient as a function of ambient pressure ratio p0/pa and area ratio ε for different γ

Thrust coefficient CF versus nozzle area ratio for γ = 1.30

The CF is a convenient parameter for seeing the effects of chamber pressure or altitude

variations in a given nozzle, or to correct sea-level results for flight altitude conditions



17 / 23

ENGINE PERFORMANCE COMPARISON (LOX/LH2)

RS-25-SSME engine (USA)

Propellants LOX/LH2Mixture ratio 6.0 : 1P Chamber 226 barIsp vacuum 453 sArea ratio 77.5 : 1

Vulcain 2 engine (Europe)

Propellants LOX/LH2Mixture ratio 6.7 : 1P Chamber 117 barIsp vacuum 429 sArea ratio 58 : 1

Vulcain engine (Europe)

Propellants LOX/LH2Mixture ratio 6.2 : 1P Chamber 102 barIsp vacuum 431 sArea ratio 45 : 1



18 / 23

ENGINE PERFORMANCE COMPARISON (LOX/RP-1)

RD-180 engine (Russia)

Propellants LOX/RP-1Mixture ratio 2.7 : 1Chamber P. 267 barIsp vacuum 338 sArea ratio 37 : 1

NK-33 engine (Russia)


F-1 engine (USA)




19 / 23

ENGINE THRUST COEFFICIENT COMPARISON (LOX/LH2)

Area ratio ε = Ae/A

t

CF

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

42 6 8 4010 20 60 10080

CFsea

= 1.60

Area ratio = 77.5

CFvac

= 1.95

SSME engineLOXLH2 O/F = 6.0p

c = 226 bar T

c = 3607 K

c* = 2292 m/sγ = 1.1915

infinity

223200

333

500

1000

100

0.35

0.15

0.40

33

2.0

2.5

5.0

3.3

10

20

50

RS-25-SSME engine (USA)Propellants LOX/LH2

Mixture ratio 6.0 : 1P Chamber 226 barIsp vacuum 453 sArea ratio 77.5 : 1


t

CF

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

42 6 8 4010 20 60 10080

CFvac

= 1.93

CFsea

= 1.42

Area ratio = 58

Vulcain 2 engineLOXLH2 O/F = 6.7p

c = 117 bar T

c = 3624 K

c* = 2222 m/sγ = 1.1902

0.35

0.15

0.40

33

2.0

2.5

5.0

3.3

10

20

50

infinity

115.5

200

333500

1000

100

Vulcain 2 engine (Europe)Propellants LOX/LH2

Mixture ratio 6.7 : 1P Chamber 117 barIsp vacuum 429 sArea ratio 58 : 1



20 / 23

ENGINE THRUST COEFFICIENT COMPARISON (LOX/RP-1)


t

CF

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

42 6 8 4010 20 60 10080

CFsea

= 1.73

Area ratio = 37

CFvac

= 1.87

RD180 engineLOXRP1 O/F = 2.7p

c = 267 bar T

c = 3886 K

c* = 1777 m/sγ = 1.2030

0.35

0.15

0.40

33

2.0

2.5

5.0

3.3

10

20

50

infinity

263.5200

333

500

1000

100

RD-180 engine (Russia)Propellants LOX/RP-1

Mixture ratio 2.7 : 1Chamber P. 267 barIsp vacuum 338 sArea ratio 37 : 1


t

CF

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

42 6 8 4010 20 60 10080

0.35

0.15

0.40

CFsea

= 1.55

F1 engineLOXRP1 O/F = 2.3p

c = 70 bar T

c = 3574 K

c* = 1770 m/sγ = 1.2169

Area ratio = 16

CFvac

= 1.78

33

2.0

2.5

5.0

3.3

10

20

infinity

50

69.1

200

333500

1000

100

F-1 engine (USA)Propellants LOX/RP-1

Mixture ratio 2.3 : 1Chamber P. 70 barIsp vacuum 304 sArea ratio 16 : 1



21 / 23

THRUST COEFFICIENT - ENGINE COMPARISON


t

CF

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

RD180 (267 bar)

SSME (226 bar)

Vulcain 2 (117 bar)

Vulcain (102 bar)

F1 (70 bar)

Merlin 1C (68 bar)

RSRM (45.6 bar)

42 6 8 4010 20 60 10080

RSRM

Merlin 1C

RSRM

F1

Merlin 1C

Vulcain

F1

Vulcain 2

Vulcain

RD180

Vulcain 2

SSME

RD180

SSME

Sea Level

Vacuum

• At sea level, the chamber pressure is strongly influencing the CF• At vacuum conditions the CF is only a function of γ and nozzle expansion ratio ε



22 / 23

THRUST COEFFICIENT AND SEPARATIONFor most applications, the rocket system has to operate over a range of altitudes; for a fixed

chamber pressure and area ratio the condition of optimum expansion (pe = pa) occurs only at

one altitude, and the nozzle will operate at either over-expanded or under-expanded conditions

The effect of separation is to increase the thrust and the thrust coefficient over the value that

they would have if separation had not occurred (but not as good as an optimum nozzle)



23 / 23

Chemical Rocket Propellant Performance Analysis · REAL NOZZLES Calculated Losses in the Space...

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