lecture 2 thrust equation, nozzles and definitionscantwell/aa284a_course_material/karabe… · –...

Lecture 2 Thrust Equation, Nozzles and Definitions

Prepared by

Arif Karabeyoglu

Mechanical Engineering KOC University

Fall 2019

MECH427/527 and AA 284a Advanced Rocket Propulsion

Stanford University

Advanced Rocket Propulsion

Stanford University

Derivation of the Static Thrust Expression

•  Force balance in the x direction

•  Assumption 1: Static firing/ External gas is at rest •  Assumption 2: No body forces acting on the rocket

( ) 0ˆˆ =⋅−++ ∫∫ xA

xA

a AdnIPAdnPT

cs

τ

Karabeyoglu

2

ForceThrustTVelocityExitUessureExitP

essureAmbientPessurePTensorStressTensorUnityI

e

e

a

::

Pr:Pr:

Pr:::

τ

x


Stanford University

Derivation of the Static Thrust Expression •  First Integral:

–  Since As+Ac and Ae+Ac are closed surfaces and Pa is constant

–  These integrals can be separated and combined to yield the following simple expression for the first integral

0ˆ =∫+

xAA

a AdnP

cs

Karabeyoglu

3

0ˆ =∫+

xAA

a AdnP

ec

eaxA

a APAdnP

s

=∫ ˆ


Stanford University

Derivation of the Static Thrust Expression •  Second Integral:

–  Assumption 3: No body forces on the working gas –  Momentum equation:

–  Assumption 4: quasi-steady operation –  Define control volume (cv) as volume covered by Ac+Ae –  Assumption 5: cv is constant in time –  Integral of the momentum eq. over the cv

( ) 0=−+⋅∇+∂

∂τρ

ρIPuu

tu

Karabeyoglu

4

( ) 0=−+⋅∇∫ dvIPuucv

τρ

DensityVectorVelocityu

::

ρ


Stanford University

•  Second Integral: –  Use Gauss’s Theorem to obtain

–  With use of the no slip condition, this equation takes the following form in the x-direction

–  We have used the following assumptions •  Assumption 6: Quasi 1D flow at the nozzle exit. Higher order averaging

terms are ignored. Velocity parallel to x-axis at the exit plane •  Assumption 7:

•  Average quantities have been introduced at the exit plane


( ) ( ) 0ˆˆ =⋅−++⋅−+ ∫∫ dAnIPuudAnIPuu

ec AA

τρτρ

( ) 0ˆ 2 =++⋅−∫ eeeeex

A

APAuAdnIP

c

ρτ

0ˆ ≅⋅∫ xA

Adn

e

τ

Karabeyoglu

5

xA

e AdPPe

∫≡ xA

e Adnuue

∫ ⋅≡ ˆ xA

e Ade

∫≡ ρρ

( ) eeeeex

A

APAuAdnIP

c

−−=⋅−∫ 2ˆ ρτ⇒


Stanford University

•  Combined to obtain the thrust force

•  Introduce the mass flow rate:

•  Two terms can be combined by introducing the effective exhaust velocity, Ve

•  Maximum thrust for unit mass flow rate requires –  High exit velocity –  High exit pressure

•  This cannot be realized. Compromise -> optimal expansion


( ) eaeeee APPAuT −+= 2ρ

( ) eaee APPumT −+= !

eVmT !=

Karabeyoglu

6

eee Aum ρ=!

RateFlowMassm :!


Stanford University

Convergent Divergent Nozzle Design Issues

Karabeyoglu

7

1@Pr:*Pr:2

=MessurePessureStagnationChamberPt


Stanford University

•  For isentropic flow - perfect gas-no chemical rxns 1. Mass flow rate relation for chocked flow

2. Area relation (Nozzle area ratio-pressure ratio relation)

3. Velocity relation

•  Solve for pressure ratio from 2 and evaluate the velocity using 3.

Convergent Divergent Nozzle Design Issues

21

11

2

212

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=−+

γγ

γγ

t

ttRTPAm!

21

1

2

1

2

11

111

211

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛ +=≡

−− γ

γγγ

γγγ

ε t

e

t

e

e

tPP

PP

AA

Karabeyoglu

8

21

1

2

2 11

2

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

−γγ

γγ

t

e

w

tue P

PMTRu

.:.:

:2

ConsGasUniversalRConsGasR

eTemperaturStagnationChamberT

u

t

HeatsSpecificofRatioRatioAreaNozzleAreaThroatNozzleA

WeightMolecularMw

t

::::

γε


Stanford University

•  Thrust equation:

•  At fixed flow rate, chamber and atmospheric pressures, the variation in thrust can be written as

•  Momentum equation in 1D

•  Substitute in the differential expression for thrust

•  Maximum thrust is obtained for a perfectly expanded nozzle

Maximum Thrust Condition ( ) eaee APPumT −+= !

( ) eeeaee dPAdAPPdumdT +−+= !

( ) eae dAPPdT −=

Karabeyoglu

9

0=+ dPAduuAρ dPAdum −=!

( )aee

PPdAdT −=

ae PP =


Stanford University

A)  Equilibrium: –  Mechanical:

–  Needed to define an equilibrium pressure –  Very fast compared to the other time scales

–  Thermal: –  Relaxation times associated with the internal degrees freedom of the gas –  Rotational relaxation time is fast compared to the other times –  Vibrational relaxation time is slow compare to the rotational. Can be important

in rocket applications. –  Calorically perfect gas assumption breaks down.

–  Chemical: –  Finite time chemical kinetics (changing temperature and pressure) –  Three cases are commonly considered:

–  Fast kinetics relative to residence time- Shifting equilibrium (Chemical composition of the gases match the local equilibrium determined by the local pressure and temperature in the nozzle)

–  Slow kinetics relative to residence time- Frozen equilibrium (Chemical composition of the gases is assumed to be fixed)

–  Nonequilibrium kinetics (Nozzle flow equations are solved simultaneously with the chemical kinetics equations)

Other Nozzle Design Issues

Karabeyoglu

10


Stanford University

B) Calorically perfect gas –  Typically not valid. Chemical composition shifts, temperature

changes and vibrational non-equilibrium.

C) Effects of Friction –  Favorable pressure gradient unless shock waves are located inside

the nozzle. –  The effects can be list as

•  Nozzle area distribution changes due to displacement layer thickness •  Direct effect of the skin friction force •  Shock boundary layer interaction-shock induced separation. •  Viscous effects are small and generally ignored for shock free nozzles

D) Multi Phase Flow Losses E) Effects of 3D flow field

–  Velocity at the exit plane is not parallel to the nozzle axis, because of the conical flow field.


Karabeyoglu

11


Stanford University

–  Types of nozzle geometries •  Conical nozzle

–  Simple design and construction –  Typical divergence angle 15 degrees (~2% Isp loss) –  3D thrust correction can be significant

•  Perfect nozzle –  Method of characteristics to minimize 3D losses –  Perfect nozzle is too long

•  Optimum nozzle (Bell shaped nozzles) –  Balance length/weight with the 3D flow losses

•  Plug nozzle and Aerospike nozzle –  Good performance over a wide range of back pressures


Karabeyoglu

12

( ) ( ) eaee APPumT −++=2cos1 α! AngleConeNozzle:α


Stanford University

Optimum Bell Nozzle - Example

Karabeyoglu

13


Stanford University

Plug and Aerospike Nozzles

Karabeyoglu

14


Stanford University

•  Thrust equation:

•  Thrust coefficient:

•  For isentropic flow and calorically perfect gas in the nozzle the thrust coefficient can be written as

Definitions-Thrust Coefficient ( ) eaee APPumT −+= !

t

e

t

a

t

e

tt

e

ttF A

APP

PP

PAum

PATC ⎟⎟⎠

⎞⎜⎜⎝

⎛−+=≡

2222

!

Karabeyoglu

15

t

e

t

a

t

e

t

eF A

APP

PP

PPC ⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛−

=

−−+

22

21

1

2

11

21

12

12 γ

γγγ

γγγ


Stanford University

•  Mass flow equation (choked and isentropic flow of a calorically perfect gas in the convergent section of the nozzle):

•  Definition of c*:

•  c* can be expressed in terms of the operational

parameters as

Definitions - c* Equation

21

11

2

212

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=−+

γγ

γγ

t

ttRTPAm!

Karabeyoglu

16

mAPc tt!2* ≡

21

211

211*

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⎟⎠⎞⎜

⎝⎛ += −

+

w

tuMTRc γ

γγ

γ


Stanford University

•  Combine the definitions of the thrust coefficient and c* to express the thrust

•  Think of nozzle as a thrust amplifier and CF as the gain •  Specific Impulse: Thrust per unit mass expelled

•  Impulse Density: Thrust per unit volume of propellant expelled

Definitions – Specific Impulse and Impulse density

FCcmT *!=

o

F

osp g

CcgmTI *=≡!

Karabeyoglu

17

Energy of the gases In the chamber

Flow in the nozzle

pspp

o

opI

Vgm

gmT

VT ρδ ==≡ !

!

!!

DensityopellantEarthonConsnalGravitatiog

p

o

Pr:.:

ρ

ppmV ρ!! ≡


Stanford University

•  The total impulse is defined as

•  Average thrust

•  Delivered Isp

Definitions – Total Impulse, Average Thrust, Delivered Isp

b

tott

b tITdt

tT

b

=≡ ∫0

1

( )op

totdelsp gM

II =

Karabeyoglu

18

∫∫ =≡bb t

osp

t

tot dtgImTdtI00

!

MassopellantTotalMTimeBurnTotalt

p

b

Pr::


Stanford University Karabeyoglu

19

2.1=γ

Overexpanded

Underexpanded at PP2

∞→at PP2

ae PP =

=at PP2

Lift off

Burn out

Thrust Coefficient Curves (from Sutton)


Stanford University

•  For the specific value of •  CF varies from 0.75 to 2.246

–  Assuming chocked flow –  CF is order 1

•  For the case of convergent nozzle –  For subsonic flow in the nozzle –  CF varies from 0.75 (optimal) to 1.25

•  For convergent divergent nozzles –  For a given pressure ratio there exists an optimum area ratio

that maximizes the thrust coefficient –  Right of the optimum is overexpanded

•  Shock structure is initially outside the nozzle •  For larger area ratios shock moves inside the nozzle and the boundary

separation takes place •  This has a positive impact on the CF

Observations on the Thrust Coefficient Curves

1=ε

2.1=γ

3.22 <at PP

Karabeyoglu

20

1>εat PP2


Stanford University

–  Left of the optimal is underexpanded •  No separation in this case

•  Optimal area ratio increases with increasing pressure ratio –  Upper stages have large nozzle area ratios (i.e.70) –  Booster stages have low area ratios (i.e. 10)

•  Vacuum Isp and sea level Isp values can be quite different –  For example:

•  Vacuum value: 1.9 •  Sea Level value: 1.2 (58% reduction in thrust)

•  All curves are enveloped by the vacuum line. Vacuum Isp is always the largest value.

•  All of the qualitative observations are valid for other gamma values.

Observations on the Thrust Coefficient Curves

Karabeyoglu

21


Stanford University

•  Summerfield Criterion: –  Danger of separation is present if

–  Calculate Pe from isentropic flow equations –  Separation is likely if

–  If can be lower –  For large nozzles modern data suggests

•  With separation flow ignores the divergent section beyond the starting point of the oblique shock wave

•  This limits the drop in the thrust coefficient •  Conical nozzles operate better at low Pe/Pa ratios for which the

separation is expected. –  Nozzle exit divergence angle determines the stability of the separation zone.

As the angle increases, the stability of the separation zone improves

Shock Induced Flow Separation in Nozzles o

e 15≈α

oe 15>α

Karabeyoglu

22

16/2 >at PP

( ) 40.0// ≅< craeae PPPP( )crae PP /

( ) 286.0// ≅< craeae PPPP

lecture 2 thrust equation, nozzles and definitionscantwell/aa284a_course_material/karabe… · –...

Documents