chemical propulsion systems - elearning.mechatronika.cool

Chemical Propulsion SystemsSpace for Education, Education for Space

Space for Education, Education for SpaceESA Contract No. 4000117400/16NL/NDe

Specialized lectures

Chemical Propulsion Systems

Vladimír Kutiš, Pavol Valko


Contents

2

1. Fluid Flow and Thermodynamics

2. Chemical Rocket Propulsion

3. Performance of Rocket Vehicle


• Introduction

• Fundamental equations

• Thermodynamics of gases

• Speed of sound

• Isentropic flow

• Nozzle fluid flow


3


• Fluid flow: – naturally three dimensional, but in some special cases can be

considered as one dimensional or quasi-one dimension

– fluid can be considered according to:

• steady-state VS transient

• turbulent VS laminar

• inviscid VS viscous fluid

1. Fluid Flow and ThermodynamicsIntroduction

4

quasi-one dimension fluid flow

inlet outlet


• System or Control Mass (CM):

– is a collection of matter of fixed identity

– it may be considered enclosed by an invisible, massless, flexible surface through which no matter can pass

– the boundary of the system may change position, size, and shape

– is also called control mass

Introduction

5


moving of control mass

inlet outlet


• Control Volume (CV):

– is arbitrary volume fixed to the coordinate system (stationary or moving)

– bounded by control surface (CS) through which fluid may pass, CV can has differential or finite size

Introduction

6


differential CV finite CV



• There are 4 fundamental equations, which must be considered:

– Continuity equation

– Momentum equation

– Energy equation

– Entropy equation

Fundamental equations

7



• Continuity equationFundamental equations

8

11

111

,

,,,

T

upA

22

222

,

,,,

T

upA

inviscid steady-state flow

0CV enters

mass Rate

CV leaves

mass Rate




9

11

111

,

,,,

T

upA

22

222

,

,,,

T

upA


0CV enters

mass Rate

CV leaves

mass Rate

222 uA 111 uA




10

11

111

,

,,,

T

upA

22

222

,

,,,

T

upA


0111222 uAuA

0CV enters

mass Rate

CV leaves

mass Rate

222 uA 111 uA




11

0CV enters

mass Rate

CV leaves

mass Rate

T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx dAAduud


uA




12

0CV enters

mass Rate

CV leaves

mass Rate

T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx dAAduud


0 uAdAduudA

uA




13

0CV enters

mass Rate

CV leaves

mass Rate

T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx dAAduud

0A

dA

u

dud


0 uAdAduudA

uA



• Momentum equationFundamental equations

14


11

111

,

,,,

T

upA

22

222

,

,,,

T

upA

CV enters

momentum

Rate

CV leaves

momentum

Rate

direction

in CVin gas

on Forces

x




15


11

111

,

,,,

T

upA

22

222

,

,,,

T

upA

2

222 uA 2

1

2211

A

ApdAApApF

CV enters

momentum

Rate

CV leaves

momentum

Rate

direction

in CVin gas

on Forces

x

2

111 uA




16


11

111

,

,,,

T

upA

22

222

,

,,,

T

upA

2

222 uA 2

1

2211

A

ApdAApApF

CV enters

momentum

Rate

CV leaves

momentum

Rate

direction

in CVin gas

on Forces

x

2

111 uA

2

111

2

2222211

2

1

uAuApdAApApA

A




17


T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx 2

dAAduud

CV enters

momentum

Rate

CV leaves

momentum

Rate

direction

in CVin gas

on Forces

x

Au2




18


T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx 2

dAAduud

CV enters

momentum

Rate

CV leaves

momentum

Rate

direction

in CVin gas

on Forces

x

Au2

uduA




19


T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx

AdpAdAAdppp

dAAdpppAF

2

1

CV enters

momentum

Rate

CV leaves

momentum

Rate

direction

in CVin gas

on Forces

x

uduA




20


T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx Adp

uduAAdp

CV enters

momentum

Rate

CV leaves

momentum

Rate

direction

in CVin gas

on Forces

x

uduA




21


T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx Adp

uduAAdp ududp

CV enters

momentum

Rate

CV leaves

momentum

Rate

direction

in CVin gas

on Forces

x

uduA

Euler’s equation



• Energy equationFundamental equations

22


11

111

,

,,,

T

upA

22

222

,

,,,

T

upA

CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate




23


11

111

,

,,,

T

upA

22

222

,

,,,

T

upA

CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate

inin QW

2

2

22222

ueAu

2

2

11111

ueAu




24


11

111

,

,,,

T

upA

22

222

,

,,,

T

upA

CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate

inin QW

ins,inp, WW

2

2

22222

ueAu

2

2

11111

ueAu




25


11

111

,

,,,

T

upA

22

222

,

,,,

T

upA

CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate

inin QW

ins,inp, WW

2

2

22222

ueAu

222

2

2111

1

1 Aup

Aup

2

2

11111

ueAu




26


11

111

,

,,,

T

upA

22

222

,

,,,

T

upA

CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate

inins,222

2

2111

1

1 QWAup

Aup

2

2

22222

ueAu

2

2

11111

ueAu

22

2

1

1

11111

2

2

2

22222inins,

upeAu

upeAuQW




27


11

111

,

,,,

T

upA

22

222

,

,,,

T

upA

CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate

22

2

11111

2

22222inins,

uhAu

uhAuQW

using enthalpy h




28


11

111

,

,,,

T

upA

22

222

,

,,,

T

upA

CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate

22

2

11111

2

22222inins,

uhAu

uhAuQW

adiabatic processno shaft work

22

2

22

2

11

uh

uh

using enthalpy h




29


CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate

T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx

2/

2duudee

dAAduud

2/2ueuA




30


CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate

T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx

2/

2duudee

dAAduud

ududeuA

2/2ueuA




31


CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate

T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx ududeuA

inin QW

ins,inp, WW

dAAduudppuAp pudAAudppAdu




32


CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate

T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx ududeuA

inins, QWpudAAudppAdu

udude

qwpd

inins,/




33


CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate

T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx ududeuA

inins, QWpudAAudppAdu

udude

qwpd

inins,/

ududhqw inins,

using enthalpy h




34


CV enters

energy

Rate

CV leaves

energy

Rate

heat and

by work CV

intotransfer

energy Rate

T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dxadiabatic processno shaft work

0ududh

ududhqw inins,



• Entropy equationFundamental equations

35

11

111

,

,,,

T

upA

22

222

,

,,,

T

upA


CV

enters

entropy

Rate

CV

leaves

entropy

Rate

ilitiesirreversib

andheat by CV

intotransfer

entropy Rate




36

11

111

,

,,,

T

upA

22

222

,

,,,

T

upA


2Sgen

2

1

in ST

Q

1S

CV

enters

entropy

Rate

CV

leaves

entropy

Rate

ilitiesirreversib

andheat by CV

intotransfer

entropy Rate




37

11

111

,

,,,

T

upA

22

222

,

,,,

T

upA


2Sgen

2

1

in ST

Q

1S

12gen

2

1

in SSST

Q

CV

enters

entropy

Rate

CV

leaves

entropy

Rate

ilitiesirreversib

andheat by CV

intotransfer

entropy Rate




38

11

111

,

,,,

T

upA

22

222

,

,,,

T

upA


2Sgen

2

1

in ST

Q

1S

12gen

2

1

in SSST

Q

reversible process 1-2

12

2

1

in SST

Q

CV

enters

entropy

Rate

CV

leaves

entropy

Rate

ilitiesirreversib

andheat by CV

intotransfer

entropy Rate




39

11

111

,

,,,

T

upA

22

222

,

,,,

T

upA


2Sgen

2

1

in ST

Q

1S

12gen

2

1

in SSST

Q

CV

enters

entropy

Rate

CV

leaves

entropy

Rate

ilitiesirreversib

andheat by CV

intotransfer

entropy Rate

adiabaticreversible process 1-2

120 SS

isentropicprocess 1-2




40


T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx

CV

enters

entropy

Rate

CV

leaves

entropy

Rate

ilitiesirreversib

andheat by CV

intotransfer

entropy Rate

SdS genin Sd

T

Q

S

SdSdT

Q

genin

reversible process

sdT

qSd

T

Q

inin ,




41


T

upA

,

,,,

dTT

dduu

dppdAA

,

,

dx

CV

enters

entropy

Rate

CV

leaves

entropy

Rate

ilitiesirreversib

andheat by CV

intotransfer

entropy Rate

SdS genin Sd

T

Q

S

SdSdT

Q

genin

dssdSd 0,0,0

adiabaticreversible process

isentropicprocess



• Continuity equation

• Momentum equation

• Energy equation

• Entropy equation

Fundamental equations

42


0A

dA

u

dud

Quasi-one dimension fluid flow equations:

ududp

0ududh

ds0 isentropic flow

adiabatic flow, no shaft work


• perfect gas

– equation of state (EOS) or

Thermodynamics of gases

43

RTp

RTpv mM MTRpv /


mM MTRp /


• perfect gas


– internal energy

– enthalpy


44

RTp

RTpv

e

pveh

Tee

Thh

dTcde v

dTcdh p

mM MTRpv /


mM MTRp /


• perfect gas


– internal energy

– enthalpy


45

RTp

RTpv

e

pveh

Tee

Thh

dTcde v

dTcdh p

Tce v

Tch p

calorically perfect gas

mM MTRpv /


mM MTRp /


• perfect gas


– internal energy

– enthalpy

– heat capacity ratio

– difference between heat capacities


46

RTp

RTpv

e

pveh

Tee

Thh

dTcde v

dTcdh p

Tce v

Tch p


vp cc /

vp ccR

mM MTRpv /


mM MTRp /


• perfect gas


– internal energy

– enthalpy

– heat capacity ratio

– difference between heat capacities


47

RTp

RTpv

e

pveh

Tee

Thh

dTcde v

dTcdh p

Tce v

Tch p


vp cc /

vp ccR 1

Rcv

1

Rcp

mM MTRpv /


mM MTRp /



• moving fluid with speed

– static parameters “measured“ in moving fluid


48

,,Tp

u





– fluid brought to rest adiabatically


49

,,Tp

u

000 ,, Tp

00 u





– fluid brought to rest adiabatically

– corresponding total enthalpy to total temperature


50

,,Tp

u

000 ,, Tp

total (stagnation) pressure, temperature, density

00 Tch p

00 u



• stagnation conditions

– adiabatic process


51

const.2

2

11

uh

2

2

110

uhh

rest of fluid

total enthalpy is constant through steady, inviscid, adiabatic flow






52

const.2

2

11

uh

2

2

110

uhh

rest of fluid


00 Tch palso total temperature is constant





– isentropic process


53

const.2

2

11

uh

2

2

110

uhh

rest of fluid


00 Tch palso total temperature is constant

00 ,p

total pressure and total density are constant






54

,,Tp

u





– fluid brought to speedof sound isentropically


55

,,Tp

u

*** ,, Tp

*au





– fluid brought to speedof sound isentropically


56

,,Tp

u

critical pressure, temperature, density

*** ,, Tp

*au



• reversible process

– added heat – closed system

– reversible process

– internal energy

– pressure work


57

deqw inin

dsTq in

dTcde v

pdvw in






– internal energy

– pressure work


58

deqw inin

dsTq in

dTcde v

pdvw in

dTcdsTpdv v

using

vc






– internal energy

– pressure work


59

deqw inin

dsTq in

dTcde v

pdvw in

dTcdsTpdv v

pvddedsTpvdpdv

dTcdsTdpv p

using

vc

using

pc






– internal energy

– pressure work


60

deqw inin

dsTq in

dTcde v

pdvw in

dTcdsTpdv v

T

pdv

T

dTcds v pvddedsTpvdpdv

dTcdsTdpv pT

vdp

T

dTcds p

using

vc

using

pc






– internal energy

– pressure work


61

deqw inin

dsTq in

dTcde v

T

pdv

T

dTcds v

T

vdp

T

dTcds p

pdvw in

isentropicprocess

T

pdv

T

dTcv 0

T

vdp

T

dTc p 0






– internal energy

– pressure work


62

deqw inin

dsTq in

dTcde v

pdvw in

T

pdv

T

dTcv 0

T

vdp

T

dTc p 0

v

Rdv

T

dTcv 0

p

Rdp

T

dTc p 0

using EOS

1

1

1

2

1

2

T

T

v

v

1

1

2

1

2

T

T

p

p

1

Rcc pv






– internal energy

– pressure work


63

deqw inin

dsTq in

dTcde v

pdvw in

1

1

1

2

1

2

T

T

v

v

1

1

2

1

2

T

T

p

p

1

2

2

11

1

2

1

2

v

v

T

T

p

p

isentropic process



• physical mechanism:

– sound propagation in gas is based on molecular motion

– energy is transfer to gas molecules, they start to move about in random fashion

– they collide with other molecules and transfer their energy to these molecules

– the process of collision repeats – energy is propagated

– macroscopic parameters are slightly varied by increased microscopic parameter – energy of molecule

Speed of sound

64

,,Tp



• parameters of fluid Speed of sound

65

wave propagates with speed a

source of disturbance

0

,,

u

Tp

dudTT

ddpp

,

,,




66

pdpp

0udu

view from “outside“ of wave

amoving wave

d




67

pdpp


a

0udu

view from wave

pdpp

au moving wave moving fluid

dua au

d

d




68

pdpp


a

0udu

view from wave

pdpp


au

d

d

0 aAAduad

continuity equation

CV

momentum equation

dpppA

AaAduad

2

dua




69

pdpp


a

0udu

view from wave

pdpp


au

d

d

0 aAAduad

addu

continuity equation

CV

dua

du

dpa

dpppA

AaAduad

2

momentum equation




70

pdpp


a

0udu

view from wave

pdpp


au

d

d

0 aAAduad

addu

continuity equation

CV

dua

du

dpa

d

dpa 2

dpppA

AaAduad

2

momentum equation




71

pdpp


a

0udu

view from wave

pdpp


au

d

d

CV

dua

d

dpa 2

11p

p

1

1pp




72

pdpp


a

0udu

view from wave

pdpp


au

d

d

CV

dua

d

dpa 2

11p

p

1

1pp

RTpp

d

dp 1

1

1




73

pdpp


a

0udu

view from wave

pdpp


au

d

d

CV

dua

d

dpa 2

RTa

11p

p

1

1pp

RTpp

d

dp 1

1

1




74

pdpp


a

0udu

view from wave

pdpp


au

d

d

CV

dua

d

dpa 2

RTa

11p

p

1

1pp

RTpp

d

dp 1

1

1

RT

u

a

uM




75

pdpp


a

0udu

view from wave

pdpp


au

d

d

CV

dua

d

dpa 2

RT

u

a

uM

T

M

Ra

m

M RTa

Gas Mm [g/mol] [] a at 0°C [m/s]

Air 28.96 1.404 331

Hydrogen 2.016 1.407 1270

Xenon 131.3 1.667 170



• Importance of isentropic flow:– isentropic flow is adiabatic in which viscous losses are negligible

– real flows are not isentropic

Isentropic flow

76

the effects of viscosity and heat transfer are restricted to thin layers near the walls

major part of the flow can be assumed to be isentropic



• Importance of isentropic flow:– isentropic flow is adiabatic in which viscous losses are negligible

– real flows are not isentropic

Isentropic flow

77

the effects of viscosity and heat transfer are restricted to thin layers near the walls

major part of the flow can be assumed to be isentropic

many flows in engineering practice can be adequately modeled by assuming them to be isentropic and also steady-state and quasi-one

dimensional flow



• relationshipsIsentropic flow

78

point 1

point 2

direction of isentropicflow

1

21

1

2

1

2

T

T

p

p





79

point 1

point 2


1

21

1

2

1

2

T

T

p

p

2

1

1

22

1

1

22

1

1

2

1

2

1

2

p

p

T

T

RT

RT

a

a





80

point 1

point 2


adiabatic energy equation 1-2

22

2

22

2

11

uh

uh




81

point 1

point 2



22

2

22

2

11

uh

uh

Tch p




82

point 1

point 2



22

2

22

2

11

uh

uh

Tch p

1

Rcp

1

RTh




83

point 1

point 2



22

2

22

2

11

uh

uh

Tch p

1

Rcp

1

RTh

RTa

1

2

ah


• relationships

1. Fluid Flow and ThermodynamicsIsentropic flow

84

point 1

point 2



22

2

22

2

11

uh

uh

2121

2

2

2

2

2

1

2

1 uaua

1

2

ah


• relationships


85

point 1

point 2



22

2

22

2

11

uh

uh

2121

2

2

2

2

2

1

2

1 uaua

critical point

2*

22

12

1

21a

ua

2

2*

2

2

12

1

2

1

1 u

a

u

a

1

2

ah


• relationships


86

point 1

point 2

direction of isentropicflow 2

2*

2

2

12

1

2

1

1 u

a

u

a

2*2

1

12

1

2

1

1

1

MM


• relationships


87

point 1

point 2


1/1

22*

2

MM

2

22*

12

1

M

MM

2

2*

2

2

12

1

2

1

1 u

a

u

a

2*2

1

12

1

2

1

1

1

MM



88

point 1

point 2


adiabatic energy equation

0

2

2h

uh

0

2

2Tc

uTc pp

• relationships



89

point 1

point 2



0

2

2h

uh

0

2

2Tc

uTc pp

T

T

Tc

u

p

02

21

• relationships



90

point 1

point 2



0

2

2h

uh

0

2

2Tc

uTc pp

T

T

Tc

u

p

02

21

1

Rcp

T

T

RT

u 02

2

11

• relationships



91

point 1

point 2



0

2

2h

uh

0

2

2Tc

uTc pp

T

T

Tc

u

p

02

21

1

Rcp

T

T

RT

u 02

2

11

T

TM 02

2

11

• relationships



92

point 1

point 2



0

2

2h

uh

0

2

2Tc

uTc pp

T

TM 02

2

11

0

100

T

T

p

p

p

pM 0

12

2

11

0

1

1

2

2

11

M

• relationships



93

point 1

point 2



0

2

2h

uh

0

2

2Tc

uTc pp

T

TM 02

2

11

0

100

T

T

p

p

p

pM 0

12

2

11

0

1

1

2

2

11

M

• relationships

if 1M

*

*

*

TT

pp


dx


94


• relationships

ududp

momentum equation


dx


95


• relationships

ududp

udupu

u

p

dp

momentum equation


dx


96


• relationships

ududp

udupu

u

p

dp

pRTa 2

speed of soundmomentum equation


dx


97


• relationships

ududp

udupu

u

p

dp

pRTa 2

speed of sound

u

du

a

u

p

dp2

2

momentum equation


dx


98


• relationships

ududp

momentum equation

udupu

u

p

dp

pRTa 2

speed of sound

u

du

a

u

p

dp2

2

u

duM

p

dp 2 • magnitude of fractional pressure change induced by a given fractionalvelocity change depends on squareof Mach number


dx


99


• relationships energy equation

0ududh


dx


100



0ududh dTcdh pcalorically perfect gas


dx


101




ududTcp


dx


102




ududTcp

duu

u

Tc

u

T

dT

p


dx


103




ududTcp

duu

u

Tc

u

T

dT

p

p

RTa 2

speed of sound


dx


104




ududTcp

duu

u

Tc

u

T

dT

p

p

RTa 2

speed of sound

u

du

ac

Ru

T

dT

p

2

2


dx


105




ududTcp

duu

u

Tc

u

T

dT

p

p

RTa 2

speed of sound

u

du

ac

Ru

T

dT

p

2

2

u

duM

c

R

T

dT

p

2


dx


106




ududTcp

duu

u

Tc

u

T

dT

p

p

RTa 2

speed of sound

u

du

ac

Ru

T

dT

p

2

2

u

duM

c

R

T

dT

p

2

u

duM

T

dT 21

• magnitude of fractional temperature change induced by a given fractional velocity changedepends on square of Mach number


dx


107


• relationships equation of state

RT

p


dx


108



RT

p

0T

dTd

p

dp

u

duM

T

dT 21

u

duM

p

dp 2


dx


109



RT

p

0T

dTd

p

dp

u

duM

T

dT 21

u

duM

p

dp 2

u

duM

u

duM

d 22 1


dx


110



RT

p

0T

dTd

p

dp

u

duM

T

dT 21

u

duM

p

dp 2

u

duM

u

duM

d 22 1

u

duM

d 2

• magnitude of fractional temperature change induced by a given fractional velocity changedepends on square of Mach number


dx


111


• relationships

u

duM

d 2

u

duM

T

dT 21

u

duM

p

dp 2

2 /

/M

udu

d

21/

/M

udu

TdT

2 /

/M

udu

pdp

• magnitude of fractional propertieschange induced by a given fractionalvelocity change depends on squareof Mach number


dx


112


• relationships

2 /

/M

udu

d

21/

/M

udu

TdT

2 /

/M

udu

pdp

• magnitude of fractional propertieschange induced by a given fractionalvelocity change depends on squareof Mach number

fractional propert. change induced by fractional velocity change of air [%]

Mach num. density temp. pressure

0.1 1 1.4 0.4

0.33 10.9 15.2 4.4

0.4 16 22.4 6.4



• governing equations for analysis of nozzleNozzle fluid flow

113

0A

dA

u

dud

ududp

0ududh

0T

dTd

p

dp

continuity

momentum

energy

state




114

0A

dA

u

dud

ududp

0ududh

0T

dTd

p

dp

u

duM

d 2




115

0A

dA

u

dud

ududp

0ududh

0T

dTd

p

dp

u

duM

d 2

02 A

dA

u

du

u

duM




116

0A

dA

u

dud

ududp

0ududh

0T

dTd

p

dp

u

duM

d 2

02 A

dA

u

du

u

duM

u

duM

A

dA12




117

u

duM

A

dA12

1. subsonic flow: 10 M

increase in velocity is associated with decrease in area

du

dA

2. supersonic flow: 1M

increase in velocity is associated with increase in area

du

dA3. sonic flow: 1M area reaches an

extremum – minimum 0dA




118

u

duM

A

dA12

increase in velocity is associated with decrease in area

du

dA

increase in velocity is associated with increase in area

du

dA

0dA

1M

10 M 1M

area reaches an extremum – minimum




119

,,uA

critical point

AuuA ***

*** ,, uA




120

,,uA

0

0

****

* u

a

u

u

A

A

critical point

AuuA ***

*** ,, uA




121

,,uA 1

1

20

2

11

M

1

1

*

0

2

11

2

22

*

2*

12

1

M

M

a

uM

critical point

AuuA ***

*** ,, uA

0

0

****

* u

a

u

u

A

A




122

,,uA 1

1

2

2

2

* 2

11

1

21

M

MA

A

area – Mach number relation

critical point

AuuA ***

*** ,, uA

0

0

****

* u

a

u

u

A

A




123

,,uA

1

1

2

2

2

* 2

11

1

21

M

MA

A

area – Mach number relation

critical point

AuuA ***

*** ,, uA

[ ]

*AA

[ ]M

15.1

05.030.1



• variation of parameters in nozzleNozzle fluid flow

124

stagnation point

streamline

00 ,Tp

gas reservoir

back pressure Bp

throat

• stagnation parameters:

• nozzle area:inlet area (location 0 m): 0.004 m2

throat area (loc. 0.05 m): 0.002 m2

exit area (loc. 0.2 m): 0.004 m2

• air:

K5000 T

MPa10 p

exitinlet

J/kgK 288R

4.1




125

[m] x

[m] x

M[-

]A

[m2]

1

1

2

2

2

* 2

11

1

21

M

MA

A




126

[m] x

[m] x

M[-

]A

[m2]

[m] x

T

TM 02

2

11

p

pM 0

12

2

11

0

1

1

2

2

11

M

0/ pp

0/

0/TT




127

[m] x

M[-

]

[m] x0/ pp

0/

0/TT

pressure difference causes fluid flow

• no pressure difference – no fluid flow• if pressure ratio pe/p0 is different

from isentropic value, the flow will be different (inside or outside the nozzle)• exit pressure for isentropic flow with

supersonic speed is pe




128

M[-

]


[m] x

p/p

0[-

]

[m] x

very low-speed subsonic flow, pB,1= pe,1

• pB,1 is reduce below p0




129

M[-

]


[m] x

p/p

0[-

]

[m] x

flow moves faster through nozzle, still subsonic flow, mass flow increases, pB,2= pe,2

• pB,2 is reduce below pB,1




130

M[-

]


[m] x

p/p

0[-

]

[m] x

• pB,3 is such, that it produces sonic flow in throat

only in throat is flow sonic, in other parts of nozzle is flow subsonic, mass flow increases and reaches max. value, pB,2= pe,2




131

M[-

]


[m] x

p/p

0[-

]

[m] x

in divergent nozzle flow is at first supersonic, than shock wave is formed and flow is subsonic, mass flow is constant – chocked flow, pB,4= pe,4





132

M[-

]


[m] x

p/p

0[-

]

[m] x

shock wave is moving toward the exit plane, pB,4= pe,4





133

M[-

]


[m] x

p/p

0[-

]

[m] x

the flow is supersonic in whole nozzle except the exit plane, pB,6= pe,6

• pB,6 is such, that shock wave is on the exit plane




134


[m] x

p/p

0[-

]



other reduction of back pressure pB:• exit pressure is constant pe

• if shock waves moves outside nozzle• if no shock waves are produced• if expansion waves are formed

outside the nozzle

eB pp

eB pp

eB pp




135



other reduction of back pressure pB:• exit pressure is constant pe

• if shock waves moves outside nozzle• if no shock waves are produced• if expansion waves are formed

outside the nozzle

eB pp

eB pp

eB pp

over-expanded flow(low altitudes)

eB pp

under-expanded flow(high altitudes)

eB pp

source: NASA


• Performance Characteristics

• Liquid Propellant Performance

• Feed System


136


• Rocket thrust chamber – combustion Performance Characteristics

137


propellant

1 2 e

*

isobaricheating in combustion chamber

isentropicexpansionin C-D nozzleheat per unit mass Rq

Tchq pR First thermodynamiclaw – isobaric heating:


• Rocket thrust chamber – combustion Performance Characteristics

138


propellant

1 2 e

*


isentropicexpansionin C-D nozzleheat per unit mass Rq

Tchq pR First thermodynamiclaw – isobaric heating:

p

R

c

qTT 0102


• Rocket thrust chamber – expansion Performance Characteristics

139


propellant

1 2 e

*


isentropicexpansionin C-D nozzle

ehh 002 stagnation enthalpy:

isentropic expansion

ee h

uh

2

2

02



140


propellant

1 2 e

*



ehh 002 stagnation enthalpy:

isentropic expansion

ee h

uh

2

2

02

ep

ee

TTc

hhu

02

02

2

2



141


propellant

1 2 e

*



ep

ee

TTc

hhu

02

02

2

2

1

0

02

02

02 1212p

pTc

T

TTcu e

pe

pe

velocity ofexhaust gases


• Rocket thrust chamber – exit velocityPerformance Characteristics

142


propellant

1 2 e

*


ep

ee

TTc

hhu

02

02

2

2

1

0

02

02

02 1212p

pTc

T

TTcu e

pe

pe

1

0

02 11

2p

pRTu e

e

1

Rcp



143


propellant

1 2 e

*


ep

ee

TTc

hhu

02

02

2

2

1

0

02 11

2p

pRTu e

e

[ ]-0

ep

p

[ ]- 02RTue

15.1

05.0

30.1



144


propellant

1 2 e

*


ep

ee

TTc

hhu

02

02

2

2

1

0

02 11

2p

pT

M

Ru e

m

Me

m

M

M

RR

1

0

02 11

2p

pRTu e

e



145


propellant

1 2 e

*


ep

ee

TTc

hhu

02

02

2

2

low molecular weight higher exit velocity

higher stag. temperature higher exit velocity

1

0

02 11

2p

pT

M

Ru e

m

Me

m

M

M

RR

1

0

02 11

2p

pRTu e

e

higher pressure ratio p0/pe higher exit velocity


• Rocket thrust chamber – mass flowPerformance Characteristics

146


propellant

1 2 e

*

mass flow

continuityequation:

*** uAm



147


propellant

1 2 e

*

mass flow

continuityequation:

1

1

*

02

2

1

** RTu

T

TM 022

2

11

2

1

02

02

02

*

2

1

1

2

RT

RTRTu

*** uAm



148


propellant

1 2 e

*

mass flow

continuityequation:

*** uAm 2

1

02

021

1

02

*

2

1

2

1

RT

RTAm

1

1

*

02

2

1

2

1

02

02

02

*

2

1

1

2

RT

RTRTu



149


propellant

1 2 e

*

continuityequation:

02

0

*1

1

1

2

RT

pAm

mass flow

*** uAm 2

1

02

021

1

02

*

2

1

2

1

RT

RTAm



150


propellant

1 2 e

*

continuityequation:

02

0

*1

1

1

2

RT

pAm

mass flow

higher stag. pressure higher mass flow

*** uAm 2

1

02

021

1

02

*

2

1

2

1

RT

RTAm


• Rocket thrust chamber – thrust Performance Characteristics

151


propellant

1 2 e

*

02

0

*1

1

1

2

RT

pAm

aeeethrust ppAumF

1

0

02 11

2p

pRTu e

e

thrust



152


propellant

1 2 e

*

thrust

0

*

1

0

21

1

0

*1

1

2

1

2

pA

ppA

p

p

pA

F aeeethrust

02

0

*1

1

1

2

RT

pAm

1

0

02 11

2p

pRTu e

e



153


propellant

1 2 e

*

thrust

0

*

1

0

21

1

0

*1

1

2

1

2

pA

ppA

p

p

pA

F aeeethrust

thrust depends only on stagnation pressure in combustion chamber

F

thrustthrust

Ccm

Ap

F

m

ApmF

*

*

0

*

0

characteristic velocity

thrust coefficient


• Rocket thrust chamber – characteristic velocityPerformance Characteristics

154


propellant

1 2 e

*

characteristic velocity –specify combustion chamber

m

Apc

*

0*

*c

m

M

M

TRc 02

1

1

*

2

11

characteristic velocity is function of combustion chamber design and propellant characteristics


• Rocket thrust chamber – thrust coefficientPerformance Characteristics

155


propellant

1 2 e

*thrust coefficient FC *

0 Ap

FC thrust

F

0

*

1

0

21

1

11

2

1

2

pA

ppA

p

pC aeee

F

thrust coefficient depends on gas property () and nozzle parameters (nozzle area ratio and pressure ratio), it is independent on combustion chamber temperature



156


propellant

1 2 e

*

0

*

1

0

21

1

11

2

1

2

pA

ppA

p

pC aeee

F

[ ] / *AAe

[ ]FC

contribution to thrust by exit velocity

*A

Ae

eM0p

pe(velocity )FC 01.0/

2.1

0

ppa



157


propellant

1 2 e

*

0

*

1

0

21

1

11

2

1

2

pA

ppA

p

pC aeee

F

[ ] / *AAe

[ ]FC

contribution to thrust by exit pressure

*A

Ae

eM0p

pe(pressure)FC 01.0/

2.1

0

ppa



158


propellant

1 2 e

*

0

*

1

0

21

1

11

2

1

2

pA

ppA

p

pC aeee

F

[ ] / *AAe

[ ]FC

01.0/

2.1

0

ppa

thrust coefficient for defined conditionsdepends on area ratio

0/ ppa



159


propellant

1 2 e

*

0

*

1

0

21

1

11

2

1

2

pA

ppA

p

pC aeee

F

[ ] / *AAe

[ ]FC

01.0/

2.1

0

ppa

optimal thrust coefficient is defined by area ratio, where exit pressure equals ambient pressure

ae pp

][/ 0 ppe



160


propellant

1 2 e

*

0

*

1

0

21

1

11

2

1

2

pA

ppA

p

pC aeee

F

[ ] / *AAe

(conv)F

F

CC

convergent part of nozzle

ep

pM 0

12

2

11

1M

FC(conv)FC2.1

0/

:parameter

ppa

1.0 025.0 01.0 0025.0

001.0

0



161


propellant

1 2 e

*

0

*

1

0

21

1

11

2

1

2

pA

ppA

p

pC aeee

F

[ ] / *AAe

(conv)F

F

CC

2.1

line thrust max.

1.0 025.0 01.0 0025.0

001.0

0

optimal thrust coefficient for individual pressure ratio form max. thrust line

0/

:parameter

ppa



162


propellant

1 2 e

*

0

*

1

0

21

1

11

2

1

2

pA

ppA

p

pC aeee

F

[ ] / *AAe

(conv)F

F

CC

2.1 1.0 025.0 01.0 0025.0

001.0

0

optimal thrust coefficient for sea level and pressure ratio 0.001 and the change of the coefficient

0/

:parameter

ppa


• performance of individual liquid propellants

Liquid Propellant Performance

163


sou

rce:

Ley

, Wit

tman

n, H

allm

ann

: H

and

bo

ok

of

spac

e te

chn

olo

gy


• There are 2 main feed systems for liquid propellant:

pressurized systems

Feed System

164


• They are usually used when:• total impulse is small• pressure in combustion chamber is

small• Disadvantages:

• walls of tanks are thicker – system is heavier

• Usage:• control of attitude and change of

orbit


• There are 2 main feed systems for liquid propellant:

turbopump systems

Feed System

165


• They are usually used when:• total impulse is large• pressure in combustion chamber is

large• Positive characteristics of system:

• pressure in tanks is lower than pressure in tanks when gas pressure feed system is used so the thickness of walls of tank is smaller

• Usage:• dominantly for boosters


• turbopump systems – 3 basic cyclesFeed System

166


Gas generator cycle - open cycle

Description:• It is the most common cycle • It is relatively simple cycle • The cycle efficiency is smaller than efficiency of

closed cycle • Small part of the propellant is consumed in small

combustion chamber for generating gas for a turbine, which drives the pump • Gas from turbine flows to separate nozzle or to

the end part of the main nozzle, where it operates as cooler of nozzle• Engines: F-1 (Saturn V) , 2 Vulcain (Ariane 5)



167


Expander cycle - closed cycle

Description:• The fuel passed through the cooling jacket of

nozzle where it picked up energy and the fuel works as coolant of nozzle• The fuel is evaporated, heated, and then fed to

low pressure-ratio turbines• at the outlet of the turbine fuel enters the

combustion chamber where it is mixed with an oxidizer • in that cycle all the fuel is burnt in combustion

chamber and the efficiency of engine is increased • Engines: RL10 (the second stage of the Delta IV) ,

Vinci (ESA)



168


Staged-combustion cycle - closed cycle

Description:• The fuel passed through the cooling jacket of

nozzle as in expander cycle • Then the fuel flows into the precombustor

where all the fuel is burnt with a part of the oxidizer, forming a high-energy gas to drive• The turbines that drive the pumps

all the gas at the outlet of the turbine flows into the combustion chamber where is mixed with remaining oxidizer • pressure in combustion chamber: up to 40 MPa• Engines: Space Shuttle Main Engine – SSME,

RD-170 (Energija)


• Static Performance

• Force-Free Motion

• Motion with Gravity

• Launch Flight Mechanics


169


• Momentum equation – written for CV:

• simplifications:

– quasi-one dimensional flow

– steady-state flow

Static Performance

170



• Momentum equation – written for CV:Static Performance

171


CV enters

momentum

Rate

CV leaves

momentum

Rate

direction

in CVin gas

on Forces

x



172


CV enters

momentum

Rate

CV leaves

momentum

Rate

direction

in CVin gas

on Forces

x

eeaethrust umppAF



173


CV enters

momentum

Rate

CV leaves

momentum

Rate

direction

in CVin gas

on Forces

x

eeaethrust umppAF

m

ppAum

ppAumF

aeee

aeeethrust



174


CV enters

momentum

Rate

CV leaves

momentum

Rate

direction

in CVin gas

on Forces

x

eeaethrust umppAF

m

ppAum

ppAumF

aeee

aeeethrust

efthrust umF

Thrust



175


efthrust umF

Thrust

Engine Thrust [MN]

F1 7.77 (vacuum)

Vulcain 2 1.35

J2 1.03

NK33 1.51



176


efthrust umF

Thrust

tFI thrustt

Total impulse



177


efthrust umF

Thrust

tFI thrustt

Total impulse

gugmum

gmFmgII

efef

thrustts

//

//

Specific impulse



178


efthrust umF

Thrust

tFI thrustt

Total impulse

gugmum

gmFmgII

efef

thrustts

//

//

Specific impulsePropellant Specific impulse Is [s]

cold gas 50

Monopropellant hydrazine

230

LOX/LH2 455

Ion propulsion >3000


• Force-free motion absence of external forcesForce-Free Motion

179


time t time t+dt

mass dm

mass Rm

Rv

RR vdv

ambientpressure ap only ambient

pressure is considered

exit pressureof nozzle ep

eu

eu

mass dm

mass Rm

Momentum: RR vdmm

eRRRR uvdmvdvm

Change of momentum in :

eRR udmvdm

dt



180


time t time t+dt

mass dm

mass Rm

Rv

RR vdv




eu

eu

mass dm

mass Rm

Pressure force: Reae iApp

Reae iApp

Total impulse in :dt dtiApp Reae



181


time t time t+dt

mass dm

mass Rm

Rv

RR vdv




eu

eu

mass dm

mass Rm

Momentum equation: dtiAppudmvdm ReaeeRR

Momentum equation in :Ri

dtAppdmudvm eaeeRR



182


time t

mass dm

mass Rm

Rvambient

pressure ap


eu


dtAppdmudvm eaeeRR

dtmdm

thrust

thrustF



183


time t

mass dm

mass Rm

Rvambient

pressure ap


eu


dtAppdmudvm eaeeRR

dtmdm

dtAppumdvm eaeeRR

thrust

thrustF



184


time t

mass dm

mass Rm

Rvambient

pressure ap


eu


dtAppdmudvm eaeeRR

dtmdm

dtAppumdvm eaeeRR

dtumdvm efRR

thrustR

R Fdt

dvm

thrust

thrustF



185


time t

mass dm

mass Rm

Rvambient

pressure ap


eu


dtAppdmudvm eaeeRR

dtmdm

dtAppumdvm eaeeRR

dtumdvm efRR

thrustR

R Fdt

vdm

thrustR

R Fdt

dvm

thrust

thrustF



186


time t

mass dm

mass Rm

Rvambient

pressure ap


eu

dtumdvm efRR

mdt

dmR

ef

R

RR u

m

dmdv

thrust

thrustF



187


time t

mass dm

mass Rm

Rvambient

pressure ap


eu

thrust

thrustF

ef

R

RR u

m

dmdv

R

RefR

m

muv 0Ln

[ ]m/s Rv

R

R

m

m 0 m/s1000efu

m/s4500efu


• Motion with gravity vertical motion in gravity field

Motion with Gravity

188


Rv

thrustF

eu

m

Rm

gmR

gmudt

dm

dt

vdm Ref

RRR

gmudt

dm

dt

dvm Ref

RRR



Motion with Gravity

189


Rv

thrustF

eu

m

Rm

gmR

gmudt

dm

dt

vdm Ref

RRR

gmudt

dm

dt

dvm Ref

RRR

gdtum

dmdv ef

R

RR



Motion with Gravity

190


Rv

thrustF

eu

m

Rm

gmR

gmudt

dm

dt

vdm Ref

RRR

gmudt

dm

dt

dvm Ref

RRR

gdtum

dmdv ef

R

RR gt

m

muv

R

RefR

0Ln


• Forces act on rocketLaunch Flight Mechanics

191


trajectory of CM

local horizont

Rv

euep

ap

Rm

The flight of rocket has 2 main phases:1. powered phase2. unpowered phase

Powered phase:• trajectory of vehicle from launch pad to

burnout point• during the phase, guidance system

control the trajectory – vehicle at burnout point should have prescribedposition and velocity Simplif.:

• all forces act on the same plane• Earth is inertial frame of reference

xy



192


trajectory of CM

local horizont

Rv

euep

ap

Rmflight path angle – angle between local horizont and velocity vector

angle of attack

Simplif.:• all forces act on the same plane• Earth is inertial frame of reference

xy



193


trajectory of CM

local horizont

Rv

euep

gmR

ap

Rm

Three forces act on rocket at each instant:1. gravitational force – applied at

the CM gmR


xy

it is function of vertical location of rocket mass of rocket is function of propellant mass flow

equation of propellant mass flow



194


trajectory of CM

Rv

eu

ap

epgmR

LF

DF

Rm


the CM

2. aerodynamic force – applied at aerodynamic center and can be decomposed into: - drag force - lift force

gmR

DF

LF


xy they are function of vertical

location and attitude of rocket

local horizont



195


trajectory of CM

Rv

eu

ap

epgmR

LF

DF

Rm


the CM


3. thrust force

gmR

DF

LF

thrustF

thrustF


xy

magnitude and direction can be controlled

local horizont



196


trajectory of CM

Rv

eu

ap

epgmR

LF

DF

Rm

Motion of vehicle in 2D:• translation motion of Center of Mass• relative rotation motion around the CM


the CM


3. thrust force

gmR

DF

LF

thrustF

thrustF


xy

local horizont



197


trajectory of CM

Rv

eu

ap

epgmR

LF

DF

Rm

thrustF


the CM


3. thrust force

gmR

DF

LF

thrustF

i

FiR Mdt

dI

2

2

x

ydynamic equations:relative rotation motion around the CM

local horizont



198


trajectory of CM

Rv

eu

ap

epgmR

LF

DF

Rm

thrustF


the CM


3. thrust force

gmR

DF

LF

thrustF

LDRthrustR

R FFgmFdt

vdm

dynamic equations:translation motion of the CM

xy

local horizont



199


trajectory of CM

Rv

eu

ap

epgmR

LF

DF

Rm

thrustF

DRthrustR

R FgmFdt

dvm sincos

LRthrustRR FgmFdt

dvm

cossin

tangent and normal decomposition

xy

LDRthrustR

R FFgmFdt

vdm

dynamic equations:translation motion of the CM

local horizont



200


trajectory of CM

Rv

eu

ap

epgmR

LF

DF

Rm

thrustF

t

R dtvx

0

cos

xy

kinematic equations:vertical and horizontaldistance

t

R dtvy

0

sin

Equations of rocket motion:• dynamic equations• kinematic equations• equation of propellant mass flow

Numerical solution of system of ODE

local horizont



201


trajectory of CM

Rv

eu

ap

epgmR

LF

DF

Rm

thrustF

xy

Gravity turn:• gravity turn trajectory – change of

flight angle due to gravity• only thrust and gravity is considered • angle of attack is zero• thrust is in axis of rocket

sin gmFdt

dvm Rthrust

RR

cos gdt

dvR

local horizont



202


trajectory of CM

Rv

eu

ap

epgmR

LF

DF

Rm

thrustF

xy

]km[ x

Gravity turn:• initial mass 90 t, propellant is

80% of mass with flow 250 kg/seffective velocity 4000 m/s• in altitude 1 km, flight angle is changed to 89.85°

sin gmFdt

dvm Rthrust

RR

cos gdt

dvR

cosRvdt

dx

sinRvdt

dy

local horizont

]km[ y

chemical propulsion systems - elearning.mechatronika.cool

Documents