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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
Chemical Propulsion SystemsSpace for Education, Education for Space
Contents
2
1. Fluid Flow and Thermodynamics
2. Chemical Rocket Propulsion
3. Performance of Rocket Vehicle
Chemical Propulsion SystemsSpace for Education, Education for Space
• Introduction
• Fundamental equations
• Thermodynamics of gases
• Speed of sound
• Isentropic flow
• Nozzle fluid flow
1. Fluid Flow and Thermodynamics
3
Chemical Propulsion SystemsSpace for Education, Education for Space
• 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
Chemical Propulsion SystemsSpace for Education, Education for Space
• 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
1. Fluid Flow and Thermodynamics
moving of control mass
inlet outlet
Chemical Propulsion SystemsSpace for Education, Education for Space
• 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
1. Fluid Flow and Thermodynamics
differential CV finite CV
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• There are 4 fundamental equations, which must be considered:
– Continuity equation
– Momentum equation
– Energy equation
– Entropy equation
Fundamental equations
7
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Continuity equationFundamental equations
8
11
111
,
,,,
T
upA
22
222
,
,,,
T
upA
inviscid steady-state flow
0CV enters
mass Rate
CV leaves
mass Rate
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Continuity equationFundamental equations
9
11
111
,
,,,
T
upA
22
222
,
,,,
T
upA
inviscid steady-state flow
0CV enters
mass Rate
CV leaves
mass Rate
222 uA 111 uA
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Continuity equationFundamental equations
10
11
111
,
,,,
T
upA
22
222
,
,,,
T
upA
inviscid steady-state flow
0111222 uAuA
0CV enters
mass Rate
CV leaves
mass Rate
222 uA 111 uA
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Continuity equationFundamental equations
11
0CV enters
mass Rate
CV leaves
mass Rate
T
upA
,
,,,
dTT
dduu
dppdAA
,
,
dx dAAduud
inviscid steady-state flow
uA
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Continuity equationFundamental equations
12
0CV enters
mass Rate
CV leaves
mass Rate
T
upA
,
,,,
dTT
dduu
dppdAA
,
,
dx dAAduud
inviscid steady-state flow
0 uAdAduudA
uA
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Continuity equationFundamental equations
13
0CV enters
mass Rate
CV leaves
mass Rate
T
upA
,
,,,
dTT
dduu
dppdAA
,
,
dx dAAduud
0A
dA
u
dud
inviscid steady-state flow
0 uAdAduudA
uA
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Momentum equationFundamental equations
14
inviscid steady-state flow
11
111
,
,,,
T
upA
22
222
,
,,,
T
upA
CV enters
momentum
Rate
CV leaves
momentum
Rate
direction
in CVin gas
on Forces
x
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Momentum equationFundamental equations
15
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Momentum equationFundamental equations
16
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Momentum equationFundamental equations
17
inviscid steady-state flow
T
upA
,
,,,
dTT
dduu
dppdAA
,
,
dx 2
dAAduud
CV enters
momentum
Rate
CV leaves
momentum
Rate
direction
in CVin gas
on Forces
x
Au2
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Momentum equationFundamental equations
18
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Momentum equationFundamental equations
19
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Momentum equationFundamental equations
20
inviscid steady-state flow
T
upA
,
,,,
dTT
dduu
dppdAA
,
,
dx Adp
uduAAdp
CV enters
momentum
Rate
CV leaves
momentum
Rate
direction
in CVin gas
on Forces
x
uduA
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Momentum equationFundamental equations
21
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
22
inviscid steady-state flow
11
111
,
,,,
T
upA
22
222
,
,,,
T
upA
CV enters
energy
Rate
CV leaves
energy
Rate
heat and
by work CV
intotransfer
energy Rate
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
23
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
24
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
25
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
26
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
27
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
28
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
29
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
30
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
31
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
32
inviscid steady-state flow
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,/
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
33
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Energy equationFundamental equations
34
inviscid steady-state flow
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,
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Entropy equationFundamental equations
35
11
111
,
,,,
T
upA
22
222
,
,,,
T
upA
inviscid steady-state flow
CV
enters
entropy
Rate
CV
leaves
entropy
Rate
ilitiesirreversib
andheat by CV
intotransfer
entropy Rate
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Entropy equationFundamental equations
36
11
111
,
,,,
T
upA
22
222
,
,,,
T
upA
inviscid steady-state flow
2Sgen
2
1
in ST
Q
1S
CV
enters
entropy
Rate
CV
leaves
entropy
Rate
ilitiesirreversib
andheat by CV
intotransfer
entropy Rate
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Entropy equationFundamental equations
37
11
111
,
,,,
T
upA
22
222
,
,,,
T
upA
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Entropy equationFundamental equations
38
11
111
,
,,,
T
upA
22
222
,
,,,
T
upA
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Entropy equationFundamental equations
39
11
111
,
,,,
T
upA
22
222
,
,,,
T
upA
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Entropy equationFundamental equations
40
inviscid steady-state flow
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 ,
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Entropy equationFundamental equations
41
inviscid steady-state flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• Continuity equation
• Momentum equation
• Energy equation
• Entropy equation
Fundamental equations
42
inviscid steady-state flow
0A
dA
u
dud
Quasi-one dimension fluid flow equations:
ududp
0ududh
ds0 isentropic flow
adiabatic flow, no shaft work
Chemical Propulsion SystemsSpace for Education, Education for Space
• perfect gas
– equation of state (EOS) or
Thermodynamics of gases
43
RTp
RTpv mM MTRpv /
1. Fluid Flow and Thermodynamics
mM MTRp /
Chemical Propulsion SystemsSpace for Education, Education for Space
• perfect gas
– equation of state (EOS) or
– internal energy
– enthalpy
Thermodynamics of gases
44
RTp
RTpv
e
pveh
Tee
Thh
dTcde v
dTcdh p
mM MTRpv /
1. Fluid Flow and Thermodynamics
mM MTRp /
Chemical Propulsion SystemsSpace for Education, Education for Space
• perfect gas
– equation of state (EOS) or
– internal energy
– enthalpy
Thermodynamics of gases
45
RTp
RTpv
e
pveh
Tee
Thh
dTcde v
dTcdh p
Tce v
Tch p
calorically perfect gas
mM MTRpv /
1. Fluid Flow and Thermodynamics
mM MTRp /
Chemical Propulsion SystemsSpace for Education, Education for Space
• perfect gas
– equation of state (EOS) or
– internal energy
– enthalpy
– heat capacity ratio
– difference between heat capacities
Thermodynamics of gases
46
RTp
RTpv
e
pveh
Tee
Thh
dTcde v
dTcdh p
Tce v
Tch p
calorically perfect gas
vp cc /
vp ccR
mM MTRpv /
1. Fluid Flow and Thermodynamics
mM MTRp /
Chemical Propulsion SystemsSpace for Education, Education for Space
• perfect gas
– equation of state (EOS) or
– internal energy
– enthalpy
– heat capacity ratio
– difference between heat capacities
Thermodynamics of gases
47
RTp
RTpv
e
pveh
Tee
Thh
dTcde v
dTcdh p
Tce v
Tch p
calorically perfect gas
vp cc /
vp ccR 1
Rcv
1
Rcp
mM MTRpv /
1. Fluid Flow and Thermodynamics
mM MTRp /
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• moving fluid with speed
– static parameters “measured“ in moving fluid
Thermodynamics of gases
48
,,Tp
u
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• moving fluid with speed
– static parameters “measured“ in moving fluid
– fluid brought to rest adiabatically
Thermodynamics of gases
49
,,Tp
u
000 ,, Tp
00 u
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• moving fluid with speed
– static parameters “measured“ in moving fluid
– fluid brought to rest adiabatically
– corresponding total enthalpy to total temperature
Thermodynamics of gases
50
,,Tp
u
000 ,, Tp
total (stagnation) pressure, temperature, density
00 Tch p
00 u
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• stagnation conditions
– adiabatic process
Thermodynamics of gases
51
const.2
2
11
uh
2
2
110
uhh
rest of fluid
total enthalpy is constant through steady, inviscid, adiabatic flow
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• stagnation conditions
– adiabatic process
Thermodynamics of gases
52
const.2
2
11
uh
2
2
110
uhh
rest of fluid
total enthalpy is constant through steady, inviscid, adiabatic flow
00 Tch palso total temperature is constant
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• stagnation conditions
– adiabatic process
– isentropic process
Thermodynamics of gases
53
const.2
2
11
uh
2
2
110
uhh
rest of fluid
total enthalpy is constant through steady, inviscid, adiabatic flow
00 Tch palso total temperature is constant
00 ,p
total pressure and total density are constant
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• moving fluid with speed
– static parameters “measured“ in moving fluid
Thermodynamics of gases
54
,,Tp
u
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• moving fluid with speed
– static parameters “measured“ in moving fluid
– fluid brought to speedof sound isentropically
Thermodynamics of gases
55
,,Tp
u
*** ,, Tp
*au
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• moving fluid with speed
– static parameters “measured“ in moving fluid
– fluid brought to speedof sound isentropically
Thermodynamics of gases
56
,,Tp
u
critical pressure, temperature, density
*** ,, Tp
*au
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• reversible process
– added heat – closed system
– reversible process
– internal energy
– pressure work
Thermodynamics of gases
57
deqw inin
dsTq in
dTcde v
pdvw in
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• reversible process
– added heat – closed system
– reversible process
– internal energy
– pressure work
Thermodynamics of gases
58
deqw inin
dsTq in
dTcde v
pdvw in
dTcdsTpdv v
using
vc
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• reversible process
– added heat – closed system
– reversible process
– internal energy
– pressure work
Thermodynamics of gases
59
deqw inin
dsTq in
dTcde v
pdvw in
dTcdsTpdv v
pvddedsTpvdpdv
dTcdsTdpv p
using
vc
using
pc
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• reversible process
– added heat – closed system
– reversible process
– internal energy
– pressure work
Thermodynamics of gases
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• reversible process
– added heat – closed system
– reversible process
– internal energy
– pressure work
Thermodynamics of gases
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• reversible process
– added heat – closed system
– reversible process
– internal energy
– pressure work
Thermodynamics of gases
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• reversible process
– added heat – closed system
– reversible process
– internal energy
– pressure work
Thermodynamics of gases
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• 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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• parameters of fluid Speed of sound
65
wave propagates with speed a
source of disturbance
0
,,
u
Tp
dudTT
ddpp
,
,,
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• parameters of fluid Speed of sound
66
pdpp
0udu
view from “outside“ of wave
amoving wave
d
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• parameters of fluid Speed of sound
67
pdpp
view from “outside“ of wave
a
0udu
view from wave
pdpp
au moving wave moving fluid
dua au
d
d
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• parameters of fluid Speed of sound
68
pdpp
view from “outside“ of wave
a
0udu
view from wave
pdpp
au moving wave moving fluid
au
d
d
0 aAAduad
continuity equation
CV
momentum equation
dpppA
AaAduad
2
dua
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• parameters of fluid Speed of sound
69
pdpp
view from “outside“ of wave
a
0udu
view from wave
pdpp
au moving wave moving fluid
au
d
d
0 aAAduad
addu
continuity equation
CV
dua
du
dpa
dpppA
AaAduad
2
momentum equation
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• parameters of fluid Speed of sound
70
pdpp
view from “outside“ of wave
a
0udu
view from wave
pdpp
au moving wave moving fluid
au
d
d
0 aAAduad
addu
continuity equation
CV
dua
du
dpa
d
dpa 2
dpppA
AaAduad
2
momentum equation
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• parameters of fluid Speed of sound
71
pdpp
view from “outside“ of wave
a
0udu
view from wave
pdpp
au moving wave moving fluid
au
d
d
CV
dua
d
dpa 2
11p
p
1
1pp
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• parameters of fluid Speed of sound
72
pdpp
view from “outside“ of wave
a
0udu
view from wave
pdpp
au moving wave moving fluid
au
d
d
CV
dua
d
dpa 2
11p
p
1
1pp
RTpp
d
dp 1
1
1
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• parameters of fluid Speed of sound
73
pdpp
view from “outside“ of wave
a
0udu
view from wave
pdpp
au moving wave moving fluid
au
d
d
CV
dua
d
dpa 2
RTa
11p
p
1
1pp
RTpp
d
dp 1
1
1
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• parameters of fluid Speed of sound
74
pdpp
view from “outside“ of wave
a
0udu
view from wave
pdpp
au moving wave moving fluid
au
d
d
CV
dua
d
dpa 2
RTa
11p
p
1
1pp
RTpp
d
dp 1
1
1
RT
u
a
uM
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• parameters of fluid Speed of sound
75
pdpp
view from “outside“ of wave
a
0udu
view from wave
pdpp
au moving wave moving fluid
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• 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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• 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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• relationshipsIsentropic flow
78
point 1
point 2
direction of isentropicflow
1
21
1
2
1
2
T
T
p
p
isentropicprocess 1-2
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• relationshipsIsentropic flow
79
point 1
point 2
direction of isentropicflow
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
isentropicprocess 1-2
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• relationshipsIsentropic flow
80
point 1
point 2
direction of isentropicflow
adiabatic energy equation 1-2
22
2
22
2
11
uh
uh
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• relationshipsIsentropic flow
81
point 1
point 2
direction of isentropicflow
adiabatic energy equation 1-2
22
2
22
2
11
uh
uh
Tch p
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• relationshipsIsentropic flow
82
point 1
point 2
direction of isentropicflow
adiabatic energy equation 1-2
22
2
22
2
11
uh
uh
Tch p
1
Rcp
1
RTh
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• relationshipsIsentropic flow
83
point 1
point 2
direction of isentropicflow
adiabatic energy equation 1-2
22
2
22
2
11
uh
uh
Tch p
1
Rcp
1
RTh
RTa
1
2
ah
Chemical Propulsion SystemsSpace for Education, Education for Space
• relationships
1. Fluid Flow and ThermodynamicsIsentropic flow
84
point 1
point 2
direction of isentropicflow
adiabatic energy equation 1-2
22
2
22
2
11
uh
uh
2121
2
2
2
2
2
1
2
1 uaua
1
2
ah
Chemical Propulsion SystemsSpace for Education, Education for Space
• relationships
1. Fluid Flow and ThermodynamicsIsentropic flow
85
point 1
point 2
direction of isentropicflow
adiabatic energy equation 1-2
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• relationships
1. Fluid Flow and ThermodynamicsIsentropic flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• relationships
1. Fluid Flow and ThermodynamicsIsentropic flow
87
point 1
point 2
direction of isentropicflow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and ThermodynamicsIsentropic flow
88
point 1
point 2
direction of isentropicflow
adiabatic energy equation
0
2
2h
uh
0
2
2Tc
uTc pp
• relationships
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and ThermodynamicsIsentropic flow
89
point 1
point 2
direction of isentropicflow
adiabatic energy equation
0
2
2h
uh
0
2
2Tc
uTc pp
T
T
Tc
u
p
02
21
• relationships
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and ThermodynamicsIsentropic flow
90
point 1
point 2
direction of isentropicflow
adiabatic energy equation
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and ThermodynamicsIsentropic flow
91
point 1
point 2
direction of isentropicflow
adiabatic energy equation
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and ThermodynamicsIsentropic flow
92
point 1
point 2
direction of isentropicflow
adiabatic energy equation
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and ThermodynamicsIsentropic flow
93
point 1
point 2
direction of isentropicflow
adiabatic energy equation
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
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
94
direction of isentropicflow
• relationships
ududp
momentum equation
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
95
direction of isentropicflow
• relationships
ududp
udupu
u
p
dp
momentum equation
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
96
direction of isentropicflow
• relationships
ududp
udupu
u
p
dp
pRTa 2
speed of soundmomentum equation
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
97
direction of isentropicflow
• relationships
ududp
udupu
u
p
dp
pRTa 2
speed of sound
u
du
a
u
p
dp2
2
momentum equation
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
98
direction of isentropicflow
• 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
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
99
direction of isentropicflow
• relationships energy equation
0ududh
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
100
direction of isentropicflow
• relationships energy equation
0ududh dTcdh pcalorically perfect gas
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
101
direction of isentropicflow
• relationships energy equation
0ududh dTcdh pcalorically perfect gas
ududTcp
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
102
direction of isentropicflow
• relationships energy equation
0ududh dTcdh pcalorically perfect gas
ududTcp
duu
u
Tc
u
T
dT
p
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
103
direction of isentropicflow
• relationships energy equation
0ududh dTcdh pcalorically perfect gas
ududTcp
duu
u
Tc
u
T
dT
p
p
RTa 2
speed of sound
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
104
direction of isentropicflow
• relationships energy equation
0ududh dTcdh pcalorically perfect gas
ududTcp
duu
u
Tc
u
T
dT
p
p
RTa 2
speed of sound
u
du
ac
Ru
T
dT
p
2
2
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
105
direction of isentropicflow
• relationships energy equation
0ududh dTcdh pcalorically perfect gas
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
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
106
direction of isentropicflow
• relationships energy equation
0ududh dTcdh pcalorically perfect gas
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
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
107
direction of isentropicflow
• relationships equation of state
RT
p
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
108
direction of isentropicflow
• relationships equation of state
RT
p
0T
dTd
p
dp
u
duM
T
dT 21
u
duM
p
dp 2
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
109
direction of isentropicflow
• relationships equation of state
RT
p
0T
dTd
p
dp
u
duM
T
dT 21
u
duM
p
dp 2
u
duM
u
duM
d 22 1
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
110
direction of isentropicflow
• relationships equation of state
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
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
111
direction of isentropicflow
• 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
Chemical Propulsion SystemsSpace for Education, Education for Space
dx
1. Fluid Flow and ThermodynamicsIsentropic flow
112
direction of isentropicflow
• 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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• governing equations for analysis of nozzleNozzle fluid flow
113
0A
dA
u
dud
ududp
0ududh
0T
dTd
p
dp
continuity
momentum
energy
state
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• governing equations for analysis of nozzleNozzle fluid flow
114
0A
dA
u
dud
ududp
0ududh
0T
dTd
p
dp
u
duM
d 2
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• governing equations for analysis of nozzleNozzle fluid flow
115
0A
dA
u
dud
ududp
0ududh
0T
dTd
p
dp
u
duM
d 2
02 A
dA
u
du
u
duM
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• governing equations for analysis of nozzleNozzle fluid flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• governing equations for analysis of nozzleNozzle fluid flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• governing equations for analysis of nozzleNozzle fluid flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• governing equations for analysis of nozzleNozzle fluid flow
119
,,uA
critical point
AuuA ***
*** ,, uA
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• governing equations for analysis of nozzleNozzle fluid flow
120
,,uA
0
0
****
* u
a
u
u
A
A
critical point
AuuA ***
*** ,, uA
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• governing equations for analysis of nozzleNozzle fluid flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• governing equations for analysis of nozzleNozzle fluid flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• governing equations for analysis of nozzleNozzle fluid flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• 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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• variation of parameters in nozzleNozzle fluid flow
125
[m] x
[m] x
M[-
]A
[m2]
1
1
2
2
2
* 2
11
1
21
M
MA
A
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• variation of parameters in nozzleNozzle fluid flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• variation of parameters in nozzleNozzle fluid flow
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• variation of parameters in nozzleNozzle fluid flow
128
M[-
]
pressure difference causes fluid flow
[m] x
p/p
0[-
]
[m] x
very low-speed subsonic flow, pB,1= pe,1
• pB,1 is reduce below p0
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• variation of parameters in nozzleNozzle fluid flow
129
M[-
]
pressure difference causes fluid flow
[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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• variation of parameters in nozzleNozzle fluid flow
130
M[-
]
pressure difference causes fluid flow
[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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• variation of parameters in nozzleNozzle fluid flow
131
M[-
]
pressure difference causes fluid flow
[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
• pB,4 is reduce below pB,3
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• variation of parameters in nozzleNozzle fluid flow
132
M[-
]
pressure difference causes fluid flow
[m] x
p/p
0[-
]
[m] x
shock wave is moving toward the exit plane, pB,4= pe,4
• pB,5 is reduce below pB,4
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• variation of parameters in nozzleNozzle fluid flow
133
M[-
]
pressure difference causes fluid flow
[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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• variation of parameters in nozzleNozzle fluid flow
134
pressure difference causes fluid flow
[m] x
p/p
0[-
]
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
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
Chemical Propulsion SystemsSpace for Education, Education for Space
1. Fluid Flow and Thermodynamics
• variation of parameters in nozzleNozzle fluid flow
135
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
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Performance Characteristics
• Liquid Propellant Performance
• Feed System
2. Chemical Rocket Propulsion
136
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – combustion Performance Characteristics
137
2. Chemical Rocket Propulsion
propellant
1 2 e
*
isobaricheating in combustion chamber
isentropicexpansionin C-D nozzleheat per unit mass Rq
Tchq pR First thermodynamiclaw – isobaric heating:
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – combustion Performance Characteristics
138
2. Chemical Rocket Propulsion
propellant
1 2 e
*
isobaricheating in combustion chamber
isentropicexpansionin C-D nozzleheat per unit mass Rq
Tchq pR First thermodynamiclaw – isobaric heating:
p
R
c
qTT 0102
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – expansion Performance Characteristics
139
2. Chemical Rocket Propulsion
propellant
1 2 e
*
isobaricheating in combustion chamber
isentropicexpansionin C-D nozzle
ehh 002 stagnation enthalpy:
isentropic expansion
ee h
uh
2
2
02
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – expansion Performance Characteristics
140
2. Chemical Rocket Propulsion
propellant
1 2 e
*
isobaricheating in combustion chamber
isentropicexpansionin C-D nozzle
ehh 002 stagnation enthalpy:
isentropic expansion
ee h
uh
2
2
02
ep
ee
TTc
hhu
02
02
2
2
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – expansion Performance Characteristics
141
2. Chemical Rocket Propulsion
propellant
1 2 e
*
isobaricheating in combustion chamber
isentropicexpansionin C-D nozzle
ep
ee
TTc
hhu
02
02
2
2
1
0
02
02
02 1212p
pTc
T
TTcu e
pe
pe
velocity ofexhaust gases
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – exit velocityPerformance Characteristics
142
2. Chemical Rocket Propulsion
propellant
1 2 e
*
velocity ofexhaust gases
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – exit velocityPerformance Characteristics
143
2. Chemical Rocket Propulsion
propellant
1 2 e
*
velocity ofexhaust gases
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – exit velocityPerformance Characteristics
144
2. Chemical Rocket Propulsion
propellant
1 2 e
*
velocity ofexhaust gases
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – exit velocityPerformance Characteristics
145
2. Chemical Rocket Propulsion
propellant
1 2 e
*
velocity ofexhaust gases
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – mass flowPerformance Characteristics
146
2. Chemical Rocket Propulsion
propellant
1 2 e
*
mass flow
continuityequation:
*** uAm
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – mass flowPerformance Characteristics
147
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – mass flowPerformance Characteristics
148
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – mass flowPerformance Characteristics
149
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – mass flowPerformance Characteristics
150
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – thrust Performance Characteristics
151
2. Chemical Rocket Propulsion
propellant
1 2 e
*
02
0
*1
1
1
2
RT
pAm
aeeethrust ppAumF
1
0
02 11
2p
pRTu e
e
thrust
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – thrust Performance Characteristics
152
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – thrust Performance Characteristics
153
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – characteristic velocityPerformance Characteristics
154
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – thrust coefficientPerformance Characteristics
155
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – thrust coefficientPerformance Characteristics
156
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – thrust coefficientPerformance Characteristics
157
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – thrust coefficientPerformance Characteristics
158
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – thrust coefficientPerformance Characteristics
159
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – thrust coefficientPerformance Characteristics
160
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – thrust coefficientPerformance Characteristics
161
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Rocket thrust chamber – thrust coefficientPerformance Characteristics
162
2. Chemical Rocket Propulsion
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• performance of individual liquid propellants
Liquid Propellant Performance
163
2. Chemical Rocket Propulsion
sou
rce:
Ley
, Wit
tman
n, H
allm
ann
: H
and
bo
ok
of
spac
e te
chn
olo
gy
Chemical Propulsion SystemsSpace for Education, Education for Space
• There are 2 main feed systems for liquid propellant:
pressurized systems
Feed System
164
2. Chemical Rocket Propulsion
• 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
Chemical Propulsion SystemsSpace for Education, Education for Space
• There are 2 main feed systems for liquid propellant:
turbopump systems
Feed System
165
2. Chemical Rocket Propulsion
• 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
Chemical Propulsion SystemsSpace for Education, Education for Space
• turbopump systems – 3 basic cyclesFeed System
166
2. Chemical Rocket Propulsion
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)
Chemical Propulsion SystemsSpace for Education, Education for Space
• turbopump systems – 3 basic cyclesFeed System
167
2. Chemical Rocket Propulsion
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)
Chemical Propulsion SystemsSpace for Education, Education for Space
• turbopump systems – 3 basic cyclesFeed System
168
2. Chemical Rocket Propulsion
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)
Chemical Propulsion SystemsSpace for Education, Education for Space
• Static Performance
• Force-Free Motion
• Motion with Gravity
• Launch Flight Mechanics
3. Performance of Rocket Vehicle
169
Chemical Propulsion SystemsSpace for Education, Education for Space
• Momentum equation – written for CV:
• simplifications:
– quasi-one dimensional flow
– steady-state flow
Static Performance
170
3. Performance of Rocket Vehicle
Chemical Propulsion SystemsSpace for Education, Education for Space
• Momentum equation – written for CV:Static Performance
171
3. Performance of Rocket Vehicle
CV enters
momentum
Rate
CV leaves
momentum
Rate
direction
in CVin gas
on Forces
x
Chemical Propulsion SystemsSpace for Education, Education for Space
• Momentum equation – written for CV:Static Performance
172
3. Performance of Rocket Vehicle
CV enters
momentum
Rate
CV leaves
momentum
Rate
direction
in CVin gas
on Forces
x
eeaethrust umppAF
Chemical Propulsion SystemsSpace for Education, Education for Space
• Momentum equation – written for CV:Static Performance
173
3. Performance of Rocket Vehicle
CV enters
momentum
Rate
CV leaves
momentum
Rate
direction
in CVin gas
on Forces
x
eeaethrust umppAF
m
ppAum
ppAumF
aeee
aeeethrust
Chemical Propulsion SystemsSpace for Education, Education for Space
• Momentum equation – written for CV:Static Performance
174
3. Performance of Rocket Vehicle
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Momentum equation – written for CV:Static Performance
175
3. Performance of Rocket Vehicle
efthrust umF
Thrust
Engine Thrust [MN]
F1 7.77 (vacuum)
Vulcain 2 1.35
J2 1.03
NK33 1.51
Chemical Propulsion SystemsSpace for Education, Education for Space
• Momentum equation – written for CV:Static Performance
176
3. Performance of Rocket Vehicle
efthrust umF
Thrust
tFI thrustt
Total impulse
Chemical Propulsion SystemsSpace for Education, Education for Space
• Momentum equation – written for CV:Static Performance
177
3. Performance of Rocket Vehicle
efthrust umF
Thrust
tFI thrustt
Total impulse
gugmum
gmFmgII
efef
thrustts
//
//
Specific impulse
Chemical Propulsion SystemsSpace for Education, Education for Space
• Momentum equation – written for CV:Static Performance
178
3. Performance of Rocket Vehicle
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Force-free motion absence of external forcesForce-Free Motion
179
3. Performance of Rocket Vehicle
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Force-free motion absence of external forcesForce-Free Motion
180
3. Performance of Rocket Vehicle
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
Pressure force: Reae iApp
Reae iApp
Total impulse in :dt dtiApp Reae
Chemical Propulsion SystemsSpace for Education, Education for Space
• Force-free motion absence of external forcesForce-Free Motion
181
3. Performance of Rocket Vehicle
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 equation: dtiAppudmvdm ReaeeRR
Momentum equation in :Ri
dtAppdmudvm eaeeRR
Chemical Propulsion SystemsSpace for Education, Education for Space
• Force-free motion absence of external forcesForce-Free Motion
182
3. Performance of Rocket Vehicle
time t
mass dm
mass Rm
Rvambient
pressure ap
exit pressureof nozzle ep
eu
Momentum equation in :Ri
dtAppdmudvm eaeeRR
dtmdm
thrust
thrustF
Chemical Propulsion SystemsSpace for Education, Education for Space
• Force-free motion absence of external forcesForce-Free Motion
183
3. Performance of Rocket Vehicle
time t
mass dm
mass Rm
Rvambient
pressure ap
exit pressureof nozzle ep
eu
Momentum equation in :Ri
dtAppdmudvm eaeeRR
dtmdm
dtAppumdvm eaeeRR
thrust
thrustF
Chemical Propulsion SystemsSpace for Education, Education for Space
• Force-free motion absence of external forcesForce-Free Motion
184
3. Performance of Rocket Vehicle
time t
mass dm
mass Rm
Rvambient
pressure ap
exit pressureof nozzle ep
eu
Momentum equation in :Ri
dtAppdmudvm eaeeRR
dtmdm
dtAppumdvm eaeeRR
dtumdvm efRR
thrustR
R Fdt
dvm
thrust
thrustF
Chemical Propulsion SystemsSpace for Education, Education for Space
• Force-free motion absence of external forcesForce-Free Motion
185
3. Performance of Rocket Vehicle
time t
mass dm
mass Rm
Rvambient
pressure ap
exit pressureof nozzle ep
eu
Momentum equation in :Ri
dtAppdmudvm eaeeRR
dtmdm
dtAppumdvm eaeeRR
dtumdvm efRR
thrustR
R Fdt
vdm
thrustR
R Fdt
dvm
thrust
thrustF
Chemical Propulsion SystemsSpace for Education, Education for Space
• Force-free motion absence of external forcesForce-Free Motion
186
3. Performance of Rocket Vehicle
time t
mass dm
mass Rm
Rvambient
pressure ap
exit pressureof nozzle ep
eu
dtumdvm efRR
mdt
dmR
ef
R
RR u
m
dmdv
thrust
thrustF
Chemical Propulsion SystemsSpace for Education, Education for Space
• Force-free motion absence of external forcesForce-Free Motion
187
3. Performance of Rocket Vehicle
time t
mass dm
mass Rm
Rvambient
pressure ap
exit pressureof nozzle ep
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Motion with gravity vertical motion in gravity field
Motion with Gravity
188
3. Performance of Rocket Vehicle
Rv
thrustF
eu
m
Rm
gmR
gmudt
dm
dt
vdm Ref
RRR
gmudt
dm
dt
dvm Ref
RRR
Chemical Propulsion SystemsSpace for Education, Education for Space
• Motion with gravity vertical motion in gravity field
Motion with Gravity
189
3. Performance of Rocket Vehicle
Rv
thrustF
eu
m
Rm
gmR
gmudt
dm
dt
vdm Ref
RRR
gmudt
dm
dt
dvm Ref
RRR
gdtum
dmdv ef
R
RR
Chemical Propulsion SystemsSpace for Education, Education for Space
• Motion with gravity vertical motion in gravity field
Motion with Gravity
190
3. Performance of Rocket Vehicle
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Forces act on rocketLaunch Flight Mechanics
191
3. Performance of Rocket Vehicle
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Forces act on rocketLaunch Flight Mechanics
192
3. Performance of Rocket Vehicle
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Forces act on rocketLaunch Flight Mechanics
193
3. Performance of Rocket Vehicle
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
Simplif.:• all forces act on the same plane• Earth is inertial frame of reference
xy
it is function of vertical location of rocket mass of rocket is function of propellant mass flow
equation of propellant mass flow
Chemical Propulsion SystemsSpace for Education, Education for Space
• Forces act on rocketLaunch Flight Mechanics
194
3. Performance of Rocket Vehicle
trajectory of CM
Rv
eu
ap
epgmR
LF
DF
Rm
Three forces act on rocket at each instant:1. gravitational force – applied at
the CM
2. aerodynamic force – applied at aerodynamic center and can be decomposed into: - drag force - lift force
gmR
DF
LF
Simplif.:• all forces act on the same plane• Earth is inertial frame of reference
xy they are function of vertical
location and attitude of rocket
local horizont
Chemical Propulsion SystemsSpace for Education, Education for Space
• Forces act on rocketLaunch Flight Mechanics
195
3. Performance of Rocket Vehicle
trajectory of CM
Rv
eu
ap
epgmR
LF
DF
Rm
Three forces act on rocket at each instant:1. gravitational force – applied at
the CM
2. aerodynamic force – applied at aerodynamic center and can be decomposed into: - drag force - lift force
3. thrust force
gmR
DF
LF
thrustF
thrustF
Simplif.:• all forces act on the same plane• Earth is inertial frame of reference
xy
magnitude and direction can be controlled
local horizont
Chemical Propulsion SystemsSpace for Education, Education for Space
• Forces act on rocketLaunch Flight Mechanics
196
3. Performance of Rocket Vehicle
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
Three forces act on rocket at each instant:1. gravitational force – applied at
the CM
2. aerodynamic force – applied at aerodynamic center and can be decomposed into: - drag force - lift force
3. thrust force
gmR
DF
LF
thrustF
thrustF
Simplif.:• all forces act on the same plane• Earth is inertial frame of reference
xy
local horizont
Chemical Propulsion SystemsSpace for Education, Education for Space
• Forces act on rocketLaunch Flight Mechanics
197
3. Performance of Rocket Vehicle
trajectory of CM
Rv
eu
ap
epgmR
LF
DF
Rm
thrustF
Three forces act on rocket at each instant:1. gravitational force – applied at
the CM
2. aerodynamic force – applied at aerodynamic center and can be decomposed into: - drag force - lift force
3. thrust force
gmR
DF
LF
thrustF
i
FiR Mdt
dI
2
2
x
ydynamic equations:relative rotation motion around the CM
local horizont
Chemical Propulsion SystemsSpace for Education, Education for Space
• Forces act on rocketLaunch Flight Mechanics
198
3. Performance of Rocket Vehicle
trajectory of CM
Rv
eu
ap
epgmR
LF
DF
Rm
thrustF
Three forces act on rocket at each instant:1. gravitational force – applied at
the CM
2. aerodynamic force – applied at aerodynamic center and can be decomposed into: - drag force - lift force
3. thrust force
gmR
DF
LF
thrustF
LDRthrustR
R FFgmFdt
vdm
dynamic equations:translation motion of the CM
xy
local horizont
Chemical Propulsion SystemsSpace for Education, Education for Space
• Forces act on rocketLaunch Flight Mechanics
199
3. Performance of Rocket Vehicle
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Forces act on rocketLaunch Flight Mechanics
200
3. Performance of Rocket Vehicle
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Forces act on rocketLaunch Flight Mechanics
201
3. Performance of Rocket Vehicle
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
Chemical Propulsion SystemsSpace for Education, Education for Space
• Forces act on rocketLaunch Flight Mechanics
202
3. Performance of Rocket Vehicle
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