Post on 04-Apr-2018
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Analysis of Rocket Propulsion
P M V Subbarao
Professor
Mechanical Engineering Department
Continuously accelerating Control Volume.
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THE TSIOLKOVSKY ROCKET EQUATION
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Force Balance on A Rocket
DgMT
dt
dVM r
rr sin
DgM
dt
dMC
dt
dVM r
rrr sin
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r
r
r
r
M
Dg
dt
dM
M
C
dt
dV sin
Let
dragrnalgravitatiorspacerrVVVV
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bt
nalgravitatiordtgV
0
sin bt
rdragr
dtM
DV
0
bt
e
rir
r
spacer Mr
MCdt
dt
dM
M
CV
0
ln
For a typical launch vehicle headed to an orbit, aerodynamic
drag losses are typically quite small, on the order of 100 to
500 m/sec.
Gravitational losses are larger, generally ranging from 700 to1200 m/sec depending on the shape of the trajectory to orbit.
By far the largest term is the equation for the space velocity
increment.
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REACHING ORBIT
The lowest altitude where a stable orbit can be maintained, is atan altitude of 185 km.
This requires an Orbital velocity approximately 7777 m/sec.
To reach this velocity from a Space Center, a rocket requires an
ideal velocity increment of 9050 m/sec.
The velocity due to the rotation of the Earth is approximately
427 m/sec, assuming gravitational plus drag losses of 1700
m/sec.
A Hydrogen-Oxygen system with an effective average exhaustvelocity (from sealevel to vacuum) of 4000 m/sec would require
Mi/ Mf= 9.7.
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Geostationary orbit
A circular geosynchronous orbit in the plane of the Earth's
equator has a radius of approximately 42,164 km (26,199
mi) from the center of the Earth.
A satellite in such an orbit is at an altitude of
approximately 35,786 km (22,236 mi) above mean sea
level.
It maintains the same position relative to the Earth's
surface.
If one could see a satellite in geostationary orbit, it would
appear to hover at the same point in the sky.
Orbital velocity is 11,066 km/hr= 3.07 km/sec (6,876
miles/hr).
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Travel Cycle of Modern Spacecrafts
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MULTISTAGE ROCKETS
With current technology and fuels, a single stage rocket toorbit is still not possible.
It is necessary to reach orbit using a multistage system
where a certain fraction of the vehicle mass is dropped off
after use thus allowing the non-payload mass carried toorbit to be as small as possible.
The final velocity of an n stage launch system is the sum of
the velocity gains from each stage.
nn VVVVV ...........321
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ANALYSIS OF MULTISTAGE ROCKETS
M0i : The total initial mass of the ith stage prior to
firing including the payload mass,ie, the mass of
i, i+1, i+2, i+3,...., n stages.
Mpi: The mass of propellant in the ith stage.
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Msi : Structural mass of the ith stage alone including the mass of its
engine, controllers and instrumentation as well as any residual
propellant which is not expended by the end of the burn.
ML : The payload mass
Mass ratio
pii
i
i
ii
MM
M
M
MR
0
0
10
0
sii
sipii
iMM
MMMR
10
10
1,
lni
ri
ispacer Mr
MCV
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100
10
100
10
ii
sii
ii
sipii
i
MMMM
MM
MMM
R
100100
10
100
101
ii
si
ii
i
ii
i
i
MMMMMM
MMM
R
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Structural coefficient
pisi
si
ii
sii
MM
M
MM
M
100
Payload ratio
100
10
ii
i
iMM
M
Ln
L
nn
n
nMM
M
MM
M
0100
10
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ii
iiR
1
Space (Ideal) velocity increment
n
iinRCV
1
ln
Payload fraction
n
LL
M
M
M
M
M
M
M
M
M
M
003
04
02
03
01
02
01
....
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nnL
M
M
1....1113
3
2
2
1
1
01
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MOMENTUM BALANCE FOR A ROCKET
Rocket mass X Acceleration = Thrust Drag -gravity effect
dragTr
r FgF
dt
dVM sin
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EFFECTIVE EXHAUST VELOCITY
The total mechanical impulse (total change of omentum)
generated by an applied force, FT, is:
riV
i
dragr
r
ti
Tii dtFgdt
dVMFI00
sin
The total propellant mass expended is
ti
ipi dtmM
0
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The instantaneous change of momentum per unit expenditure
of propellant mass defines the effective exhaust velocity.
ii
Ti
pi
i
i Sm
F
dM
dI
C
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Rocket Principles
High pressure/temperature/velocity exhaust gases
provided through combustion and expansion through
nozzle of suitable fuel and oxidiser mixture.
A rocket carries both the fuel and oxidiser onboard
the vehicle whereas an air-breatherengine takes in
its oxygen supply from the atmosphere.
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Criteria of Performance
Specific to rockets only. thrust
specific impulse
total impulse
effective exhaust velocity
thrust coefficient
characteristic velocity
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Thrust (F)
For a rocket engine:
m
ambeeejectsejectsT ppAUmF
Where:
= propellant mass flow rate
pe = exit pressure, paamb = ambientpressure
Uejects = exit plane velocity, Ae = exit area
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Specific Impulse (I or Isp)
The ratio of thrust / ejects mass flow rate is used to define arockets specific impulse-best measure of overall performance of
rocket motor.
In SI terms, the units of I are m/s or Ns/kg.
In the US:
with units of seconds - multiply by g (i.e. 9.80665 m/s2) in
order to obtain SI units of m/s or Ns/kg.
Losses mean typical values are 92% to 98% of ideal values.
ejects
T
m
F
spI
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Total Impulse (Itot)
Defined as:
where tb = time of burning
If FT is constant during burn:
bt
TdtF0
totalI
bT tF totalI
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Thus the same total impulse may be obtained byeither :
high FT, short tb (usually preferable), or
low FT, long tb
Also, for constant propellant consumption (ejects) rate:
bejects
ejects
T tmm
F
totalI
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Effective Exhaust Velocity (c)
Convenient to define an effective exhaustvelocity (c), where:
cmF ejectsT
I
ejects
T
m
Fc
ejects
e
mpc
eambe
ApU
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Thrust Coefficient (CF)
Defined as:
t
TF
A
FC
cP
where pc = combustion chamber pressure,
At = nozzle throat area
Depends primarily on (pc/pa) so a good indicator of
nozzleperformance dominated by pressure ratio.
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Characteristic Velocity (c*)
Defined as:
(6)ejects
t*
m
AC
cP
Calculated from standard test data.
It is independent of nozzle performance and is
therefore used as a measure ofcombustion
efficiency dominated by Tc (combustion chamber
temperature).
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Thermodynamic Performance - Thrust
Parameters affecting thrust are primarily: mass flow rate
exhaust velocity
exhaust pressure nozzle exit area
http://www.fas.org/man/dod-101/sys/missile/agm-119-980721-N-5961C-001.jpg7/29/2019 rocket .ppt
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Thermodynamic Performance- Specific Impulse
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Thermodynamic Performance
- Specific Impulse
Variable Parameters - Observations
Strong pressure ratio effect - but rapidly diminishing returns
after about 30:1.
High Tc value desirable for high I - but gives problems with
heat transfer into case walls and dissociation of combustion
products practical limit between about 2750 and 3500 K,
depending on propellant.
Low value of molecular weight desirable
favouring use ofhydrogen-based fuels.
Low values of desirable.
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31
Thrust Coefficient (CF) Maximum thrust when exhausting into a
vacuum (e.g. in space), when: (11a
)
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Thrust Coefficient (CF)
- Observations
More desirable to run a rocket under-expanded
(to left of optimum line) rather than over-expanded.
Uses shorter nozzle with reduced weight and size.
Increasing pressure ratio improves performance
but improvements diminish above about 30/1.
Large nozzle exit area required at high pressureratios implications for space applications.
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Actual Rocket Performance
Performance may be affected by any of the followingdeviations to simplifying assumptions:
Properties of products of combustion vary with static
temperature and thus position in nozzle.
Specific heats of combustion products vary with temperature. Non-isentropic flow in nozzle.
Heat loss to case and nozzle walls.
Pressure drop in combustion chamber due to heat release.
Power required for pumping liquid propellants.
Suspended particles present in exhaust gas.
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Internal Ballistics
Liquid propellant engines store fuel and oxidiserseparately - then introduced into combustion chamber.
Solid propellant motors use propellant mixture
containing all material required for combustion.
Majority of modern GW use solid propellant rocket
motors, mainly due to simplicity and storage
advantages.
Internal ballisticsis study of combustion process of
solid propellant.
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Solid Propellant Combustion
Combustion chamber is high pressure tankcontaining propellant charge at whosesurface burning occurs.
No arrangement made for its control chargeignited and left to itself so must self-regulateto avoid explosion.
Certain measure of control provided bycharge and combustion chamber design andwith inhibitor coatings.