50406734 propulsion class note draft 2
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Critical Ratios for Choking to Occur
Once the choking condition is prescribed, the evaluation of
various characteristic parameters becomes so simple that we do not
need the differential form of the equations (in particular, themomentum equations) to obtain the desired result.
We have the energy equation for steady one
dimensional flow
Assuming no heat addition, this becomes
By definition of total conditions, u2 = 0 and T2 = To
Hence, the above equation becomes

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The above equation can be written in the following
form.
Hence

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By applying the energy equation to the nozzle throat,
ct
t hu
h =+2
2
ct
t TCpu
TCp =+2
2
1
2
1;
2
1;;
+=
+===
kk
t
c
t
ctt
t
tt
k
P
Pk
T
TkRTu
RT
P
Mass flow through nozzle at the choked flow condition
may be written as,
tttn Aum =
We have

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1
2
1;
2
1;;
+
+=
+===
kk
t
c
t
ctt
t
tt
k
P
Pk
T
TkRTu
RT
P
1
2
1;
2
1;;
+
+=+
===k
k
t
c
t
ctt
t
tt
k
P
Pk
T
TkRTu
RT
P
tt
t
tn
AkRT
RT
Pm =
Rearranging the above equation we get,
tc
t
c
c
t
c
n APT
T
P
P
RT
km =
At the chocked flow condition we have,1
2
1;
2
1;;
+
+=
+===
kk
t
c
t
ctt
t
tt
k
P
Pk
T
TkRTu
RT
P
1
2
1;
2
1;;
+=
+===
kk
t
c
t
ctt
t
tt
k
P
Pk
T
TkRTu
RT
P
Substituting we get
tc
t
c
c
t
c
n APT
T
P
P
RT
km =
)1(21
1
2 +
+=
kk
c
tcn
kRT
kAPm
( )11
1
2 +
+=
kk
c
tcn
kRT
kAPm
Mass flow coefficient,

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)1(21
1
2 +
+=
kk
c
dkRT
kC
( )11
1
2 +
+
=k
k
c
d
kRT
kC
The reciprocal of mass flow coefficient, Cd has
dimensions of velocity and is termed as characteristic
velocity, C*
Basic Performance Relations
The basic performance relation is derived from the principle of conservation of matter.
The propellant mass burned per unit time has to equal the sum of the change in gas mass per
unit time in the combustion chamber grain cavity and the mass flowing out through the
exhaust nozzle per unit time (Assuming negligible / nil igniter mass flow).
( )11
1
2)( +
++=
kk
c
tc
cg
bpkRT
kAP
dt
VdrA
tcd
cgn
cbp APCdt
VdaPA +=
)(
At steady state condition the above equation
reduced to,
tcd
n
cbp APCaPA =
Solving for Pc we get,
11
=
n
bp
tdc
aA
ACP
Or

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n
td
bp
cAC
aAP
=
11
Rocket motor chamber pressure during the burning time can be evaluated
approximately using the following equation.
n
t
bp
cA
caAP
=
11
where,
p = density of the propellant
Ab = Burning surface area of the graina = Constant obtained from the burn rate law )( ncPar= n = Burn rate index obtained from the burn rate law
)( ncPar=
c* = Characteristicvelocity (1/Cd)
At = Nozzle throat area

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Exit velocity
Exit velocity assumes importance from the
point of view of thrust ie., it describe the
extent to which thermal energy is converted
to a form that produces useful work.
Starting from energy equation again, we
work for exit conditions,
cp
e
ep TCu
TC =+2
2
=
c
ecpe
T
TTCU 12
=
1
12c
ecpe
P
PTCu
Complete expansion to vacuum (Pe =0) will
give the limiting exhaust velocity. i.e.,

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cpe TCu 2= dictated solely by the chamber
temperature, Tc
The nozzle geometry comes into picture
through the pressure ratio Pe/Pc .Writing by mass conservation eeettt uAuA = and
substituting for densities from equation of
state and for ue and ut from expressions
derived earlier, i.e.,
=
1
12c
ecpe
P
PTCu
te TRu =
One gets,
2111
)1(2
1
11
2
1
2
+
=
+
c
e
c
e
t
e
P
P
P
PA
A
Therefore if the choked flow prevails, Pe/Pc is
a unique function of Ae/At. So if one neglects
the influence of Pc on Tc, the nozzle exit

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velocity does not change with Pc at all, though
the mass flow rate increases linearly with Pc.
2111
)1(2
1
11
2
1
2
+
=
+
c
e
c
e
t
e
P
P
P
PA
A

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Thrust of a Rocket Motor
eaee APPumF )( +=
We have,( )1
1
1
2 +
+
=
c
tc
RT
APm
tc
t
e
c
a
c
ee AP
A
A
P
P
P
PumF
+=
=
1
12c
ecpe
P
PTCu

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+
+
=
+
tc
c
a
c
e
t
e
c
ecp
tc
c
APP
P
P
P
A
A
P
P
R
TRCAP
RTF
1
)1(2
1
121
2
tc
c
a
c
e
t
e
c
ecp
c
APP
P
P
P
A
A
P
P
R
TRC
RTF
+
+
=
+
1
)1(2
1
121
2
We have
1=
R
Cp
tc
c
a
c
e
t
e
c
e APP
P
P
P
A
A
P
PF
+
+=
+
1
)1(2
1
11
2
1
2
F = CF Pc At
+
+
=
+
c
a
c
e
t
e
c
eF
P
P
P
P
A
A
P
Pc
1
)1(2
1
11
2
1
2
When Pe = Pa , CF = CF0 and it is
referred to as optimum expansion or
the optimum nozzle or adapted nozzle
+=
+
1
)1(2
1
0 11
21
2c
eF
PPC

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For a set nozzle geometry (i.e., Ae/At), the
pressure ratio Pc/Pe is unique under choked
flow condition and hence CF0
is a constant. Itdoes not depend on chamber pressure (Pc)
also. The dependence of CF on Pc comes
through the term (Pa/Pc) only.
Thrust coefficient CF as a function of Pressure Ratio, Nozzle
Area Ratio, and Specific Heat Ratio for optimum expansion
conditions (Pe = Pa)

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r = 1.3
Ae/At = 1, 2, 4, 5, 6,10
Pc/Pe = 20
CF = 1.2, 1.365, 1.39, 1.37, 1.31,
+
+=
+
c
a
c
e
t
e
c
eF
PP
PP
AA
PPc
1
)1(2
1
11
21
2

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Flight Performance
Rocket propulsion systems provide forces to a flight vehicleand cause it to accelerate (or decelerate), overcome drag
forces, or change flight direction. They are usually applied to
several different flight regimes:

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(1) Flight within the atmosphere (air to surface missile
or sounding rockets)
(2) Nearspace environment (earth satellites)
(3) Lunar and planetary flights etc.
Gravityfree, Dragfree Space Flight
This simple rocket flight analysis applies to an outer space
environment, where there is no air (thus no drag) essentially
no significant gravitational attraction. The flight direction is
same as the thrust direction (along the axis of the nozzle) ,
namely, a onedimensional, straightline acceleration path;the propellant mass flow and thrust remain constant for the
propellant burning duration tp . For a constant propellant
flow the flow rate is mp/tp , where mp is the total usable
propellant mass. From Newtons second law and for an
instantaneous vehicle mass m and a vehicle velocity u,
F = m du/dtF = mue + (PePa) Ae
F/m = ue + (PePa) Ae / m
= c (defined as effective exhaust
velocity)
F = c m
Instantaneous vehicle mass m = mo m t

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After integration we will get the maximum vehicle velocity at
propellant burnout up in a gravity free environment
(vacuum).
up = c ln (Minitial / Mfinal)
up = c ln (mo / mf)
Powered Vertical Flight
Let u be the velocity of the rocket attained after time tfromtakeoff (considering the gravitational pull too). The
corresponding altitude is given by,
= dtuZ
=t
o
t
f
insp dttgdtM
MIgZ
0
ln
= t
o
t
pin
insp dttgdt
tmM
MIgZ
0
ln
= t
o
t
pinsp
t
insp dttgdttmMIgdtMIgZ00
)(ln
2
02
1)(lnln tgdttmMIgMtIgZ p
t
inspinsp =
Integrating the second term separately,

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t
pin
p
pint
pin tdttmM
m
tmMdttmM
0
0
)(ln)(ln
=
in
p
inpin
p
in M
tm
MttmM
m
Mt ln)(ln
+
=
Substituting it in the original equation for altitude, we get
2
2
1
lnln tgIgtmM
M
m
M
tmM
M
tIgZ sppin
in
p
in
pin
in
sp +
=
At the end of powered flight or burn out, t = tp
p
fpin
ZZ
MtmM
=
=
Therefore above equation reduces to,
2
2
1ln pp
f
in
p
inpspp tgt
M
M
m
MtIgZ
+
=
Introducing the propellant mass ratio (zeta) = mp / Min
ppp
in
tmm
M
==
The above equation yields a simpler relation

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2
2
1ln
111 p
f
inpp tg
M
MtCZ
+=
in
f
M
Mwhere = 1,
The above equation shows that the powered flight
(rocket/missile) altitude is a function of effective jet velocity,
burning time and propellant mass ratio.
A missile has a maximum flight speed to jet speed ratio of
0.2105 and specific impulse equal to 203.88 seconds.
Determine for a burn out time of 8 seconds.
(a) Effective jet velocity
(b) Mass ratio and propellant mass fraction
(c) Maximum flight speed, and
(d) Altitude gain during powered and coasting flights
Solution:
(a) C = g Isp = 9.81 x 203.88 = 2000.06 m/s
(b) Up = C ln (Min/Mf) g tp
Up/C = ln (Min/Mf) g tp/C
0.2105 = ln (Min/Mf) 9.81 x 8/2000.06ln (Min/Mf) = 0.2105 + 0.0392 = 0.2497
Mass ratio = Mf/Min = 0.78
Propellant mass fraction, = 1 Mf/Min

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= 1 0.78 = 0.22
(c ) Up/C = 0.2105
Up = 0.2105 x 2000.06 = 2121 m/sUp = 1515.65 kmph
(d )2
2
1ln
111 p
f
inpp tg
M
MtCZ
+=
= 1.594 km
Zc = 0.5 x 4212 / 9.81 x 1000 = 9.0336 km
Starting Transient of Solid Rocket Motor
*
)(
c
APrA
mmdt
md
tcbp
outinc
=
=
When nozzle is choked, the gaseous mass in the
chamber per volume can be approximated by the
perfect gas equation as,
c
cccc
TR
MVPm =
If we assume Tc to be a constant during theoperational time (transient or steady state), then

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dTdP
TRVM
dtdV
dt
dPV
dt
dVP
RT
M
dt
dm
c
c
cccg
cc
cc
c
cc
+=
+=
Note that rate of change in free volume is equal to the
volume rate of propellant consumption.
dt
dP
TR
VMrA
dt
dm c
c
ccbg
c +=
Substituting it in the mass balance equation, we get
( ) = cAP
rAdt
dP
TR
VM tcbgp
c
c
cc

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Basic Relations of Motion
For a vehicle that flies within the proximity of the earth, the gravitational of all other
heavenly bodies may usually be neglected. Let it be assumed that the vehicle ismoving in rectilinear equilibrium flight and that all control forces, lateral forces, and
moments that tend to turn the vehicle are zero. The trajectory is twodimensional and
is contained in a fixed plane. The vehicle has wings that are inclined to the flight pathat an angle of attack and that give a lift in a direction normal to the flight path. The

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direction of flight does not coincide with the direction of thrust. Below figure shows
these conditions schematically.
Twodimensional freebody force diagram for
flying vehicle with wings and fins
Let be the angle of the flight path with the horizontal
and (psi) the angle of the direction of thrust with the
horizontal. In the direction of the flight path the product of
the mass and acceleration has to equal the sum of all forces,
namely the propulsive, aerodynamics, and gravitational
forces.
From the above figure we have,
The acceleration
perpendicular to the flight path is
dt
du
; for a constant value
sin)cos( mgDFdt
dum =

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of u and the instantaneous radius R of the flight path it isR
u 2
.
The equation of motion in a direction normal to the
flight velocity is
cos)sin( mgLFdt
dmu +=
We have,2
21 uACL L =
2
21 uACD D =
The drag D is the aerodynamic force in a direction opposite
to the flight path due to the resistance of the body to motion
in a fluid. The lift L is the aerodynamic force acting in a
direction normal to the flight path. Thhey are expressed as a
function of the flight speed u, the mass density of the fluid in
which the vehicle moves , and a typical surface area A. CL
and CD are lift and drag coefficient, respectively. Forairplanes and winged missiles the area A is understood to
mean the wing area. For wingless missiles or space launch
vehicles it is the maximum crosssectional area normal to the
missile axis. The lift and drag coefficients are primarily
functions of the vehicle configuration, flight Mach number,
and angle of attack, which is the angle between the vehicle
axis (or the wing plane) and flight direction. The value ofthese coefficients reach a maximum value near a Mach
number of unity. For wingless vehicle the angle of attack is
usually very small (0 < < 10 ). The density of the earths
atmosphere can vary by a factor up to two (for altitude of
300 to 1200 km) depending on solar activity and nighttoday

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temperature variations. This introduces a major unknown in
the drag. The aerodynamic forces are affected by the flow
and pressure distribution of the rocket exhaust gases.
For space launch vehicles and ballistic missiles the dragloss, when expressed in terms of u, is typically 5 to 10% of
the final vehicle velocity increment. This relatively low value
is due to the fact that the air density is low at high altitudes,
when the velocity is high, and at low altitudes the air density
is high but the flighyt velocity and thus the dynamic
pressure are low.
Gravitational attraction is exerted upon a flying spacevehicle by all planets, stars, the moon, and the sun. Gravity
force pull the vehicle in the direction of the center of mass of
the attracting body. Within the immediate vicinity of the
earth, the attraction of other planets and bodies is negligibly
small compared to the earths gravitational forces. This is
the weight.
If the variation of gravity with the geographical features and
the oblate shape of the earth are neglected, the acceleration
of gravity varies inversely as the square of the distance from
the earths center. If Ro is the radius of the earths surface
and go the acceleration on the earths surface at the earths
effective radius Ro , the gravitational attraction g is
[ ]2
2
)/(
)/(
hRRg
RRgg
ooo
oo
+=
=
Where h is the altitude. At the equator the earths radius is
6378.388 km and the standard value of go is 9.80665 m/sec2.

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At a distance as far away as the moon, the earths gravity
acceleration is only about 3.3 x 104 go
By substituting the expression for drag and lift in the aboveequations, we get
sin2
)cos( 2 gAum
C
m
F
dt
du D =
cos2
)sin( 2 gum
C
m
F
dt
du L +=
No general solution can be given to these equations, since tp ,
u, CD, CL, p, or can vary independently with time,
mission profile, or altitude. Also CD, and CL are functions of
velocity or Mach number.
In a more sophisticated analysis other factors may be considered, such as the
propellant used for nonpropulsive purposes (e.g., altitude control or flight
stability).
Rocket Equation for vertical trajectory
this derivation neglects the effect of air resistance.
for a rocket drifting in space, gtis not applicable and can be omitted.

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LIQUID ROCKET THRUST CHAMBERS
The thrust chamber is the key subassembly of a rocket engine. Here the liquid propellants are
metered, injected, atomized, vaporized, mixed, and burned to form hot reaction gas
products, which in turn are accelerated and ejected at high velocity. A rocket thrust chamber
assembly has an injector, a combustion chamber, a supersonic nozzle, and mounting provisions.

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All have to withstand the extreme heat of combustion and the various forces, including the
transmission of the thrust force to the vehicle. There also is an ignition system if non
spontaneously ignitable propellants are used. Some thrust chamber assemblies also have
integrally mounted propellant valves and sometimes a thrust vector control device.
LIQUID ROCKET COMBUSTION CHAMBER
Admittedly, combustion in a liquid rocket is never perfectly smooth; some fluctuations of
pressure, temperature, and velocity are always present. When these fluctuations interact with the
natural frequencies of the propellant feed system (with and without vehicle structure) or the
chamber acoustics, periodic superimposed oscillations, recognized as instability, occur. In
normal rocket practice smooth combustion occurs when pressure fluctuations during steady
operation do not exceed about + 5% of the mean chamber pressure.
Combustion instability
descriptionFrequency Range (Hz) Cause Relationship
Low frequency, called
chugging or feed system
instability
10400
Linked with pressure
interactions between propellant
feed system, if not the entire
vehicle, and combustion
chamber
Intermediate frequency, called
acoustic, buzzing or entropy
waves
4001000
Linked with mechanicalvibrations of propulsion
structure, injector manifold,
flow eddies, fuel/oxidizer ratio
fluctuations, and propellant
feed system resonances
High frequency, called
screaming, screeching, or
squealing
Above 1000
Linked with combustion
process forces (pressure waves)
and chamber acoustical
resonance properties
Use of the word acoustic stems from the fact the frequency of the oscillations is related to
combustion chamber dimensions and velocity of sound in the combustion gas.
DESIGN CONSIDERATIONS OF LIQUID ROCKET
COMBUSTION CHAMBER

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Volume and Shape:
 Volume large enough for adequate mixing,
evaporation, and complete combustion
 Volume depends on speed of reaction of propellants
and stay time
 Cylindrical chamber with flat injector is preferred
Cooling of Thrust Chamber is needed to maintain the combustion chamber and nozzlewall temperatures at acceptable level
The walls of the Thrust chambers are required to withstand chamber pressure, flight
loads, ignition pressure surge, thrust loads, and thermal shock due to rapid starting.
TYPES OF INJECTORS
Doublet impinging stream pattern
Triplet impinging stream pattern
Self impinging stream pattern
Shower head stream pattern
Hollow post and sleeve element
Variable injector area concentric tube injector

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Fig. Schematic diagrams of several injector types
FACTORS INFLUENCING INJECTOR BEHAVIOUR
Propellant combination
Injector orifice pattern and size
Transient conditions
Hydraulic characteristics
Heat transfer
Structural design
The injector hole pattern, impingement pattern, hole distribution and pressure drop have a
strong influence on combustion stability.

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PROPELLANT FEED SYSTEMS
The propellant feed system has two principal functions: to raise the pressure of the propellant
and to feed them to one or more thrust chambers. The energy for these functions comes either
from a highpressure gas, centrigugal pumps, or a combination of the two. The selection of aparticular feed system and its components is governed primarily by the application of the rocket,
duration, number or type of thrust chambers, past experience, mission, and by general
requirements of simplicity of design, ease of manufacture, low cost, and minimum inert mass.
All feed systems have piping, a series of valves, provisions for filling and removing (draining
and flushing) the liquid propellants, and control devices to initiate, stop, and regulate their flow
and operation.
VALVES AND PIPE LINES
Valves control the flows of liquids and gases. Pipes conduct these fluids to the intended
components. There are no rocket engines without them. There are different types of valves. All
have to be reliable, light weight, leak proof, and must withstand intensive vibrations and very
loud noises. Often the design details, such as clearance, seat materials, or opening time delay
present development difficulties. Any leakage or valve failure can cause a failure of the rocket
unit itself. All valves are tested for two qualities prior to installation; they are tested for leaks
through the seat and also through the glands and for functional soundness or performance.
Two valves commonly used in pressuried feed system are isolation valves (when shut,
they isolate or shut off a portion of the propulsion system) and latch valves; they require power
for brief period during movements, such as to open or shut, but need no power when latched or
fastened into position. A simple and very light valve is a burst diaphragm.

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PROPELLANT TANKS
Liquid bipropellant rocket engine separate oxidizer and fuel tanks
Liquid monopropellant rocket engine one propellant tank
Common tank materials are aluminum, stainless steel, titanium, alloy steel and fibre
reinforced plastics
Optimum shape spherical
Vehicle integrated tanks are mostly cylindrical with half ellipses at the ends.

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TANK PRESSURIZATION
Typical pressure ranges of pressurizedfeed systems is from 1.3 to 9.0 MPa
Typical pressure ranges of turbopump feed systems is from 0.07 to 0.34 MPa
Inert gases such as helium or nitrogen are the most common method of pressurization
Dynamic loads on Liquid Rockets Propellant Slosh
 Free surface oscillations of Fluid
 Lateral loads and C.G shift
 uncover the tank outlet
Propellant Hammer
 Similar to Water Hammer due to sudden closure or opening of valves
Geysering effect and its elimination
 Caused by rising bubbles in vertical propellant pipes due to heating
 Introduction of Geysering inhibitor pipe

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Rockets and Mission Analysis
Concept Quiz
It may be shown that the thrust of a rocket scales with area and weight scales with volume, both
based on the same characteristic length scale, L. This is expressed below as:
Thrust, T T scales with thethroat area, At
2
~~ LAT t
Weight,
W
W scales with the
volume, V
3~~ LVW
1. If we double T by increasing the characteristic length scale, how does T/W vary?T/W ~ 1/L ~ 1/T ~ T
Double T, T/W changes by 1/2
2. If we double T by increasing the characteristic length scale, how does W vary?W ~ T3/2
2(3/2) = 2.83
W goes up by factor of 2.83
HOME WORK!
1. Differentiate solid, liquid and hybrid rockets
2. Derive an expression for coasting time and coasting altitude of a rocket with verticaltakeoff.
3. What is the purpose of thrust vector control in rockets?
4. Name two liquid propellants and two solid propellants.
5. Why is it better to launch a spaceship from near the equator?
6. Derive rocket equation to evaluate the relative velocity (V) in terms of effectiveexhaust velocity and mass ratio.
7. Derive an expression for the exit velocity of a rocket nozzle.
8. Prove that powered rocket altitude is a function of effective jet velocity, burning time
and mass ratio.
9. A 5,000 kg spacecraft is in Earth orbit traveling at a velocity of 7,790 m/s. Its engineis burned to accelerate it to a velocity of 12,000 m/s placing it on an escape trajectory.
The engine expels mass at a rate of 10 kg/s and an effective velocity of 3,000 m/s.
Calculate the duration of the burn.
10. Consider two rockets:
Rocket 1: n=3 stages, each stage with identical payload ratio, =0.25, exit
velocity, Ue=3,000 m/s, and structural coefficient,
=0.1. Rocket 2: n=5 stages, each stage with identical payload ratio, =0.5, exit
velocity, Ue=4,000 m/s, and structural coefficient, =0.2.
Which of these rockets has a larger overall mass ratio, Mo1/ML?
11. A rocket has a maximum flight speed to jet speed ratio of 0.2105 and specific
impulse equal to 204 seconds. Determine for a burn out time of 8 seconds.

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(i) Effective jet velocity, (ii) Mass ratio and propellant mass fraction,
(iii) Maximum flight speed, and (iv) Altitude gain during powered and
coasting flights.12. Assume a rocket of total mass 100 tons, carrying a spacecraft payload of 1 ton. The
engines develop a constant exhaust velocity of 3,000 m/s. The structural mass is
assumed to be 10% of the fuel mass.1) Determine the velocity of this configuration as a single stage rocket2) If the rocket is divided into two smaller stages, each with half the fuel, and the
structural mass also shared equally, and the payload being the same,
determine the total velocity increment for the two stage configuration.
13. Repeat part (2), assuming 3 stages. What do you notice about the total velocityincrement as you add more and more stages? As an engineer, how would you
determine how many stages to use? .
14. A rocket engine burning liquid oxygen and liquid hydrogen operates at a
combustion chamber pressure of 75 atmospheres. If the nozzle is expanded to operate
at sea level, calculate the exhaust gas velocity relative to the rocket. Use the designcharts providing optimum mixture ratio, adiabatic flame temperature, gas molecular
weight, and specific heat ratio for liquid oxygen and liquid hydrogen.
15. What are the technological challenges for the design of multistage rocketswith both solid propellant motors and liquid propellant engines? Discuss about the
selection criteria of propellants, important hardware components, solid propellant grain
design and liquid engine design considerations, vectoring, cooling rocket systems, andmerits and demerits of solid and liquid rockets.