Download - 50406734 Propulsion Class Note Draft 2

8/3/2019 50406734 Propulsion Class Note Draft 2

1/41

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


11/41

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


13/41

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)


14/41

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


15/41


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18/41

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) Near-space environment (earth satellites)

(3) Lunar and planetary flights etc.

Gravity-free, Drag-free 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 one-dimensional, straight-line 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 + (Pe-Pa) Ae

F/m = ue + (Pe-Pa) 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 tfromtake-off (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,


21/41

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


22/41

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


24/41

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


25/41


26/41

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 two-dimensional 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


27/41

direction of flight does not coincide with the direction of thrust. Below figure shows

these conditions schematically.

Two-dimensional free-body 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 =


28/41

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 cross-sectional 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 night-to-day


29/41

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.


30/41

At a distance as far away as the moon, the earths gravity

acceleration is only about 3.3 x 10-4 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.


31/41

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.


32/41

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

10-400

Linked with pressure

interactions between propellant

feed system, if not the entire

vehicle, and combustion

chamber

Intermediate frequency, called

acoustic, buzzing or entropy

waves

400-1000

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


33/41

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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35/41


36/41

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.


37/41

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 high-pressure 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.


38/41

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.


39/41

TANK PRESSURIZATION

Typical pressure ranges of pressurizedfeed systems is from 1.3 to 9.0 MPa

Typical pressure ranges of turbo-pump 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


40/41

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.

Download - 50406734 Propulsion Class Note Draft 2

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