Analysis of Rocket Propulsion

P M V SubbaraoProfessor

Mechanical Engineering Department

Continuously accelerating Control Volume….

THE TSIOLKOVSKY ROCKET EQUATION

Force Balance on A Rocket

DgMTdt

dVM rr

r sin

DgMdt

dMCdt

dVM rrr

r sin

r

r

r

r

MDg

dtdM

MC

dtdV

sin

Let

dragrnalgravitatiorspacerr VVVV

bt

nalgravitatior dtgV0

sin bt

rdragr dt

MDV

0

bt

e

rir

rspacer Mr

MCdtdt

dMMCV

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 to 1200 m/sec depending on the shape of the trajectory to orbit. By far the largest term is the equation for the space velocity increment.

REACHING ORBIT

• The lowest altitude where a stable orbit can be maintained, is at an 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 exhaust velocity (from sealevel to vacuum) of 4000 m/sec would require Mi/ Mf = 9.7.

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).

Travel Cycle of Modern Spacecrafts

MULTISTAGE ROCKETS

• With current technology and fuels, a single stage rocket to orbit 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 to orbit 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

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.

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 ratiopii

i

i

ii MM

MMMR

0

0

10

0

sii

sipiii MM

MMMR

10

10

1,

lni

riispacer Mr

MCV

100

10

100

10

ii

sii

ii

sipii

i

MMMM

MMMMM

R

100100

10

100

101

ii

si

ii

i

ii

i

i

MMM

MMM

MMM

R

Structural coefficient

pisi

si

ii

sii MM

MMM

M

100

Payload ratio

100

10

ii

ii MM

M

Ln

L

nn

nn MM

MMM

M

0100

10

ii

iiR

1

Space (Ideal) velocity increment

n

iin RCV1

ln

Payload fraction

n

LL

MM

MM

MM

MM

MM

003

04

02

03

01

02

01

....

n

nL

MM

1....

111 3

3

2

2

1

1

01

MOMENTUM BALANCE FOR A ROCKET

Rocket mass X Acceleration = Thrust – Drag -gravity effect

dragTr

r FgFdt

dVM sin

EFFECTIVE EXHAUST VELOCITY

The total mechanical impulse (total change of omentum) generated by an applied force, FT, is:

riV

idrag

rr

ti

Tii dtFgdt

dVMFI00

sin

The total propellant mass expended is

ti

ipi dtmM0

The instantaneous change of momentum per unit expenditure of propellant mass defines the effective exhaust velocity.

ii

Ti

pi

ii S

mF

dMdIC

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-breather engine takes in its oxygen supply from the atmosphere.

Criteria of Performance• Specific to rockets only.

– thrust– specific impulse– total impulse– effective exhaust velocity– thrust coefficient– characteristic velocity

Thrust (F)For a rocket engine:

m

ambeeejectsejectsT ppAUmF

Where:

= propellant mass flow rate

pe = exit pressure, paamb = ambient pressure

Uejects = exit plane velocity, Ae = exit area

Specific Impulse (I or Isp)

• The ratio of thrust / ejects mass flow rate is used to define a rocket’s 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

mF

spI

Total Impulse (Itot)

• Defined as:

where tb = time of burning

• If FT is constant during burn:

bt

T dtF0

totalI

bT tF totalI

• Thus the same total impulse may be obtained by either :

• high FT, short tb (usually preferable), or

• low FT, long tb

• Also, for constant propellant consumption (ejects) rate:

bejectsejects

T tmm

F

totalI

Effective Exhaust Velocity (c)

• Convenient to define an effective exhaust velocity (c), where:

cmF ejectsT I

ejects

T

mFc

ejects

e

mpc

eamb

eApU

Thrust Coefficient (CF)

• Defined as:

t

TF A

FCcP

where pc = combustion chamber pressure,

At = nozzle throat area

• Depends primarily on (pc/pa) so a good indicator of nozzle performance – dominated by pressure ratio.

Characteristic Velocity (c*)

• Defined as:

(6)ejects

t*

mAC

cP

•Calculated from standard test data.

• It is independent of nozzle performance and is therefore used as a measure of combustion efficiency – dominated by Tc (combustion chamber temperature).

Thermodynamic Performance - Thrust

• Parameters affecting thrust are primarily:– mass flow rate– exhaust velocity– exhaust pressure– nozzle exit area

Thermodynamic Performance - Specific Impulse

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 of hydrogen-based fuels.

• Low values of desirable.

31

Thrust Coefficient (CF)

• Maximum thrust when exhausting into a vacuum (e.g. in space), when:

(11a)

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 pressure ratios – implications for space applications.

Actual Rocket Performance • Performance may be affected by any of the following

deviations 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.

Internal Ballistics• Liquid propellant engines store fuel and oxidiser

separately - 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 ballistics is study of combustion process of solid propellant.

Solid Propellant Combustion• Combustion chamber is high pressure tank

containing propellant charge at whose surface burning occurs.

• No arrangement made for its control – charge ignited and left to itself so must self-regulate to avoid explosion.

• Certain measure of control provided by charge and combustion chamber design and with inhibitor coatings.