chapter 2n+êdefinitions and parametersn+ë (2)

8/2/2019 Chapter 2n+definitions and parametersn+ (2)

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2 Definitions and Basic Parameters

2.1 Main Performance Parameters of Thrust Chamber

2.1.1 Thrust

Fig.2- 1shows schematically the external pressure acting uniformly on the outer surface and the internal gas

pressure acting on the inside surface of a typical rocket engine thrust chamber.

Fig.2- 1 Pressure distribution on chamber and nozzle inside and outer surfaces1Thrust F

eaee AppumF )( += &

Discussion:

2Characteristic thrust cFeF mu&c

3Effective exhaust velocity (an imaginary exhaust velocity)efue

ae

eef Am

ppuu

&

+=

4Altitude performance of rocket engines/Motors2.1.2 Nozzle Exhaust Velocity

1. Nozzle Exhaust velocity eu

=

k

k

c

efe

p

pRT

k

ku

1

11

2

or

2


3/12

=

k

k

c

efe

p

pT

m

R

k

ku

1

0 11

2

Discussion: main factors

combustion temperature average molecular weight specific heat ratio pressure expansion ratio

2. Effective Exhaust velocity efue

aeeef

Am

ppuu

&

+=

3. Maximum theoretical value of the nozzle outlet velocity, i.e. maximum exhaust velocity maxuffpL RT

k

kTcHu

1

222 0

===

2.1.3 Mass Flow Rate and Characteristic Velocity

1. Mass flow rate m&

f

tc

RT

Apm =&

)1(21

1

2 +

+=

kk

kk

2. Coefficient of mass flow rate DCtcD ApCm =&

f

DRT

c

=

3. Characteristic Velocity *c*c

Apm tc=&

f

D

RT

Cc ==

1*

=)(/1800~1500

)(/2400~1550*

motorssolidforsm

chambersthrustrocketliquidforsmc

2.1.4 Thrust Coefficient

1. Thrust coefficient FC 3


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c tFF C p A=

+

=

c

a

c

e

t

ek

k

c

eF

p

p

p

p

A

A

p

p

k

kC

1

11

2

2. Characteristic thrust coefficient0

FC

=

k

k

c

eF

p

p

k

kC

1

0 11

2

3. Nozzle area expansion ratio (De Laval nozzle) =

t

e

A

A

4. Relationship between the nozzle area ratiot

e

A

Aand the nozzle pressure ratio

c

e

p

p

+=

+

k

k

c

ek

c

e

k

k

t

e

p

p

k

k

p

p

kk

A

A

11

)1(2

1

11

2

1

2

5. Maximum thrust and Maximum thrust coefficientmaxF max,FCFor any fixed pressure ratio between the chamber pressure

cp and the nozzle exit pressure ep the

thrust coefficient has a maximum value whenFC e ap p= . This value is also known as the optimum thrust

coefficient. This maximum value can be derived by differentiation of the expression of the thrust coefficient

with respect to the pressure ratioFC c ap p ,namely, ed d(F cC p p ) , and then setting the derivative

equal to zero, that is:

ed d(F cC p p ) =0

Corresponding to the maximum thrust coefficient, the gas expansion in the nozzle is called optimum

expansion(See Fig.2- 2).

Three working states of nozzle:

Underexpansion

Optimum expansion

Overexpansion

The thrust coefficient is plotted in Fig.2- 2 as a function of the pressure ratio and the area ratio for =1.30.

The set of curves are useful in solving various nozzle problems, for they permit an evaluation of under- and

overexpansion. The values given in this figure is ideal and do not consider any losses such as nozzle

divergence, friction, or internal shock waves.

k

4


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Fig.2- 2 Thrust coefficient versus area ratioFC =

t

e

A

Afor =1.30k

2.1.5 Total Impulse and Specific Impulse

1. Action time at2. Burning time(duration) (for solid motors)bt

Fig.2- 3 Action time and Burning time

3. Total Impulse tIDefinition:

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=at

FdtI0

Average thrust

a

t

t

FdtF

a

=

0

atFI =

If thrust is a constant,then

I = Fta

, ,=at

efdtumI0

& =at

pMdtm0&

efu constant

I=Mpuef

4.

Specific Impulse sI

Definition:s

p

II

M= N s/kg or m/s

=

)(/2600~2100

)

ound),3562m/s(grcuum)4464m/s(va:SSMEassuch

s,propellantenergygh4000m/s(hiabovetoup3300m/s,~2500

motorssolidforsm

chambersthrustrocketliquidforIs

5. Specific thrust sFDefinition:

m

FFs

&= N(kgs)] or [ms]

2.1.6 Primary Relations Among the Main Performance Parameters of Thrust Chambers

1. Relation Between andsI efu

Is = uef

2. Relation Between and andsI *c FCIs = c

*CF

3. Other relations

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*

*

ac

*c

c

c =

id

Its value is high. *c

=0.94~0.99 approximately.

Propellants burn to varying degrees of completeness depending on the fuel, the oxidizer, their ratios,

the energy losses, and the environmental within the engine or the motor. For solid rocket motors,

propellants with non-metal fuels usually burn with an efficiency of 97 or 98% as contrasted to 90 to 96%

for propellants with aluminum powder as the fuel. The solid particles in the exhaust do not contribute to

the gas expansion, require energy to be accelerated and two-phase flow is less efficient.

2. Impulse coefficient of the nozzle (or nozzle efficiency)FC

,ac

,idF

F

C

F

C

C =

FC =0.88~0.97.

3. Impulse coefficient of the thrust chamber

s

,ac

,id

s

I

s

I

I =

*s FI Cc

=

sI =0.82~0.96.

2.3 Main Engine Parameters

This section is mainly used for liquid rocket engine.

2.3.1 Thrust of Engine

The thrust of an engine is defined as:F

1

n

i tcF F F F= + +

2.3.2 Total impulse of Engine

The total impulse of an engine I is defined as:

=at

dtFI0

2.3.3 Specific impulse

The specific impulse of an engine s,I is defined as:

=

++= n

i

tci

s

mmm

FI

1

,

&&&

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2.3.4 Thrust-to-mass ratio

The thrust-to-mass ratio of an engine is defined as:,F mr

,

0

F m

Fr

m

=

0m loaded propulsion system mass which usually consists of engine hardware mass (engine hardware

plus hardware necessary to store propellant) and the loaded propellant mass. may be called the wet mass

of engines.

0m

,

1.0~1.3kN/kg(for moden liquid engines, at chamber pressure of 15~20MPa)

0.05(for low thrust units)~2kN/kg(for high thrust units)F mr

=

The Thrust-to-weight ratio is also used as follows:GFr ,

grr mFGF

,, =

FOR CHEMICAL ROCKET ENGINE (SOLID OR LIQUID BIPROPELLANT), IS APPROXIMATELY WITHIN THE

RANGE OF 10

,F Gr

-2~100(REFERRING TO FULL PROPULSION SYSTEM SEA LEVEL WEIGHT WITH PROPELLANTS ,

BUT WITHOUT PAYLOAD).

2.3.5 Specific propellant consumption

Specific propellant consumptions

1mF I= =&

2.3.6 Mass ratio of a vehicle or a particular vehicle stage or a engine

The mass ratio of a vehicle or a particular vehicle stage or a engineis defined as:

0m

mf=

2.3.7 Propellant mass fraction of a vehicle or a particular vehicle stage or a engine

The propellant mass fraction of a vehicle or a particular vehicle stage or a engine is defined as:

0m

mp=

=1-

2.3.8 Impulse-to-mass ratio

s

pf

t Imm

Ft

m

I

+=

0

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10

A high value indicates an efficient design. As the fraction of propellant becomes very large (

approaches 1.0), the value of the impulse-to-mass ratio approaches that of the specific impulse. The value of

the impulse-to-mass ratio cannot exceed the value of the specific impulse.

2.4 Energy and Efficiencies of Rocket Engines

2.4.1 Kinetic energy of the ejected matter

The kinetic energy of the ejected matter is defined as:jetE

2

jet e

1

2 E mu=

2.4.2 Power of the jet

Thepowerof the jet is defined as:jetP

jet 2

jet e

d 1

d 2

EP m

t= = & u

For the nozzle with optimum expansion ratio, there is

jet 2 2

jet e s s e

d 1 1 1 1

d 2 2 2 2

EP mu mI FI

t= = = = =& & Fu

2.4.3 Specific power

0

jetP

m

2.4.4 Power input to a chemical engine

The power input to a chemical engine is defined as:chemP

RchemP mQ= &

2.4.5 Power transmitted to the vehicle

The Power transmitted to the vehicle is defined as:vehicleP

FvPvehicle =

vvehicle velocity.

2.4.6 Internal efficiency

The Internal efficiencyint is defined as:


11/12

chem

jet

P

P=int

2.4.7 Propulsive efficiency

At the condition of optimum expansion, thepropulsive efficiency p is defined as:

22 )/(1

/2

)(2

1

energyjetkineticresidual

energyjetkineticresidualenergyvehicle

energyvehicle

ef

ef

ef

vehicle

vehicle

p

uv

uv

vumFv

Fv

P

P

+=

+=

+=

+=

&

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

80

90

100

Propusive efficiency calculated

Velocity ratio

Propusive efficiency-velocity ratio

Fig.2- 5 Propulsive efficiency at varying velocities

2.4.8 Overall efficiency

The overall efficiency overall is defined as:

2

2

1

enginethefromvehiclethetoavailableenergymaximum

powervehicle

vmP

P

pchem

vehcle

overall

&+=

=

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chapter 2n+êdefinitions and parametersn+ë (2)

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