Rocket Propulsion Fundamentals

Transcript

ROCKET PROPULSION SYSTEMS
Functions
•
Thrust generation by acceleration of an internally stored
propellant for:
– launch & orbit insertion
– orbit maneuvering & maintenance
– attitude control
Basic Elements
• Propellant(s) & Propellant Storage
• Propellant Feed System
• Energy Source
• Energy Conversion
• Accelerator
Main Technologies & Energy Sources
Cold Gas Rocket
• Cold Gas Rocket Propulsion Systems (CGRPSs), thermal
• Liquid Rocket Propulsion Systems (LRPSs ), chemical
• Solid Rocket Propulsion Systems (SRPSs), chemical
• Hybrid Rocket Propulsion Systems (HRPSs), chemical
• Nuclear Rocket Propulsion Systems (NRPSs), nuclear
• Electrical Rocket Propulsion Systems (ERPSs), electrical (photovoltaic, nuclear, etc.)
Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11.
LIQUID ROCKET PROPULSION SYSTEMS
System Architecture
SOLID & HYBRID ROCKET PROPULSION SYSTEMS
System Architecture
ELECTRIC ROCKET PROPULSION SYSTEMS
System Architecture
Arcjets
Magnetoplasmadynamic (MPD) Thrusters
Electrostatic Thrusters
Hall Thrusters
KEY PROPULSION TECHNOLOGIES
Operational Comparison
ROCKET MASS BALANCE
Rocket Continuity Equation
•
AS
Notations and assumptions:
– control volume V , the rocket itself, bounded by:
AS the rocket external surface (impermeable, u ! dS = 0 )
Ae the nozzle exit area
– inertial frame moving at the instantaneous rocket speed:
rocket velocity v = 0 (but in general dv dt ! 0 )
flow velocity u
– rocket mass m
– propellant mass flow rate m!
– nearly uniform flow at the nozzle exit (index e)
•
Then:
!
" dV + # "u $ dS +
Ae
! t #V
where:
dm !
= # " dV
dt ! t V
m! = # !u " dS
Ae
#
AS
"u $ dS = 0
!
v
m
Ae , ! e
ue
dm
+ m! = 0
dt
is the rate of change of the rocket mass m
is the propellant flow rate ( m! ! "eue Ae in the 1D approximation)
ROCKET FORCES
pa
Rocket Momentum Equation
•
AS
v
– uniform ambient pressure pa and gravitational body force g
•
D
In an inertial frame where v = 0 (but in general dv dt ! 0 ):
mg
!
& $ "uu $ ( p $ pa ) 1 + % () * dS + # " g dV ,
"udV = #
Ae + AS '
V
! t #V
Ae , pe , !e
d
dv
dm
ue
!
= F + FA + mg
( mv ) = m + v
dt
dt
dt
v=0
where the use of the relative pressure p ! pa is justified because a uniform pressure pa
yields no net force on closed surfaces, and:
" !g dV
F ! " % #uu $ dS " % ( p " p ) dS
!#"#$ !##"##
$
is the rocket weight
FA ! " # ( p " pa ) dS + # $ % dS
AS
AS
!
##"##
$ !
#
"#
$
is the aerodynamic force ( u ! dS = 0 on AS )
mg =
V
Ae
Ae
momentum thrust
A/D pressure force
•
a
is the rocket thrust ( ! " 0 on Ae )
pressure thrust
A/D viscous force
Notice that, projecting in the forward direction, in the 1D approximation:
! e + ( pe " pa ) Ae
F ! mu
ROCKET POWERS
AS
Rocket Energy Equation
•
•
m
– free rocket, adiabatic conditions ( g = pa = q! = Q! = 0 )
– hR , m R formation enthalpy and mass of reacting propellants
– hP ! h!P + c pP ( T " T° ) formation enthalpy of combustion products
v
ue
Ae , ! e , Te , pe
From the energy equation for the usual control volume V :
!
!p
"
"
h
dV
+
dS
$
u
"
h
=
dV
=
0
!
( mRhR + mBOhBO ) + %Ae dS # u$ htP & 0
t
t
!# S
#V ! t
! t #V
"t
and, since mBO hBO is constant and by continuity ! m R ! t = " m! = " % #u $ dS :
Ae
1
%
(
!hR $ "u # dS + $ ' h!P + c pP ( T ! T° ) + u # u * "u # dS + 0
! Pc = Q! j + W! j
Ae
Ae
2
&
)
where:
P!c = m! (hR ! h"P )
(combustion power)
! pP ( Te " T° )
(exhaust thermal power)
Q! e ! c pP ( T " T° ) #u $ dS ! mc
%
Ae
1
1 2
1 2F 1
!
u
!
u
#
u
!
dS
$
u
m
=
ue $ Fc
e
"Ae 2
2
2 c 2
$ # 1) Me2 2
W! e
ue2 2
(
!=
"
"
"1
P!c c pP ( T # T° ) + ue2 2 1 + ($ # 1) Me2 2
W! e =
(exhaust mechanical power)
(rocket engine efficiency)
FREE ROCKET PERFORMANCE
Thrust Performance Parameters
•
Effective exhaust velocity:
! e + ( pe " pa ) Ae
( p " pa ) Ae
F mu
c= !
= ue + e
m!
m!
m!
Notice:
– c is a function of the ambient pressure pa
– usually ( pe ! pa ) Ae m! << ue and therefore c ! ue (exhaust velocity)
•
Specific impulse and volume specific impulse (or density impulse):
F
F
F
and
Isp =
! c = g0 Isp
Id = =
= ! g0 Isp
! 0
mg
V! m! !
Notice:
– Isp
–
–
–
•
Id
g0
V!
mostly relevant to mass optimization
mostly relevant to volumetric and aerodynamic optimization
is the gravity acceleration at sea level
is the volumetric flow rate of the stored propellant
Problem: Show that for rockets operating with two propellants (fuel F and oxidizer O ):
m! O + m! F
Id = ! Isp g0
! "=
m! O "O + m! F " F
FREE ROCKET PERFORMANCE
Tsiolkovsky’s Free Rocket Equation
•
•
•
•
Assumptions:
– straight trajectory
– no body forces, no atmosphere ( g = pa = 0 )
– constant effective exhaust velocity c
F
v
m
From the rocket mass and axial momentum balances:
Dm
Dv
and
= ! m!
m
=F
Dt
Dt
ṁ
! and eliminating m! with the continuity equation:
Using F = mc
ue
Dv
Dm
dm
! = !c
m
= mc
! dv = "c
Dt
Dt
m
Integrating between the initial and burn-out masses m0 , mBO the rocket velocity change is:
m
(Tsiolkovsky’s free rocket equation)
! v = "c ln BO
m0
Compare with Breguet eq’n for leveled cruise with lift-to-drag ratio L D and velocity v0 :
L m
! x = "v0 Isp ln BO
D m0
Problem: Why are both equations logarithmic in the vehicle mass and linear in the
propulsive (and aerodynamic) performance parameters Isp (and L D )?
ROCKET THRUST PERFORMANCE
Equations and Parameters
•
Total impulse:
I=
!
tb
0
where tb is the thrust duration or burn time
Fdt
For operation at constant specific impulse:
tb
! = cmP
I = Isp g0 ! mdt
where mP is the propellant mass
0
•
Thrust profile:
" progressive dF dt > 0 (SRPS's)
$
F ( t ) ! #regressive
dF dt < 0 (SRPS's, HRPS's)
$neutral
dF dt = 0 (LRPS's, ERPS's, NRPS's, SRPS's)
%
•
Thrust-to-weight ratio:
– for free flight:
v!
F
(acceleration in g-No.)
=
g0 mg0
– for vertical take-off:
v!
F
=
!1 "
g0 mg0
F
mg0
>1
take-off
PERFORMANCE OF KEY PROPULSIVE TECHNOLOGIES
Performance Comparison
Exhaust Velocity v/s Vehicle Acceleration
Specific Impulse v/s Thrust
MAJOR LAUNCH VEHICLE FAMILIES
SINGLE-STAGE ROCKETS
Rocket Masses
•
Let:
mP = propellant mass
mT = tank mass
mE = engine & propellant management mass
m L = pay-load mass (all other masses)
•
•
•
Then:
mS = mE + mT
(structural mass)
m0 = mS + mP + m L (initial mass)
m0 ! mP = mS + m L (burn-out mass)
Define:
(tankage fraction, dependent on propellant type and feed system)
! = mT mP
! = mS ( mS + mP ) (structural coefficient, ! constant for equal rocket technology)
(payload ratio)
! = m L m0
Then, using the free rocket equation mP m0 = 1 ! e ! " v c , obtain:
(
)
m E m0 + " 1 # e # $ v c
mS
mE + mT
!=
=
=
mS + mP mE + mT + mP mE m0 + " 1 # e # $ v c + 1 # e # $ v c
(
m L m0 ! m P
mS m0 ! m L
=
!
m0
m0
mS + m P m0
)
e! $v c ! %
$v
" #=
> 0 " c > cmin = !
1! %
ln %
MULTISTAGE ROCKETS
Multistage Rocket Performance
•
For missions with high propellant masses ( mP comparable to m0 ):
– large tank and engine masses need to be accelerated
– final thrust and acceleration become excessive
Hence staging can be advantageous
•
For each stage i = 1, 2,... n :
!i = e " # vi
ci
" $ i (1 " !i ) % # vi = "ci ln &'$ i + !i (1 " $ i ) ()
where:
ci i th exhaust velocity
! i i th structural coefficient
!i i th payload ratio
•
For n stages:
– total velocity change:
! v = " ! vi
i
– overall payload ratio:
! = " !i
i
MULTISTAGE ROCKETS
Multistage Rocket Optimization
•
Alternative approaches:
– minimum m0 for given ! v and m L ! max {!} for given ! v
!i
– maximum ! v for given m0 and m L ! max {" v} for given !
!i
•
(new design)
(new mission)
Overall velocity change and payload ratio constraint for a rocket with i = 1, 2,... n stages:
! v = " i ! vi = # " i ci ln &'$ i + %i (1 # $ i ) ()
! = " i !i
and
# ln ! = $ i ln !i
For the constrained optimization of ! v consider the augmented objective function:
(
F = ! ( i ci ln $%" i + #i (1 ! " i ) &' ! k ln # ! ( i ln #i
)
( k is a Lagrangian multiplier)
Hence, for F to be extremum for some values of the !i ’s:
(1 # $ i ) ci + k = 0
!F
=#
!"i
$ i + "i (1 # $ i ) "i
! "i =
# i (1 $ # i )
ci k $ 1
where k is determined using the constraint on the payload ratio:
# (1 $ # i )
( n th order polynomial for 1 k ! k )
! = " !i = " i
c
k
$
1
i
i
i
In general need to choose the best maximum corresponding to the n solutions for k = k j :
*
! v = max +" ) ci ln %&# i + $ij (1 " # i ) '( .
j
, i
/
where
!ij =
" i (1 # " i )
;
ci k j # 1
j = 1, 2,... n
MULTISTAGE ROCKETS
Multistage Rocket Optimization (continued)
•
For equal exhaust velocities ( c1 , c2 ,... cn = c ):
%1
c
#i (
!1= ' $
k
& " i 1 ! # i *)
1n
"i %
1 # "i (
!i =
!
$
1 # " i '& i " i *)
1n
1n
,
&
1 " #i ) /
! v = "c2 ln .# i + # i ( $ %
+* 1
#
'
.i
10
i
i
•
For similar stages ( c1 , c2 ,... cn = c and ! 1 , ! 2 ,... ! n = ! ):
(equal payload ratios)
!i = ! 1 n
! v = "nc ln %&# + $ 1 n (1 " # ) '(
(equal velocity changes)
and:
% e " # v nc " $ (
!='
& 1 " $ *)
n
In particular, for an infinite number if similar stages:
lim # v = $c lim n ln '(% + & 1 n (1 $ % ) )* = $c (1 $ % ) ln &
n!"
n!"
MULTISTAGE ROCKETS
Multistage Free Rockets
Problem
•
Carry out the payload fraction optimization (1st approach) of multistage rockets.
ELECTRIC ROCKET OPTIMIZATION
Optimum Specific Impulse
•
Assumptions:
– electric thruster effective beam velocity c
– mission requirement ! v << c (not too unrealistic)
– propellant mass flow rate m! = mP t BO
– nearly uniform (1D) beam properties
– electric thruster efficiency ! = W! e P!el
– spacecraft specific mass ! = mS P!el
•
In the above assumptions:
1 2 1 mP 2
! =
W! e ! mc
c
2
2 t BO
P!el
! c2
!
mS = ! Wel = !
=
mP
" 2"t BO
mP
"v
for
= 1 ! e! "v c #
! v << c
m0
c
#
! c2 &
m0 = m P + mS + m L = % 1 +
mP + m L
2"t BO ('
$
exhaust beam mechanical power
spacecraft structural mass
propellant mass ratio
spacecraft initial mass
ELECTRIC ROCKET OPTIMIZATION
Optimum Specific Impulse (continued)
•
Pay-load ratio:
%
%1
mL
# c 2 ( mP
#c (
!=
= 1 " '1 +
+
1
"
+
'& c 2$t *) , v - ".
m0
2$t BO *) m0
&
BO
as
#%0
c! $
%&"
( mP ! " )
( P! , m
el
S
! ")
Hence, differentiating w.r.t. c , the pay-load ratio is maximum for:
d! & 1
$ )
"( 2 #
,v = 0
+
dc ' c
2%t *
BO
corresponding to:
copt ! g0 Ispopt =
2"t BO
#
From this result:
!opt " 1 # $ v
2%
&t BO
or:
t BOopt
•
2! & # v )
=
(
+
" '1$ %*
2
Notice that t BOopt is a strong function of ! " 1 , of ! v and, to a lesser extent, of ! and ! .
PROPULSION SYSTEM DESIGN PROCESS
PROPULSION SYSTEM DESIGN PROCESS (continued)

Rocket Propulsion Fundamentals

Transcript

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