16.522, Space Propulsion Lecture 1b
Prof. Manuel Martinez-Sanchez Page 1 of 6
16.522, Space Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 1b: Review of Rocket Propulsion
The only practical way to accelerate something in free space is by reaction. The idea
is the same as in air breathing propulsion (to push something backwards) but in
rockets the “something” must be inside and is lost.
Here is a revealing derivation of the thrust equation for vacuum:
() () () () () ()
t
0
Mom.t=Mtvt+ mt' vt'-ct' dt'=Constant??
??∫
i
()dMom
=0
dt
()
dv dM
M+v +mv-c=0
dt dt
i
But
dM dv
m=- M =mc
dt dt
→
ii
call this thrust, F.
Notice
()
()
dMv
=F-vm=m c-v
dt
ii
which can be (+) or (-)
Using the same technique, the kinetic energy of the system rocket-jet is
() () ()
t 2
2
0
11
KE = Mv + m t' v t' - c t' dt'
22
??
??∫
i
So,
()
222 2
dKE dv 1 dM 1 1
=Mv + v m v +c -2vc mc
dt dt 2 dt 2 2
+=
ii
mcv
i
m?
i
So, if thermal (or electrical) energy is expended internally at the rate E,
i
and
converted to total kinetic energy with efficiency
th e1.
(or )η η ;
i.e.,
no drag, no interaction
of molecules with
ambient air
16.522, Space Propulsion Lecture 1b
Prof. Manuel Martinez-Sanchez Page 2 of 6
()
th
dKE
=E,
dt
η
i
then
2
th
1
E= mc
2
η
ii
jet kinetic power
Note that we counted both vehicle and wake KE as produced by E
i
, and this is
unambiguous. If we try to define “useful Propulsive work” as Fv = mcv,
i
then we find
that the “propulsive efficiency”
prop
2
th
F.v mvc 2v
==
1
c
E mc
2
=η
η
i
i i
is arbitrarily high!
pr.
c
If v > , >1 .
2
??
??
??
η
For this reason,
prop
η is not used for rockets. But it is still true that thrusting at high
speed increases kinetic energy more
f
2
220
f0 0
f
m11
? mv = m v +c ln - v
22
??
??
??
??
??
??
??
??
??
In the presence of external air, some modification is needed, leading to the well-
known formula
()
eeea
F=mu +A P -P mc≡
ii
defines
e
cu=null Jet speed far from exhaust
For finite P
a
, in thermal rockets, increasing A
e
increases u
e
(towards a limit
max
ep0
u2cT? ), but it eventually makes
eae
(P - P )A negative.
The best A
e
is such as to make P
e
= P
a
.
The thrust coefficient c
F
is used to quantify the performance of nozzles. Starting from
()
eeea
F=mu +A P -P
i
(or finite P
e
)
22
2 00
f0
ff
mm1
=m c ln +v c ln
2m m
??
??
??
16.522, Space Propulsion Lecture 1b
Prof. Manuel Martinez-Sanchez Page 3 of 6
Isentropic flow
r
pp∝
r
V
p = constant
and assuming ideal gas,
1
-1
tt t 0 0 t
22
m= uA = RT A
+1 -1
γ
?? ??
ρρ
?? ??
γγ
?? ??
i
0
00
P
RT
0t
*
PA
m=
c
?
i
with
()
0*
RT
c= ,
Γγ
()
+1
2-1
2
=
+1
γ
γ
??
Γγ
??
γ
??
eeea
0t *
t0 0
uAPP
F=PA + -
APPc
????
→
????
????
non-dimensional c
F
So
F0 t
F=cPA
Usually,
F
c 1.5-2~
16.522, Space Propulsion Lecture 1b
Prof. Manuel Martinez-Sanchez Page 4 of 6
Specific impulse (inverse of specific fuel consumption) is defined as
*
F
sp
ccFc
I= = =
gg
mg
i
We want high I
sp
(high c), since
0
f
mdv dm
m=- c?V=c ln
dt dt m
→
Using
0PSL
m=m+m+m,
()
PSL
m = propellant, m = structure, m = payload
?V
P c
LSL0S?V
c
m
m = - m or m = m e - m
e-1
?
So, for given m
P
, m
S
, the higher c (I
sp
), the higher m
L
; this dependence is at least
linear (for small
LP S
?Vc
,m m -m
c ?V
null ), but it becomes exponentially fast for high
energy missions (
?V
1
c
null ).
For chemical rockets, c is limited by chemical energy/mass in fuel
max
c2E.=
In general, since
2
e
th th
1
mu = E= mE
2
ηη
iii
e
th
u2E= η
(Accounting for work of expansion leads to a replacement of E by H, the enthalpy.)
For ideal expansion, Brayton cycle, so
-1
e
0
th
P
=1-
P
γ
γ
??
??
??
η
and
H=c
P
T
0
(ideal gas)
-1
e
eP0
0
P
u= 2cT 1-
P
γ
γ
??
??
??
??
??
??
16.522, Space Propulsion Lecture 1b
Prof. Manuel Martinez-Sanchez Page 5 of 6
This shows we want high P
0
/P
e
. This means large area ratio
e
*
A
A
, not necessarily high
P
0
. In fact, for vacuum operation, higher P
0
simply means higher P
e
, with no increase
of u
e
(or I
sp
). In a closer look, higher P
0
can increase T
0
by inhibiting
dissociation higher I
sp
.
But the main reason to go to high P
0
is to reduce weight for a given thrust (also for
boosting, where P
e
cannot be much lower than P
a
, so high P
0
/P
e
means high P
0
).
Roughly
2
2w
0
F0 t
4Rt M
2 Rt = R p
FcPA
πρ
πσ πnull
0
R
t=p
2σ
2
20w
t2
F0
4RPRρM
A = K R
F 2cPK R
π
σ
null
2w
F0
F F0
M2 F
R F = c P K R R =
FKc cPK
ρπ
σ
null
()
w
32 32
0
F
M2 1
=
PF
Kc
ρπ
σ
Thus, for a given thrust level, the engine mass scales like
0
1
P
.
For a given total impulse (not thrust), it may be better to reduce P
0
, reduce thrust,
operate longer. In boosters, there is a complex tradeoff, involving gravity losses,
drag penalties, improved I
sp
at high P
0
, etc. In general, for boosters it is found
advantageous to go to high P
0
, limited mostly (in liquid rockets) by turbomachinery.
For space engines the result is less clear, but they do tend to optimize at much lower
P
0
.
The power of a rocket can be extremely high. For the Shuttle
7
F3000 ton=3×10Nt , null
c 3300 m/secnull
27310
111
mc Fc 3×10 3.3 10 =5 10 watt =50 GW!!
222
== ×× ×
i
Thus, if one step in the power chain involves electrical power, the engine is likely to
be very heavy. Why then electrical? Because it breaks the u
e
limit, allowing any I
sp
,
or, in other words, it gives very fuel efficient rockets. With EM forces one can
increase u
e
almost arbitrarily; however looking at the power requirements,
16.522, Space Propulsion Lecture 1b
Prof. Manuel Martinez-Sanchez Page 6 of 6
cc
11 Fc
Pmc
22
==
ηη
i
cc
PFc ac
MM2 2
==
ηη
0.5 watt/Kg 2 Kg/watt = 2000 Kg/KW→
-2
50 watt/Kg 2 10 20→× →
So, with
reasonable
mass/power
ratios for electric
power one gets
very low
accelerations
If one pushes I
sp
in an electric thruster, M
P
is reduced for given ?V, but M
s
increases due to the high P, unless a is very low (which may be impossible
because of mission duration constraints). Thus, an optimum I
sp
is found to
exist (~2000 sec, depending on conditions).
Reference
Martinez-Sanchez, M., and J. E. Pollard. “Spacecraft Electric Propulsion – An
Overview.” Journal of Propulsion and Power 14, no. 5 (September–October
1998): 688-699. [A publication of the American Institute of Aeronautics and
Astronautics, Inc.]