16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 3: Ideal Nozzle Fluid Mechanics
Ideal Nozzle Flow with No Separation (1-D)
-
-
Quasi 1-D (slender) approximation
Ideal gas assumed
F = m ue + ( Pe − Pa ) Ae
CF ≡
F
Pc At
Optimum expansion: Pe = Pa
-
For less
Ae
, Pe > Pa , could derive more forward push by additional
At
expansion
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 3
Page 1 of 10
-
For more
Ae
, Pe < Pa , and the extra pressure forces are a suction,
At
backwards
Compute m = ρ uA at sonic throat:
1
⎛ 2 ⎞ γ −1
γ RgTc
m = ρc ⎜
⎟
⎝ γ + 1⎠
⎛ 2 ⎞
⎜
⎟ At
⎝ γ + 1⎠
γ +1
=
⎛ 2 ⎞ 2 (γ −1) Pc At
g⎜
⎟
RgTc
⎝ γ + 1⎠
call Γ
call c* =
RgTc
2
R
M
3
(“characteristic velocity”) → m =
Γ (γ )
; Rg =
Pc At
c*
⎛P ⎞
A
Can express ue , Pe , Ae , etc in terms of either Me or ⎜⎜ e ⎟⎟ or e :
At
⎝ Pc ⎠
Pe
=
Pc
1
γ −1 2⎞
⎛
⎜ 1 + 2 Me ⎟
⎝
⎠
γ
;
γ −1
Ae ⎛ Pt ⎞ ⎛ ut ⎞ ⎛ Pt ⎞ 1
= ⎜⎜ ⎟⎟ ⎜⎜
⎟⎟ = ⎜⎜ ⎟⎟
At
⎝ Pe ⎠ ⎝ ue ⎠ ⎝ Pe ⎠ Me
Pe ⎛ Te
= ⎜⎜
Pc
⎝ Tc
and
Te
=
Tc
⎞
⎟⎟
⎠
γ
γ −1
γ −1 2
⎛
Me
1+
Tt
1 ⎜
2
=
⎜
γ +1
Te
Me ⎜
⎜
2
⎝
1+ 1
γ −1
⎞2
⎟
⎟
⎟⎟
⎠
,
1
1+
γ +1
2 ( γ −1)
γ −1
2
Because c pTe +
Me2
,
ue2
M2
γ
γ
RgTe + e γ RTe =
RT
= c pTc →
γ −1
γ −1 g c
2
2
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 3
Page 2 of 10
CF =
⎛ P − Pa
m
ue + ⎜⎜ e
Pc At
⎝ Pc
Me γ Rg
ue
c
*
=
⎞ Ae
⎛P
u
P
= *e + ⎜⎜ e − a
⎟⎟
c
⎠ At
⎝ Pc Pc
⎞ Ae
⎟⎟
⎠ At
Tc
1+
γ −1
2
γ +1
Me2
⎛ 2 ⎞ 2 (γ −1)
=γ⎜
⎟
⎝ γ + 1⎠
RgTc
Γ
Me
1+
γ −1
2
Me2
In vacuum,
( Pa
= 0)
γ +1
CFv =
(CF )v
ue
c
*
+
γ +1
⎞ 2 (γ −1)
Pe Ae ⎛ 2
=⎜
⎟
Pc At
⎝ γ + 1⎠
γ +1
⎞ 2 (γ −1)
⎛ 2
=⎜
⎟
⎝ γ + 1⎠
1+
γ Me +
1+
γ Me
2
Me2
γ
−
1
2
1
Me
γ −1
2
γ −1
−
γ − 1 2 ⎞ 2 (γ −1) γ −1
⎛
1+
Me ⎟
⎜
2
1 ⎝
⎠
+
γ +1
Me
⎛ γ + 1 ⎞ 2 (γ −1)
Me ⎜
⎟
⎝ 2 ⎠
Me2
and otherwise,
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 3
Page 3 of 10
⎛ A ⎞ Pa
CF = CF v − ⎜⎜ e ⎟⎟
⎝ At ⎠Me Pc
Note:
For Pe = Pa ,
(CF )Matched
=
ue
c
*
For Pe = Pa = 0
γ +1
⎛ 2 ⎞ 2 (γ −1)
=⎜
⎟
⎝ γ + 1⎠
(CF )Max,Vac
γ Me
1+
γ −1
2
=γ
γ −1
2
Me2
γ +1
⎛ 2 ⎞ 2 (γ −1)
⎜
⎟
⎝γ + 1⎠
Choice of Optimum Expansion For a Rocket Flying Through an Atmosphere
( Pa varying)
The thrust coefficient CF =
CF = CFvac −
Pa
Pc
F
was derived in class in the form
Pc At
⎛ Ae ⎞
⎜⎜
⎟⎟
⎝ At ⎠
(1)
γ +1
CFvac
⎛ 2 ⎞ 2 (γ −1)
=⎜
⎟
⎝ γ + 1⎠
γ Me + 1
1+
Me
γ −1
2
(2)
Me2
and we also found
γ −1 2
⎛
1+
Me
Ae
1 ⎜
2
=
⎜
γ +1
At
Me ⎜
⎜
2
⎝
γ +1
⎞ 2 (γ −1)
⎟
⎟
⎟⎟
⎠
(3)
The thrust-derived velocity increment is
ΔVF =
∫
tb
0
F
dt = Pc At
m
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
∫
tb
0
CF
dt
m
(4)
Lecture 3
Page 4 of 10
where CF = CF ( t ) due only to the variation of Pa in (1), while m = m ( t ) because of
mass burnout. The quantities CFvac and
Ae
depend on Me (or nozzle geometry), but
At
are time-invariant. Substituting (1), (2) and (3) into (4),
⎡
ΔVF = Pc At ⎢CFvac
⎣⎢
∫
tb
0
dt ⎛ Ae ⎞
−⎜
⎟
m ⎜⎝ At ⎟⎠
∫
tb
0
Pa dt ⎤
⎥
Pc m ⎦⎥
or
ΔVF
Pc At
∫
dt
m
tb
0
= CFvac −
∫
Pa dt
Pc m Ae
tb dt
At
0 m
tb
0
∫
(5)
We now make the approximation that the trajectory will change little when we vary
A
Me (and hence CFvac , e ). We can then regard the time integrals in (5) as fixed
At
quantities while we optimize Me . Define the non-dimensional variables
v =
ΔVF
Pc At
∫
tb
0
dt
m
;
p=
∫
Pa dt
Pc m
tb dt
0 m
tb
0
∫
(6)
so that (5) becomes
⎛A ⎞
v = CFvac ( Me ) − p ⎜⎜ e ⎟⎟ ( Me )
⎝ At ⎠
(7)
and we can now differentiate v w.r.t Me (holding p=const.)
⎛A
⎞
∂⎜ e ⎟
∂CFvac
A
∂v
t
⎠ =0
=
−p ⎝
∂Me
∂Me
∂Me
(8)
γ +1
⎛ 2 ⎞ 2 (γ −1)
appears in both terms of (8) and can be
From (2) and (3), the factor ⎜
⎟
⎝ γ + 1⎠
ignored. We then have
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 3
Page 5 of 10
⎛
⎜ γ Me + 1 M
∂ ⎜
e
∂Me ⎜
γ −1 2
Me
⎜ 1+
2
⎝
⎞
⎟
⎟= p ∂
⎟
∂Me
⎟
⎠
γ +1
⎡
⎢ ⎛ 1 + γ − 1 M 2 ⎞ 2 (γ −1)
e ⎟
⎢ ⎜⎝
2
⎠
⎢
M
⎢
e
⎢
⎣⎢
⎤
⎥
⎥
⎥
⎥
⎥
⎦⎥
γ +1
γ −1
2Me
⎛
1 ⎞ 1
2
− ⎜⎜ γ Me +
⎟⎟
3
Me ⎠ 2 ⎛
γ −1 2 ⎝
γ −1 2 ⎞ 2
Me
1+
⎜1 + 2 Me ⎟
2
⎝
⎠
γ− 1
Me2
3
γ −1 2⎞
⎛
Multiply times ⎜1 +
Me ⎟
2
⎝
⎠
2
, and note that
γ − 1 2 ⎞2(γ −1)
⎛
⎜1 + 2 Me ⎟
⎠
= p⎝
Me
γ +1
γ
1
+ =
2 ( γ − 1) 2 γ − 1
γ −1 2⎞
⎛
⎜1 + 2 Me ⎟
⎛
γ −1 2⎞ γ −1
1 ⎞⎛
⎠
γ Me2 + 1 = p ⎝
Me ⎟ −
⎜ γ − 2 ⎟ ⎜1 +
2
⎜
⎟⎝
2
2
M
M
⎠
e ⎠
e
⎝
(
γ −1
⎡
⎤
2Me
⎢ γ +1
1 ⎥
2
−
⎢
⎥
⎢ 2 ( γ − 1) 1 + γ − 1 M2 Me ⎥
e
⎥⎦
2
⎣⎢
γ
γ −1
)
⎡
γ − 1 2 ⎞⎤
⎛
2
Me ⎟ ⎥
⎢( γ + 1) Me − ⎜1 +
2
⎠⎦
⎝
⎣
Expand & simplify
γ −1 2⎞
⎛
⎜1 + 2 Me ⎟
γ ( γ − 1) 2
1
γ − 1 γ ( γ − 1) 2 γ − 1
⎝
⎠
−
= p
Me − 2 −
Me −
γ +
2
2
2
2
Me
Me2
1444444444442444444444443
1−
Cancel the factor
Me2 − 1
Me2
1
Me2
=
γ
γ −1
(M
2
e
)
−1
Me2 − 1
Me2
( Me = 1 is clearly not an optimum!)
γ γ −1
γ −1 2⎞
⎛
Me ⎟
1 = p ⎜1 +
2
⎝
⎠
or
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 3
Page 6 of 10
1+
γ −1
2
⎛1⎞
Me2 = ⎜ ⎟
⎝ p⎠
γ −1
γ
Me
OPT
=
γ −1
⎡
⎤
2 ⎢⎛ 1 ⎞ γ
⎥
− 1⎥
⎜ ⎟
⎢
γ −1 ⎝ p⎠
⎢⎣
⎥⎦
(9)
Notice that the exit pressure is given by
Pe
=
Pc
1
γ −1 2⎞
⎛
⎜ 1 + 2 Me ⎟
⎝
⎠
γ
(10)
γ −1
and so the optimum exit pressure turn out to be
⎛ Pe
⎜⎜
⎝ Pc
⎞
=p
⎟⎟
⎠OPT
However, if p < 0.4
(11)
Pao
Pc
, this would imply Pe < 0.4Pao , and there would be flow
separation at the highest Pa (on the ground). To avoid this, the optimality condition
must be amended to
⎛ Pe
⎜⎜
⎝ Pc
Pa ⎪⎫
⎞
⎪⎧
= Greater of ⎨ ρ , 0.4 o ⎬
⎟⎟
Pc ⎪⎭
⎪⎩
⎠OPT
(12)
with a similar expression for Me :
⎧
γ −1
⎡
⎤
⎪
2 ⎢⎛ 1 ⎞ γ
⎪
⎥
= Least of ⎨
− 1⎥ ,
⎜ ⎟
⎢
⎪ γ − 1 ⎢⎝ p ⎠
⎥⎦
⎣
⎪⎩
Me
OPT
⎡
Pc
2 ⎢⎛
⎢⎜ 2.5
⎜
γ − 1 ⎢⎝
Pao
⎣⎢
⎞
⎟
⎟
⎠
γ −1
γ
⎤⎫
⎥ ⎪⎪
− 1⎥ ⎬
⎥⎪
⎦⎥ ⎪⎭
(13)
The limiting condition in which the whole burn occurs at Pa o is simple.
We then obtain
p=
∫
tb
0
Pao dt
Pao
Pc m
=
tb dt
Pc
0 m
∫
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
(14)
Lecture 3
Page 7 of 10
and the optimality condition (12) yields ( Pe )OPT = Pao , i.e., the nozzle should be
pressure-matched, as expected.
As more and more of the burn shifts to higher altitudes, p decreases from
long as it still remains above 0.4
Pao
Pc
Pao
Pc
. As
, equation (11) gives some intermediate
optimum design, and if p drops below 0.4
Pao
Pc
, the nozzle should be designed to be
on the verge of separation on the ground.
Nozzle Flow Separation Effects
Rule of thumb (to be explored later):
Flow separates at the point in the nozzle where
P
0.4Pa (Summerfield criterion)
So, if Pe > 0.4Pa (even if Pe < Pa ), no separation
After separation, roughly parallel flow, at P = Pa (no strong p gradients in “dead
water” region to turn flow).
So zero thrust contribution
Performance with separation at that of a nozzle
with exit pressure Pe' = 0.4Pa
So,
(a) Pa <
Pe (full nozzle)
,
0.4
CF = CFvac −
Pa Ae
Po At
f ( Me )
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
g ( Me )
Lecture 3
Page 8 of 10
(b) Pa >
Pe (full nozzle)
,
0.4
(
⎧M ' = M P ' = 0.4 P
e
a
⎪⎪ e
calculate ⎨ A'
'
A
⎪ e = e Me'
At
⎪⎩ At
)
( )
( )
then CF = CFvac Me' −
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Pa Ae'
Po At
Lecture 3
Page 9 of 10
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 3
Page 10 of 10