16.512, Rocket Propulsion
Prof. Manuel MartinezSanchez
Lecture 8: Convective Heat Transfer: Other Effects
Overall Heat Loss and Performance Effects of Heat Loss
(1)
Overall Heat Loss
The local heat loss per unit area is
(
)
w
p
aw
w
q
uc
T
T
=
−
ρ
t
S
, and using
, the
integrated heat loss is
2
m
u R
=
π
i
ρ
L
2
w
w
x 0
dR
Q
q 2 R ds ; ds
1
dx
dx
dx
=
⎛
⎞
π
=
+
⎜
⎟
⎝
⎠
∫
±
±
(small angles)
(1)
(
)
(
)
L
L
w
p
aw
w
t
p
aw
w
t
2
0
0
m
d
Q
c
T
T
S 2 R dx
m c
T
T
S 2
R
R
−
π
=
−
π
∫
∫
i
i
±
x
(2)
For an approximate evaluation, assume the quantity
(
)
p
aw
w
t
c
T
T
S
−
is a weak
function of x, and treat it as a constant. We then obtain
(
)
(
)
L
L
aw
w
w
t
c
c
0
0
p c
T
T
Q
T
dx
dx
2S
1
2S
T
T
R x
R x
mc T
⎛
⎞
−
−
⎜
⎟
⎜
⎟
⎝
⎠
∫
i
±
±
w
t
∫
(3)
For many rockets,
(
)
L
eff
0
L
d
R
R
⎛
⎞
≡
⎜
⎟
⎝
⎠
∫
x
x
is of the order of 610, and
w
c
T
1
1
T
4
3
−
∼
, so the
ratio
w
p c
Q
mc T
i
(heat loss divided by total enthalpy flux) is of the order of 816 times
16.512, Rocket Propulsion
Lecture 8
Prof. Manuel MartinezSanchez
Page 1 of 6
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the Stanton number. As we found before,
is itself ~ 0.001, leading to fraction
heat losses of the order of 12%. While this is a small fraction, its absolute value
may be large, because the total thermal power is enormous. As an example, for the
SSME engine
t
S
6
p c
p c
F
2 10
N
J
mc T
c T
2770
3600K
C
4500 m s
KgK
×
=
×
i
±
×
,
or
(the output power of four large power stations).
9
p c
mc T
4.4 10 W
=
×
i
A 1.5% fraction of this means 66 MW lost to the walls (some 80,000 HP).
(2)
Effect on Performance
As a starting guess, we could imagine that all of the losses
(
)
w
Q
are reflected in an
equal amount of kinetic energy loss in the exhaust. If
is the exist velocity with
no losses,
e
0
u
⎛
⎞
⎜
⎟
−
⎜
⎟
⎝
⎠
i
±
2
2
e
0
e
w
u
u
m
2
2
Q
(4)
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 Fall '05
 ManuelMartinezSanchez
 Energy, Kinetic Energy, Heat, Propulsion, Trigraph, tc, Prof. Manuel MartinezSanchez

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