16.50 Propulsion Systems
Spring 2012
Homework 3: Two-Position Nozzle
a) ܿ ‫ כ‬ൌ
ඥோ்೎
(1)
୻ሺஓሻ
ംశభ
Ȟൌ
ଶ మሺംషభሻ
ξߛ ቀఊାଵቁ
ଶ
ଶǤଶ
Ȟ ൌ ξͳǤʹ ቀ ቁ
ܴൌ
మǤమ
మ‫כ‬బǡమ
଼Ǥଷଵସ௃ൗ௠௢௟ି௄
଴Ǥ଴ଵ଼௞௚ൗ௠௢௟
(2)
ൌ ͲǤ͸Ͷͺͷ
‫ܬ‬
ൌ Ͷ͸ͳǤͻ ൗ݉‫ ݈݋‬െ ‫ܭ‬
(3)
ሶ
Solving for ࢉ‫ כ‬and ࢓:
ܿ‫ כ‬ൌ
݉ሶ ൌ
ξସ଺ଵǤଽ‫כ‬ଷଷ଴଴
ൌ ͳͻͲ͵Ǥ͹݉Ȁ‫ݏ‬
଴Ǥ଺ସ଼ହ
௉೎ ஺೟
଻଴‫כ‬ଵǤ଴ଵଷாହ‫כ‬଴Ǥଵ
ൌ
ൌ ͵͹ʹǤͷ݇݃Ȁ‫ݏ‬
௖‫כ‬
ଵଽ଴ଷǤ଻
For the inner bell:
௉೐భ
௉ೌబ
ൌ ͲǤͶ
Therefore, the exit Mach number is given by:
଻଴
଴Ǥସ
ൌ ቀͳ ൅
ଵǤଶିଵ
ଶ
ଶ
‫ܯ‬௘ଵ
ቁ
భǤమ
బǤమ
‫ܯ‬௘ଵ ൌ ͵Ǥ͸ͻͷ
The area ratio is then:
ംశభ
ംషభ
మ
మሺംషభሻ
ଵା మ ெ೐భ
ଵ
ቆ ംశభ
ቇ
ெ೐భ
஺೐భ
஺೟
ൌ
஺೐భ
஺೟
ൌ ͳͺǤʹ͵Ͷ
(4)
మ
‫ܣ‬௘ଵ ൌ ͳǤͺʹ͵Ͷ݉ଶ
The vacuum thrust is:
‫ܨ‬௩ଵ ൌ ݉‫ݑ‬
ሶ ௘ଵ ൅ ܲ௘ଵ ‫ܣ‬௘ଵ
(5)
ംషభ
‫ݑ‬௘ଵ ൌ ඨʹ
ఊ
ܴܶ௖
ఊିଵ
‫ݑ‬௘ଵ ൌ ඨʹ
ଵǤଶ
ቈͳ െ ቀ
௉೐భ ം
ቁ
௉೎
቉
(6)
బǤమ
଴Ǥଶ
଴Ǥସ భǤమ
଻଴
Ͷ͸ͳǤͻ ‫ ͲͲ͵͵ כ‬ቈͳ െ ቀ ቁ ቉ ൌ ͵ʹͶͳ݉Ȁ‫ݏ‬
1
On the ground:
‫ܨ‬଴ ൌ ‫ܨ‬௩ଵ െ ܲ௔଴ ‫ܣ‬௘ଵ
(7)
‫ܨ‬଴ ൌ ͳǤʹͺͶʹ‫ܧ‬͸ െ ͳǤͲͳ͵‫ܧ‬ͷ ‫ͳ כ‬Ǥͺʹ͵Ͷ ൌ ͳǤͲͻͻͷ‫ܧ‬͸ܰ
ሶ ௘ଵ ൅ ܲ௘ଵ ‫ܣ‬௪ଵ ൌ ͵͹ʹǤͷ ൈ ͵Ͷʹͳ ൅ ͲǤͶ ൈ ͳǤͲͳ͵݁ͷ ൈ ͳǤͺʹ͵Ͷ ൌ ͳǤʹͺͶʹ݁͸ܰ
‫ܨ‬௩ଵ ൌ ݉‫ݑ‬
ௗ௩
b) ݉ ௗ௧ ൌ ‫ ܨ‬െ ݉݃ ൌ ‫ܨ‬௩ െ ܲ௔ ‫ܣ‬௘ െ ݉݃
ௗ௩
ௗ௧
ൌ
ிೡ
௠
െ݃െ
஺೐ ௉ೌ
௠
ሶ
݉ ൌ ݉଴ െ ݉‫ݐ‬
݀݉ ൌ െ݉݀‫ݐ‬
ሶ
(8)
(9)
(10)
(11)
Putting equations together:
݀‫ ݒ‬ൌ െ
ிೡ ௗ௠
௠ ௠
െ ݃݀‫ ݐ‬െ
஺೐ ௉ೌ
݀‫ݐ‬
௠
(12)
Integrating between ‫ ݐ‬ൌ Ͳ and ‫ ݐ‬ൌ ‫ݐ‬ଵ (with ‫ܨ‬௩ ൌ ‫ܨ‬௩ଵ and ‫ܣ‬௘ ൌ ‫ܣ‬௘ଵ ) :
‫ݒ‬ଵ ൌ
ிೡభ
௠
݈݊ ቀ బ ቁ െ ݃‫ݐ‬ଵ
௠ሶ
௠భ
௧ ௉ೌ ሺ௧ሻ
݀‫ݐ‬
௠ሺ௧ሻ
െ ‫ܣ‬௘ଵ ‫׬‬଴ భ
(13)
Integrating between ‫ ݐ‬ൌ ‫ݐ‬ଵ and ‫ ݐ‬ൌ ‫ݐ‬௕ (with ‫ܨ‬௩ ൌ ‫ܨ‬௩ଵ and ‫ܣ‬௘ ൌ ‫ܣ‬௘ଶ ):
‫ݒ‬௕ െ ‫ݒ‬ଵ ൌ
ிೡమ
௠
݈݊ ቀ భ ቁ െ ݃ሺ‫ݐ‬௕
௠ሶ
௠್
௧ ௉ೌ ሺ௧ሻ
݀‫ݐ‬
భ ௠ሺ௧ሻ
െ ‫ݐ‬ଵ ሻ െ ‫ܣ‬௘ଶ ‫׬‬௧ ್
(14)
Adding equations (13) and (14):
‫ݒ‬௕ ൌ
ிೡభ
௠
ி
௠
݈݊ ቀ బ ቁ ൅ ೡమሶ ݈݊ ቀ భ ቁ െ ݃‫ݐ‬௕
௠ሶ
௠భ
௠
௠್
௧ ௉ೌ ሺ௧ሻ
݀‫ݐ‬
௠ሺ௧ሻ
െ ‫ܣ‬௘ଵ ‫׬‬଴ భ
Here, of course:
݉ଵ ൌ ݉଴ െ ݉‫ݐ‬
ሶ ଵ
(16)
So ‫ݐ‬ଵ appears in ݉ଵ and in the limits of integration.
ௗ௩
ௗ௠భ
ൌ െ݉ሶ
ௗ௧భ
௉ ሺ௧ ሻ
௉ ሺ௧ ሻ
‫ܣ‬௘ଵ ೌ భ ൅ ‫ܣ‬௘ଶ ೌ భ
௠భ
௠భ
To optimize, set ௗ௧ ್ ൌ Ͳ, using
భ
ିிೡభ ଵ
ሺെ݉ሻ
௠ ௠భ
൅
ிೡమ ଵ
௠ ௠భ
ሺെ݉ሻ െ
ൌͲ
‫ܨ‬௩ଵ െ ‫ܨ‬௩ଶ ൅ሶ ሺ‫ܣ‬௘ଶ െ ‫ܣ‬௘ଵ ሻܲ௔ሶ2ሺ‫ݐ‬3ଵ ሻ ൌ Ͳ
ሶ
ܲ௔ ሺ‫ݐ‬ଵ ሻ ൌ
ிೡమ ିிೡభ ሶ
(for
஺೐మ ି஺೐భ
optimum ‫ݐ‬ଵ )
If this ‫ݐ‬ଵ is chosen, the thrust is:
Just before the transition:
‫ܨ‬ଵ ሺ‫ݐ‬ଵ െ ߝሻ ൌ ‫ܨ‬௩ଵ െ
ிೡమ ିிೡభ
‫ܣ‬
஺೐మ ି஺೐భ ௘ଵ
ൌ
ிೡభ ஺೐మ ିிೡమ ஺೐భ
஺೐మ ି஺೐భ
(17)
Just after the transition:
2
௧ ௉ೌ ሺ௧ሻ
݀‫ݐ‬
భ ௠ሺ௧ሻ
െ ‫ܣ‬௘ଶ ‫׬‬௧ ್
(15)
‫ܨ‬ଶ ሺ‫ݐ‬ଵ ൅ ߝሻ ൌ ‫ܨ‬௩ଶ െ
ிೡమ ିிೡభ
஺೐మ ି஺೐భ
‫ܣ‬௘ଶ ൌ
ிೡభ ஺೐మ ିிೡమ ஺೐భ
஺೐మ ି஺೐భ
(18)
Therefore,
‫ܨ‬ଵ ሺ‫ݐ‬ଵ െ ߝሻ ൌ ‫ܨ‬ଶ ሺ‫ݐ‬ଵ ൅ ߝሻ There is no discontinuity in the thrust.
c) If we impose now ܲ௘ଶ ൌ ͲǤͶܲ௔ ሺ‫ݐ‬ଵ ሻ (incipient separation on the extended nozzle), we must have:
ܲ௘ଶ ൌ ͲǤͶ
ிೡమ ିிೡభ
஺೐మ ି஺೐భ
ൌ ͲǤͶ
ሺ௠௨
ሶ ೐మ ା௉೐మ ஺೐మ ሻିிೡభ
஺೐మ ି஺೐భ
(19)
Here, ݉,ሶ ‫ܨ‬௩ଵ , and ‫ܣ‬௘ଵ are already known (from part a), and all the other quantities (‫ݑ‬௘ଶ , ܲ௘ଶ , ‫ܣ‬௘ଶ )
depend on a single parameter like ‫ܯ‬௘ଶ . Hence the equation above determines ‫ܯ‬௘ଶ , and all other
quantities. Equation (19) can be solved by trial and error as follows:
a) Guess ‫ܯ‬௘ଶ
ംశభ
ംషభ
b) Calculate ‫ܣ‬௘ଶ ൌ
మ
మሺംషభሻ
ଵା
ெ೐మ
஺೟
మ
ቆ ംశభ
ቇ
ெ೐మ
మ
c) Calculate ܶ௘ଶ ൌ
்೎
ംషభ మ
ெ೐
మ
ଵା
, then ‫ݑ‬௘ଶ ൌ ‫ܯ‬௘ଶ ඥߛܴܶ௘ଶ
ം
d) Calculate ܲ௘ଶ ൌ ܲ௖ ቀ
்೐మ ംషభ
ቁ
்೎
e) Calculate the right hand side of equation (19), compare to ܲ௘ଶ from the left hand side.
Results of this process are plotted in Figures 1a and 1b, and the solution is seen to be ‫ܯ‬௘ଶ ൌ ͶǤͺͳͷ.
From this, we find:
஺೐మ
஺೟
ൌ ͻͲǤͳ͵
‫ܣ‬௘ଶ ൌ ͻǤͲͳ͵݉ଶ
‫ݑ‬௘ଶ ൌ ͵ǡͷ͹ͷ݉Ȁ‫ݏ‬
ܲ௘ଶ ൌ ͲǤͲͷʹ͵͸ܽ‫݉ݐ‬
ሶ ௘ଶ ൅ ܲ௘ଶ ‫ܣ‬௘ଶ ൌ ͳǤ͵͹ͻͷ‫ܧ‬͸ܰ
‫ܨ‬௩ଶ ൌ ݉‫ݑ‬
ிೡమ ିிೡభ
ൌ
஺೐మ ି஺೐భ
ଵ
͸Ǥͺ݈݊ ቀ
ቁ
଴Ǥଵଷ଴଼
ܲ௔ ሺ‫ݐ‬ଵ ሻ ൌ
ͲǤͳ͵Ͳͺܽ‫݉ݐ‬
‫ݖ‬ଵ ൌ
ൌ ͳ͵Ǥͺ͵݇݉
The thrust at transition (from either nozzle 1 or 2) is:
‫ܨ‬ሺ‫ݐ‬ଵ ሻ ൌ ‫ܨ‬௩ଵ െ ܲ௔ ሺ‫ݐ‬ଵ ሻ‫ܣ‬௘ଵ ൌ ͳǤʹͷ͹Ͳ‫ܧ‬͸ܰ
c) For the ideally expanded (“rubber”) nozzle, ܲ௘ ൌ ܲ௔ ሺ‫ݖ‬ሻ, and then:
ംషభ
‫ݑ‬௘ ൌ ඨʹ
ఊ
ܴܶ௖
ఊିଵ
ቈͳ െ ቀ
௉ೌ ሺ௭ሻ ം
ቁ
௉೎
቉
‫ ܨ‬ൌ ݉‫ݑ‬
ሶ ௘
3
For ‫ ݖ‬ൌ Ͳǡ ‫ ݖ‬ൌ ‫ݖ‬ଵ , and ‫ ݖ‬՜ λሺܲ௔ ൌ Ͳሻ, and for the three types of nozzles, Table 1 collects the values of
thrust:
Table 1: Thurst calculations for three nozzles at varying altitudes.
Thrust (MN)
Nozzle
‫ݖ‬՜λ
‫ݖ‬ൌͲ
‫ ݖ‬ൌ ‫ݖ‬ଵ ൌ ͳ͵Ǥͺ͵݇݉
Fixed Geometry
1.0995
1.2570
1.2842
Two-Position
1.0995
1.2570
1.3795
Ideally Expanded
1.1348
1.2836
1.5931
Notice:
a) Thrust increases with altitude in all cases.
b) The two-position nozzle is equivalent to the fixed nozzle at ‫ݖ‬ଵ , but clearly superior at the vacuum
condition.
c) The ideally expanded nozzle outperforms the others at all altitudes.
Figure 2 compares the thrust profiles of the thrust nozzles. The two-position nozzle would follow the
curve for nozzle 1 up to ‫ ݖ‬ൌ ‫ݖ‬ଵ , then follow the nozzle 2 curve.
Notice how the ideally expanded curve touches that for nozzle 1 at ‫ ݖ‬ൎ ͸Ǥʹ͵݇݉; that is where this
ଵ
ଵ
଴Ǥସ
଴Ǥ଴ହଶଷ଺
nozzle is matched ሺ͸Ǥͺ݈݊ ቀ ቁ ൌ ͸Ǥʹ͵ሻ. It also touches that of nozzle 2 at ‫ ݖ‬ൌ ͸Ǥͺ݈݊ ቀ
ʹͲǤͳ݇݉, where nozzle 2 is matched.
4
ቁൌ
Figure 1a) Solving equation (19).
Figure 1b) Solving equation (19) at a higher resolution. Solution is ࡹࢋ૛ ൌ ૝Ǥ ૡ૚૞.
5
Figure 2: Thrust profiles for the three nozzles.
6
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