16.50 Propulsion Systems
Spring 2012
Homework 5: Thermochemistry Exploration Using CEA Code
The CEA output is very detailed and takes a total of 12 pages, so only one case (equilibrium, O/F=2.6)
ܲ
will be displayed here. The results are for ܲ௖ ൌ ͹Ͳܽ‫݉ݐ‬ǡ ܲ௘ ൌ ͲǤͶܽ‫ ݉ݐ‬ቀ ௖ൗܲ ൌ ͳ͹ͷቁ, using RP-1 fuel
௘
and 02L as oxidizer.
1)For all cases run, we need the “ground jet velocity,”namely, that for ܲ௔ ൌ ͳܽ‫݉ݐ‬. The code supplies
only the jet velocity (“specific impulse”) for vacuum and for matched conditions, which we will call
ܿ଴ ǡ ܿ௠௔௧௖௛ , respectively. In general:
(1)
‫ܨ‬଴ ൌ ‫ܨ‬௩௔௖ െ ܲ௔ ‫ܣ‬௘
ܿൌ
ி
௠ሶ
ܿ଴ ൌ
ி
ܿ‫כ‬
௉೎ ஺೟ ௉ ஺
ܿ௩௔௖ െ ೌ ೐ ܿ ‫כ‬
௉೎ ஺೟
ൌ
(2)
(3)
An alternative, more convenient formulation is:
ܿൌ
ி
௠ሶ
ൌ
ி
ఘ೐ ௨೐ ஺೐
ܿ଴ ൌ ܿ௩௔௖ െ
௉ೌ
ఘ೐ ௨೐
(4)
ൌ ܿ௩௔௖ െ
࣬
௉ೌ ቀ ൗெ೐ቁ்೐
௉೐
௨೐
(5)
All quantities are listed as output. Here,
௉ೌ
௉೐
ൌ
ଵ௔௧௠
଴Ǥସ௔௧௠
ൌ ʹǤͷ. Notice that the exit velocity ‫ݑ‬௘ is actually
equal to ܿ௠௔௧௖௛ , and can be read directly from the output.
For the Equilibrium case we then find:
Table 1: Equilibrium Case
ܿ௩௔௖ ሾ݉Ȁ‫ݏ‬ሿ
O/F
2
3205.8
2.2
3286.0
2.4
3341.4
2.6
3374.2
2.8
3384.5
‫ݑ‬௘ ሾ݉Ȁ‫ݏ‬ሿ
3030.6
3099.1
3142.3
3161.4
3160.6
ܶ௘ ሾ‫ܭ‬ሿ
1355.5
1577.5
1810.4
2051.0
2278.1
For the Frozen Flow case (nfz=2, frozen after throat), we find:
1
‫ܯ‬௘ ሾ݇݃Ȁ݉‫݈݋‬ሿ
0.02123
0.02265
0.02406
0.02545
0.02677
ܿ଴ ሾ݉Ȁ‫ݏ‬ሿ
2767.9
2818.9
2843.7
2844.4 (opt)
2824.9
Table 2: Frozen Flow Case
ܿ௩௔௖ ሾ݉Ȁ‫ݏ‬ሿ
O/F
2
3145.4
2.2
3188.0
2.4
3196.1
2.6
3183.0
2.8
3159.1
‫ݑ‬௘ ሾ݉Ȁ‫ݏ‬ሿ
2891.7
3017.8
3022.7
3008.5
2984.8
ܶ௘ ሾ‫ܭ‬ሿ
1232.3
1364.2
1451.9
1504.2
1531.8
‫ܯ‬௘ ሾ݇݃Ȁ݉‫݈݋‬ሿ
0.02099
0.02209
0.02303
0.02382
0.02450
ܿ଴ ሾ݉Ȁ‫ݏ‬ሿ
2736.2
2762.7 (opt)
2762.6
2746.7
2723.2
Some observations:
a) ሺܿ଴ ሻ௢௣௧ is 2844 m/s for equilibrium and 2763 m/s for frozen flow, a difference of 2.8%. For a rocket of
large dimensions and this high pressure, the actual performance is likely to be close to equilibrium.
b) ሺܱȀ‫ܨ‬ሻ௢௣௧ is about 2.52 for equilibrium, but only 2.3 for frozen flow. This can be understood
qualitatively: the reason an optimum exists in any case is the trade-off between higher ܶ௖ at higher ܱȀ‫ܨ‬
(closer to stoichiometric), but also higher ‫ܯ‬at higher ܱȀ‫( ܨ‬less extra ‫ܪ‬ଶ around). There is a third effect,
though: higher ܱȀ‫ܨ‬, with its higher ܶ௖ , produces more dissociation in the chamber; if the flow is in
equilibrium, most of this dissociation is reversed during the expansion, and the corresponding energy is
recovered (partially) as kinetic energy. This does not happen in a frozen expansion, and so in the
equilibrium case there is more of an incentive to go on to higher ܱȀ‫ܨ‬, as observed.
2) For ைி ൌ ʹǤ͸, equilibrium, we read off ܶ௖ ൌ ͵͸͹ͶǤʹ‫ ܭ‬and, at the throat, ߛ ൌ ͳǤͳ͵ͶͲǡ ࣧ ൌ
ͲǤͲʹ͵ͺʹ݇݃Ȁ݉‫݈݋‬Ǥ Using these as constants, we can calculate:
ംశభ
ଶ మሺംషభሻ
ൌ ͲǤ͸͵ͷͶ
Ȟ ൌ ξߛ ቀ ቁ
ఊାଵ
଼Ǥଷଵସ
‫ܬ‬
ൌ ͵ͶͻǤͲ ൗ݇݃ ‫ܭ כ‬
ܴൌ
଴Ǥ଴ଶଷ଼ଶ
ܿ‫ כ‬ൌ
ඥோ்೎
୻
ൌ ͳ͹ͺʹǤʹ
(6)
(7)
(8)
CEA: 1793.8 m/s
ം
௉
For the exit Mach number, we use ௉೎ ൌ ቀͳ ൅
೐
ఊିଵ ଶ ംషభ
‫ܯ‬௘ ቁ ,
ଶ
ܲ௖ ൌ ͹Ͳܽ‫݉ݐ‬ǡ ܲ௘ ൌ ͲǤͶܽ‫݉ݐ‬.
Therefore:
‫ܯ‬௘ ൌ ͵ǤͷͶ͵
(CEA: 3.536)
For exit temperature:
ܶ௘ ൌ
்೎
ംషభ మ
ெ೐
మ
ଵା
ൌ ͳͻͻ͸‫ܭ‬
(9)
(CEA: 2051K)
2
ംషభ
ଶ
௉
ം
ቈቀ ೎ ቁ
ఊିଵ ௉೐
or ‫ܯ‬௘ ൌ ඨ
െ ͳ቉, where
For area ratio:
஺೐
஺೟
ംషభ
ൌ
ംశభ
ଵା మ ெ೐మ మሺംషభሻ
ଵ
ቆ ംశభ
ቇ
ெ೐
ൌ ʹͳǤ͹͵
(10)
మ
(CEA: 20.67)
For exit velocity (or matched specific impulse):
‫ݑ‬௘ ൌ ‫ܯ‬௘ ඥߛܴܶ௘ ൌ ͵ͳͶͻ݉Ȁ‫ݏ‬
(CEA: 3162 m/s)
(11)
For vacuum specific impulse:
ܿ௩௔௖ ൌ ‫ݑ‬௘ ൅
௉ೌ ஺೐ ‫כ‬
ܿ
௉బ ஺೟
ൌ ͵͵͹Ͳ െ
ଵ
ൈ
଻଴
ʹͳǤ͹͵ ൈ ͳ͹ͺʹǤʹ ൌ ʹͺͳ͹݉Ȁ‫ݏ‬
(12)
(CEA: 2844 m/s)
The simple model agrees to better than 5% in all the important quantities with the full equilibrium
model. But you need hindsight in the choices of ߛ and ‫ܯ‬.
ଷଶ௫
3) Atom conservation: The reactants are ‫ܪܥ‬ଵǤଽ଻ହ ൅ ܱܺଶ , and imposing ܱȀ‫ ܨ‬ൌ ʹǤ͸, ଵଶାଵǤଽ଻ହ ൌ ʹǤ͸ ՜
‫ ݔ‬ൌ ͳǤͳ͵ͷ. Since the total quantity is arbitrary, only relative atomic amounts matter. We have then in
the reactants:
௠ಹ
ଵǤଽ଻ହ
ൌ
ൌ ͲǤͳ͸ͷ
௠಴
ଵଶ
௠బ
ଵǤଵଷହൈଷଶ
ൌ
ൌ ͵ǤͲʹ͹
௠಴
ଵଶ
(13)
(14)
For the products we read for this case the mole fractions at exit:
‫ݕ‬஼ை ൌ ͲǤʹ͸ͶͲ
‫ݕ‬஼ைమ ൌ ͲǤʹͶͳͻ
‫ݕ‬ுమ ൌ ͲǤͲͻͳͺ
‫ݕ‬ுమ ை ൌ ͲǤͶͲͲ͸
‫ݕ‬ு ൌ ͲǤͲͲͳͳ
‫ݕ‬ைு ൌ ͲǤͲͲͲ͸
With very minor amounts of other molecules. Thus, the mass fractions at exit are:
௠ಹ
௠಴
ൌ
௠బ
௠಴
ൌ
௬ಹమ ൈଶା௬ಹమೀ ൈଶା௬ಹ ൈଵା௬ೀಹൈଵ
ൌ ͲǤͳ͸ʹ
௬಴ೀ ൈଵଶା௬಴ೀమ ൈଵଶ
௬಴ೀ ൈଵ଺ା௬಴ೀమ ൈଷଶା௬ಹమೀ ൈଵ଺ା௬ೀಹ ൈଵ଺
௬಴ೀ ൈଵଶା௬಴ೀమ ൈଷଶ
(compare to 0.165)
ൌ ͵ǤͲʹͺ
(compare to 3.027)
Entropy conservation: Since ܶ௘ ൌ ʹͲͷͳ‫ܭ‬, we need to extrapolate slightly from the given table of
standard molar entropies; we get:
ι
‫ݏ‬ǁ ஼ை
ൌ ʹͷͻǤ͸ͺ
௃
௠௢௟‫כ‬௄
3
ι
‫ݏ‬ǁ ஼ை
ൌ ͵ͳͲǤͻ͵
మ
‫ݏ‬ǁ ுι మ ൌ ͳͺͻǤ͵ʹ
‫ݏ‬ǁ ுι మை
ൌ
௃
௠௢௟‫כ‬௄
௃
௠௢௟‫כ‬௄
௃
ʹ͸͸ǤͲͶ
௠௢௟‫כ‬௄
Then, for each molecule ࢙෤࢏ ൌ ‫ݏ‬ǁ ௜ι െ ݈࣬݊ܲ௜ ሺܽ‫݉ݐ‬ሻ ൌ ‫ݏ‬ǁ ௜ι െ ݈࣬݊ሺ‫ݕ‬௜ ܲ௘ ሺܽ‫݉ݐ‬ሻሻ. This gives:
௃
௠௢௟‫כ‬௄
௃
‫ݏ‬ǁ ஼ைమ ൌ ͵͵ͲǤ͵ͷ
௠௢௟‫כ‬௄
௃
‫ݏ‬ǁ ுమ ൌ ʹͳ͸Ǥ͹ͻ
௠௢௟‫כ‬௄
௃
‫ݏ‬ǁ ுమை ൌ ʹͺͳǤʹ͸
௠௢௟‫כ‬௄
‫ݏ‬ǁ ஼ை ൌ ʹ͹ͺǤ͵͹
Finally, the specific entropy (per unit mass) is:
σ ௬೔ ௦೔
ܵ௘ ൌ σ ೔
೔ ௬೔ ெǁ ೔
ൌ
σ೔ ௬೔ ௦೔
ഥ
ெ
ǁ4
ഥ௘ ൌ ͲǤͲʹͷͶͷ
Using ‫ܯ‬
ܵ௘ ൌ ͳͳǡʹ͵Ͳ
(15)
௞௚
௠௢௟
(CEA) and the four mole fractions ሺ‫ܱܥ‬ǡ ‫ܱܥ‬ଶ ǡ ‫ܪ‬ଶ ǡ ‫ܪ‬ଶ ܱሻ, this gives:
௃
௞௚‫כ‬௄
Compared to ܵ௖ ൌ ܵ௘ ൌ ͳͳǡͲ͸ͺ
௃
from
௞௚‫כ‬௄
CEA. This is a bit high, not clear why.
4
5
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