16.50 Propulsion Systems
Spring 2012
Homework : Mean-Line Compressor Design
a) At the compressor inlet (station 2), the flow area is:
ଶ
ଶ
‫ܣ‬ଶ ൌ ߨሺ‫்ݎ‬ଶ
െ ‫ݎ‬ுଶ
ሻ
(1)
The mean radius (same everywhere) is:
‫ݎ‬ҧ ൌ
௥೅మ ା௥ಹమ
ଶ
(2)
Hence:
మ
‫ܣ‬ଶ ൌ ߨ‫ݎ‬ҧ ଶ
మ
మ
௥೅మ
ି௥ಹమ
మ
ೝ శೝ
ቀ ೅మ మ ಹమቁ
ൌ Ͷߨ‫ݎ‬ҧ ଶ
ೝ
ଵିቆ ಹ
మቇ
ೝ೅
మ
మ
ೝಹ
൬ଵା൬ ೝ ൰ ൰
೅ మ
(3)
࢘
Using ࡭૛ ൌ ૙Ǥ ૚૜ૡ૞࢓૛ and ቀ ࡴ ቁ ൌ ૙Ǥ ૝:
࢘ࢀ ૛
ͲǤͳ͵ͺͷ ൌ
ଵି଴Ǥସమ
Ͷߨ‫ݎ‬ҧ ଶ
ሺଵା଴Ǥସሻమ
‫ݎ‬ҧ ൌ ͲǤͳ͸ͲͶ݉
ͲǤͳ͸ͲͶ ൌ ‫்ݎ‬ଶ
ೝ
ଵା൬ ಹ൰
ೝ
೅ మ
ଶ
ൌ ͲǤ͹‫்ݎ‬ଶ
‫்ݎ‬ଶ ൌ ͲǤʹʹͻͳ݉
‫ݎ‬ுଶ ൌ ͲǤͶ‫்ݎ‬ଶ ൌ ͲǤͲͻͳ͸Ͷ݉
Blade height:
݄ଶ ൌ ‫்ݎ‬ଶ െ ‫ݎ‬ுଶ ൌ ͲǤͳ͵͹ͷ݉
b) We can calculate the axial velocity at station 2:
‫ ݓ‬ൌ ‫ܯ‬ଶ ඥߛܴܶଶ ൌ ‫ܯ‬ଶ ට
ఊோ்೟మ
ംషభ మ
ெమ
మ
ଵା
ൌ ͲǤͶට
ଵǤସൈଶ଼଻ൈଶ଻ଽǤ଼
ଵା଴Ǥଶൈ଴Ǥସమ
ൌ ͳͻʹǤͲʹ݉Ȁ‫ݏ‬
Keeping ࢝૛ ൌ ࢝૜ ൌ ࢝:
஺య
஺మ
ൌ
௉೟మ
௉೟య
ൌ
௉మ
௉య
ംషభ
ൌ
భ
మ ംషభ
௉೟మ ଵା మ ெయ
ቆ ംషభ మቇ
௉೟య ଵା
ெమ
௉೟య ்೟య
௉೟మ ்೟మ
(5)
మ
(6)
We also have:
‫ܯ‬ଷ ඥܶଷ ൌ ‫ܯ‬ଶ ඥܶଶ
(7)
1
(4)
ெయ
ெమ
ൌ
்
ට்మ
య
ெయ
ംషభ మ
ଵା
ெయ
మ
ெయ
ଵା଴Ǥଶெయమ
ംషభ
ൌ
మ
் ଵା మ ெయ
ඨ ೟మ ംషభ
்೟య ଵା
ெమమ
(8)
మ
ൌ
்೟మ
ெమమ
ംషభ మ
்೟య ଵା
ெమ
(9)
మ
ൌ
଴Ǥସమ
ଶ଻ଽǤ଼
ହହଽǤ଻ ଵା଴Ǥଶൈ଴Ǥସమ
ൌ ͲǤͲ͹͹ͷͲ͸
‫ܯ‬ଷଶ ൌ ͲǤͲ͹͹ͷͲ͸ ൅ ͲǤͲͳͷͷͲͳ‫ܯ‬ଷଶ
‫ܯ‬ଷ ൌ ͲǤʹͺͲ͸
Then:
஺య
஺మ
ൌቀ
ଶ଻ଽǤ଼ ଷǤହିଵ ଵା଴Ǥଶൈ଴Ǥଶ଼଴଺మ
ቁ
ቀ
ቁ
ହହଽǤ଻
ଵା଴Ǥଶൈ଴Ǥସమ
ଶ
ଶǤହ
ൌ ͲǤͳ͸ͻͺ
‫ܣ‬ଷ ൌ ͲǤͲʹ͵ͷʹ݉
As in part a),
஺య
ସగ௥ҧ మ
ൌ
ೝ
ଵି൬ ಹ൰
ೝ
మ
೅ య
మ
ೝಹ
൬ଵା൬ ೝ ൰ ൰
೅ య
ൌ ͲǤͲ͹ʹ͹Ͷͺ
࢘
To solve for ቀ ࢘ࡴ ቁ ൌ ࢞ǣ
ࢀ
૜
ͳ െ ‫ ݔ‬ଶ ൌ ͲǤͲ͹ʹ͹Ͷͺሺͳ ൅ ʹ‫ ݔ‬൅ ‫ ݔ‬ଶ ሻ
‫ ݔ‬ൌ ͲǤͺ͸Ͷ͵͹ ൌ
௥ಹయ
௥೅య
Then:
ͲǤͳ͸ͲͶ ൌ ‫்ݎ‬ଷ
ଵା଴Ǥ଼଺ସଷ଻
ଶ
‫்ݎ‬ଷ ൌ Ͳǡͳ͹ʹͳ݉
‫ݎ‬ுଷ ൌ ͲǤͳͶͺ͹݉
Blade height:
݄ଷ ൌ ‫்ݎ‬ଷ െ ‫ݎ‬ுଷ ൌ ͲǤͲʹ͵͵͹݉
c) Euler’s equation for one stage gives:
௩మ ି௩భ
ቁ
ఠ௥ҧ
ܿ௣ ሺοܶ௧ ሻଷ ൌ ߱‫ݎ‬ሺ‫ݒ‬
ҧ ଶ െ ‫ݒ‬ଵ ሻ ൌ ሺ߱‫ݎ‬ሻҧ ଶ ቀ
(10)
Since we will use repeating stage, the factor
௩మ ି௩భ
,
ఠ௥ҧ
which depends on angles only, will be the same for all
stages. In addition, ߱‫ݎ‬ҧ is also the same, so ሺοܶ௧ ሻଷ will be the same for all stages. Using N stages,
ሺοܶ௧ ሻଷ ൌ
்೟య ି்೟మ
ே
ൌ
ହହଽǤ଻ିଶ଻ଽǤ଼
ே
ൌ
ଶ଻ଽǤଽ
ே
(11)
The velocity triangle can be sketched from the condition that the rotor and the stator provide the same
flow turning and, in their frame, the same deceleration:
2
‫ݒ‬ଶ െ ‫ݒ‬ଵ ൌ ο‫ݒ‬
‫ݒ‬ଵ ൌ ‫ߚ ƒ– ݓ‬ଵ
‫ݒ‬ଶ ൌ ߱‫ݎ‬ҧ െ ‫ߚ ƒ– ݓ‬ଵ
ο‫ ݒ‬ൌ ߱‫ݎ‬ҧ െ ʹ‫ߚ ƒ– ݓ‬ଵ
–ƒ ߚଵ ൌ
ҧ
ఠ௥ିο௩
ଶ௪
(12)
(13)
(14)
(15)
(16)
We can now try values of N and follow these steps:
ଶ଻ଽǤଽ
ே
௖೛ ሺο்೟ ሻయ
οܶ௧ଷ ൌ
ଵ଴଴ସǤହሺο்೟ ሻయ
ൌ
ఠ௥ҧ
ଷ଴଴
ଷ଴଴ିο௩
–ƒ ߚଵ ൌ
ଶൈଵଷଶǤ଴
ଵଷଶ
‫ݒ‬ଶᇱ ൌ ‫ݒ‬ଵ ൌ
ୡ୭ୱ ఉభ
ο‫ ݒ‬ൌ
‫ݒ‬ଶ ൌ ͵ͲͲ െ ͳ͵ʹ –ƒ ߚଵ
‫ݒ‬ଵᇱ ൌ ‫ݒ‬ଶ ൌ ඥ‫ ݓ‬ଶ ൅ ‫ݒ‬ଶଶ
The diffusion factor (the same for stator and rotor) is:
‫ ܦ‬ൌͳെ
௩మᇲ
௩భᇲ
൅
ο௩
ଶఙ௩భᇲ
ሺߪ ൌ ͳǤͷሻ
Some results follow:
ܰ
8
9
10
ሺοܶ௧ ሻଷ ሾ‫ܭ‬ሿ
34.99
31.10
27.99
ο‫ݒ‬ሾ݉Ȁ‫ݏ‬ሿ
117.2
104.1
93.7
‫ݒ‬ଶᇱ ሾ݉Ȁ‫ݏ‬ሿ
160.6
164.6
167.5
ߚଵ ሾιሿ
34.71
36.57
38.00
These are all probably acceptable designs. We select N = 9.
3
‫ݒ‬ଶ ሾ݉Ȁ‫ݏ‬ሿ
208.6
202.1
196.9
‫ݒ‬ଵᇱ ሾ݉Ȁ‫ݏ‬ሿ
246.8
241.4
237.0
‫ܦ‬
0.5077
0.4628
0.4250
d)
Figure 1: Nondimensional cross-section of compressor (dimensions in m)
Figure 2: Velocity triangles (any stage) and blade profiles
Concept Questions
1) The temperature increments are uniform. The stage temperature ratios are then:
߬௦ ൌ ͳ ൅
ሺο்೟ ሻೞ
்೟ೞǡ೔೙
ം
ംషభ
ߨ௦ ൌ ߬௦
As ܶ௧௦ǡ௜௡ increases, ߬௦ and ߨ௦ decrease. So the first stage has the highest pressure ratio across it, and the
last stage has the lowest.
4
2) Actually, the statement of the question is not correct, unless it refers to points along the compressor
operating line. The problem with this interpretation is that as ݉ሶ is reduced along the operating line, so is
߱‫ݎ‬,ҧ and no problem arises. Problems do arise if one reduces ݉ሶ along a constant speed line, like along
DA in the sketch. To see this, take the design velocity triangle, leave the base ߱‫ݎ‬ҧ unchanged, but reduce
the height ‫ݓ‬: The incidence angle to a rotor blade is seen to increase, and the same happens to the
stator. At some point, the blades stall.
5
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