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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 శ ቀ మ మ ಹమቁ ൌ Ͷߨݎҧ ଶ ଵିቆ ಹ మቇ మ మ ಹ ൬ଵା൬ ൰ ൰ మ (3) Using ൌ Ǥ ૡ and ቀ ቁ ൌ Ǥ : ͲǤͳ͵ͺͷ ൌ ଵିǤସమ Ͷߨݎҧ ଶ ሺଵାǤସሻమ ݎҧ ൌ ͲǤͳͲͶ݉ ͲǤͳͲͶ ൌ ்ݎଶ ଵା൬ ಹ൰ మ ଶ ൌ ͲǤ்ݎଶ ்ݎଶ ൌ ͲǤʹʹͻͳ݉ ݎுଶ ൌ ͲǤͶ்ݎଶ ൌ ͲǤͲͻͳͶ݉ Blade height: ݄ଶ ൌ ்ݎଶ െ ݎுଶ ൌ ͲǤͳ͵ͷ݉ b) We can calculate the axial velocity at station 2: ݓൌ ܯଶ ඥߛܴܶଶ ൌ ܯଶ ට ఊோ்మ ംషభ మ ெమ మ ଵା ൌ ͲǤͶට ଵǤସൈଶ଼ൈଶଽǤ଼ ଵାǤଶൈǤସమ ൌ 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 ଵାǤଶெయమ ംషభ ൌ మ ் ଵା మ ெయ ඨ మ ംషభ ்య ଵା ெమమ (8) మ ൌ ்మ ெమమ ംషభ మ ்య ଵା ெమ (9) మ ൌ Ǥସమ ଶଽǤ଼ ହହଽǤ ଵାǤଶൈǤସమ ൌ ͲǤͲͷͲ ܯଷଶ ൌ ͲǤͲͷͲ ͲǤͲͳͷͷͲͳܯଷଶ ܯଷ ൌ ͲǤʹͺͲ Then: య మ ൌቀ ଶଽǤ଼ ଷǤହିଵ ଵାǤଶൈǤଶ଼మ ቁ ቀ ቁ ହହଽǤ ଵାǤଶൈǤସమ ଶ ଶǤହ ൌ ͲǤͳͻͺ ܣଷ ൌ ͲǤͲʹ͵ͷʹ݉ As in part a), య ସగҧ మ ൌ ଵି൬ ಹ൰ మ య మ ಹ ൬ଵା൬ ൰ ൰ య ൌ ͲǤͲʹͶͺ To solve for ቀ ቁ ൌ ǣ ͳ െ ݔଶ ൌ ͲǤͲʹͶͺሺͳ ʹ ݔ ݔଶ ሻ ݔൌ ͲǤͺͶ͵ ൌ ಹయ య Then: ͲǤͳͲͶ ൌ ்ݎଷ ଵାǤ଼ସଷ ଶ ்ݎଷ ൌ Ͳǡͳʹͳ݉ ݎுଷ ൌ ͲǤͳͶͺ݉ Blade height: ݄ଷ ൌ ்ݎଷ െ ݎுଷ ൌ ͲǤͲʹ͵͵݉ c) Euler’s equation for one stage gives: ௩మ ି௩భ ቁ ఠҧ ܿ ሺοܶ௧ ሻଷ ൌ ߱ݎሺݒ ҧ ଶ െ ݒଵ ሻ ൌ ሺ߱ݎሻҧ ଶ ቀ (10) Since we will use repeating stage, the factor ௩మ ି௩భ , ఠҧ 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 ே ൌ ହହଽǤିଶଽǤ଼ ே ൌ ଶଽǤଽ ே (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 ݓଵ ݒଶ ൌ ߱ݎҧ െ ߚ ݓଵ ο ݒൌ ߱ݎҧ െ ʹߚ ݓଵ ߚଵ ൌ ҧ ఠିο௩ ଶ௪ (12) (13) (14) (15) (16) We can now try values of N and follow these steps: ଶଽǤଽ ே ሺο் ሻయ οܶ௧ଷ ൌ ଵସǤହሺο் ሻయ ൌ ఠҧ ଷ ଷିο௩ ߚଵ ൌ ଶൈଵଷଶǤ ଵଷଶ ݒଶᇱ ൌ ݒଵ ൌ ୡ୭ୱ ఉభ ο ݒൌ ݒଶ ൌ ͵ͲͲ െ ͳ͵ʹ 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 0,72SHQ&RXUVH:DUH KWWSRFZPLWHGX ,QWURGXFWLRQWR3URSXOVLRQ6\VWHPV 6SULQJ )RULQIRUPDWLRQDERXWFLWLQJWKHVHPDWHULDOVRURXU7HUPVRI8VHYLVLWKWWSRFZPLWHGXWHUPV