16.50 Homework 1 (Solutions)

16.50 Propulsion Systems Spring 2012 Homework 1: Preliminary Design of a Satellite Launcher a) Velocity Calculations: Point A: Start of ascent trajectory Point B: Apogee of ascent trajectory Conservation of Angular Momentum: ‫ݒ‬ଵ ‘• ߙ ܴா ൌ ‫ݒ‬௔ ܴ௖ Rearranging becomes: ‫ݒ‬௔ ൌ ‫ݒ‬ଵ ோಶ ோ೎ ‘• ߙ (1) Conservation of Energy: ௩భమ ଶ െ ఓಶ ோಶ ͳ ଶ ‫ ݒ‬ቆͳ ʹ ଵ ௩ೌమ ఓ ௩మ ோ െ ಶൌ భቀ ಶ ଶ ோಶ ଶ ோ೎ ଶ ܴா ߤா െ ଶ ‘• ଶ ߙቇ ൌ ܴா ܴ௖ ଶ ൌ ଶఓಶ ቌ ோಶ ‫ݒ‬ଵ ൌ ඩ ೃ ଵି ಶ ೃ಴ ೃమ ଵି ಶ ୡ୭ୱమ ఈ ೃమ ಴ ‘• ߙቁ െ െ Values, Constants, and Given Parameters: ܴா ൌ ͸͵͹Ͳ݇݉ ܴ஼ ൌ ͸ͺ͹Ͳ݇݉ ఓಶ ோಶ ߤா ܴ௖ ߤா ൌ ͵Ǥͻͺ݁ͳͶ ௠య ௦య ߙ ൌ ʹͲι ቍ (2) Substituting Values: Using equation (2) to find ‫ݒ‬ଵ : ‫ݒ‬ଵ ൌ ඨቀ ଶ‫כ‬ଷǤଽ଼௘ଵସ ଺Ǥଷ଻௘ହ ‫ݒ‬ଵ ൌ ͸ͳͶͷ ቁቆ Orbital Velocity: ଵି଺ଷ଻ൗ଺଼଻ మቇ ଺ଷ଻ ଵି൫ ൗ଺଼଻ ୡ୭ୱ ଶ଴ι൯ ఓಶ ‫ݒ‬௖ ൌ ට ௠ ௦ ‫ݒ‬௔ ൌ ͸ͳͶͷ ൈ ‫ݒ‬௔ ൌ ͷ͵ͷͶ ௠ ௦ ଺ଷ଻ ଺଼଻ ‫ݒ‬௖ ൌ ‘• ʹͲι 1 (3) ଷǤଽ଼௘ଵସ ଺Ǥ଼଻௘଺ ௠ ͹͸ͳͳ ௦ ‫ݒ‬௖ ൌ ට Using equation (1) to find ‫ݒ‬௔ : ோ಴ b) Stage οࢂ Calculations: We allocate the full “apogee kick” to the third stage: οܸଷ ൌ ‫ݒ‬௖ െ ‫ݒ‬௔ (4) οܸଷ ൌ ͹͸ͳͳ οܸଷ ൌ ʹʹͷ͹ ௠ ௦ ௠ ௠ ௦ െ ͷ͵ͷͶ  ௦ The initial ascent velocity ‫ݒ‬ଵ contains the velocity increments of the 1 st and 2nd stages, plus the contribution from Earth rotation, minus the losses from gravity and drag: ‫ݒ‬ଵ ൌ οܸଵ ൅ οܸଶ ൅ ߱ா ܴா ‘• ߙ െ οܸீ െ οܸ஽ (5) By design: ଵ ଶ οܸଵ ൌ οܸଶ ൌ ሺ‫ݒ‬ଵ െ ߱ா ܴா ‘• ߙ ൅ οܸீ ൅ οܸ஽ ሻ To solve: ߱ா ܴா ‘• ߙ ൌ ͹Ǥʹ͸ͺ݁ሺെͷሻ ߱ா ܴா ‘• ߙ ൌ ௠ Ͷ͵ͷ ௦ ௥௔ௗ ௦ ൈ ͸Ǥ͵͹݁͸݉ ൈ ‘• ʹͲι To estimate the gravity loss (1st stage only) we assume: •‹ ߛ ൌ ͳ െ ௧ ௧್భ (6) ( •‹ ߛis linear between ߛ ൌ ͳ at ‫ ݐ‬ൌ Ͳ and ߛ ൌ Ͳ at ‫ ݐ‬ൌ ‫ݐ‬௕ଵ ) Determining relations among parameters: ‫ݐ‬௕ଵ ൌ ெ೛భ ெబభ ெ೛భ ௠ሶ భ ൌͳെ ൌ ெ೛భ ቀிభൗ௖ቁ ெ೑భ ெబభ ൌ ெ೛భ ெబభ ‫כ‬ଷ௚ ܿ οೇభ ೎ ൌ ͳ െ ݁ି ௠ We need to make a preliminary guess at οࢂ૚ . Take for now οܸଵ ൌ ͵ʹͲͲ ௦ . ெ೛భ ெబభ షయమబబ ൌ ͳ െ ݁ వǤఴ‫כ‬మళబ ൌ ͲǤ͹Ͳͳ͸ ‫ݐ‬௕ଵ ൌ ଴Ǥ଻଴ଵ଺ ሺͻǤͺ ‫כ‬ ଷ‫כ‬ଽǤ଼ ʹ͹Ͳሻ ൌ ͸͵Ǥͳͷ‫ݏ‬ Solving for οࢂࡳ : ௧ ௧್భ ଵ ݃‫ݐ‬௕ଵ ‫׬‬଴ ሺͳ െ ‫ݖ‬ሻ݀‫ݖ‬ Making the substitution ‫ ݖ‬ൌ ௧ οܸீ ൌ ‫׬‬଴ ್భ ݃ •‹ ߛ ݀‫ ݐ‬ൌ ଵ ଶ ൌ ݃‫ݐ‬௕ଵ (7) As a first approximation: ଵ ଶ οܸீ ൌ ൈ ͻǤͺ ൈ ͸͵Ǥͳͷ οܸீ ൌ ͵Ͳͻ ௠ ௦ 2 We can now calculate a better οࢂ૚ : ଵ ଶ οܸଵ ൌ ሺ͸ͳͶͷ െ Ͷ͵ͷ ൅ ͵Ͳͻ ൅ ͳͷͲሻ οܸଵ ൌ ͵Ͳͺͷ ௠ ௦ Refine other quantities: ఓ೛భ ఓబభ యబఴఱ ൌ ͳ െ ݁ ିమలరల ൌ ͲǤ͸ͺͺͶ ଴Ǥ଺଼଼ସ ʹ͹Ͳ ൌ ͸ͳǤͻͷ‫ݏ‬ ଷ ଵ ௠ οܸீ ൌ ൈ ͻǤͺ ൈ ͸ͳǤͻͷ ൌ ͵ͲͶ ଶ ௦ ௠ οܸଵ ൌ ͵Ͳͺʹ ௦ ௠ οܸଶ ൌ οܸଵ ൌ ͵Ͳͺʹ ௦ ‫ݐ‬௕ଵ ൌ These values are close enough to the first approximation, and we accept them as converged. c) Calculation of Stage Masses: For each stage: ఓ೛ೌ೤ǡ೔ ఓబǡ೔ ൌ ݁ି οೇ೔ ೎ െ ߝ (8) We apply this first to the 3rd stage, for which ݉௣௔௬ǡଷ ൌ ݉௣௔௬ ൌ ͵݇݃Ǥ ‫ܯ‬଴ଷ ൌ ଷ ௘ ష మమఱళ మలరల ି଴Ǥଵ ൌ ଷ ଴Ǥଷଷ଺ଵ ൌ ͻǤʹͲ݇݃ The structural mass of the third stage is then: ‫ܯ‬௦ଷ ൌ ͲǤͳ‫ܯ‬଴ଷ ൌ ͲǤͻʹ݇݃ The propellant mass is: మమఱళ ‫ܯ‬௣ଷ ൌ ͻǤʹͲ ቀͳ െ ݁ ିమలరలቁ ൌ ͷǤʹͺ݇݃ As a check: ‫ܯ‬௦ଷ ൅ ‫ܯ‬௣ଷ ൅ ‫ܯ‬௣௔௬ଷ ൌ ͲǤͻʹ ൅ ͷǤʹͺ ൅ ͵ ൌ ͻǤʹͲ݇݃ ൌ  ‫ܯ‬଴ଷ (as it should) For the second stage: ‫ܯ‬௣௔௬ଶ ൌ ‫ܯ‬଴ଷ ൌ ͻǤʹͲ݇݃ ‫ܯ‬଴ଶ ൌ ଽǤଶ଴ ௘ ష యబఴమ మలరల ି଴Ǥଵ ൌ ଽǤଶ଴ ଴Ǥଶଵଶ଴ ൌ Ͷ͵Ǥ͵ͻ݇݃ ‫ܯ‬௦ଶ ൌ ͲǤͳ ൈ Ͷ͵Ǥ͵ͻ ൌ ͶǤ͵Ͷ݇݃ యబఴమ ‫ܯ‬௣ଶ ൌ Ͷ͵Ǥ͵ͻ ቀͳ െ ݁ ିమలరల ቁ ൌ ʹͻǤͺͷ݇݃ Again, we check that: ‫ܯ‬௦ଶ ൅ ‫ܯ‬௣ଶ ൅ ‫ܯ‬௣௔௬ଶ ൌ ͶǤ͵Ͷ ൅ ʹͻǤͺͷ ൅ ͻǤʹͲ ൌ Ͷ͵Ǥ͵ͻ݇݃ ൌ  ‫ܯ‬଴ଶ 3 For the first stage: ‫ܯ‬௣௔௬ସ ൌ ‫ܯ‬଴ଶ ൌ Ͷ͵ǤͶͻ݇݃ ‫ܯ‬଴ଵ ൌ ସଷǤଷଽ ௘ ష యబఴమ మలరల ି଴Ǥଵ ൌ ସଷǤଷଽ ଴Ǥଶଵଶ଴ ൌ ʹͲͶǤ͸͹݇݃ ‫ܯ‬௦ଵ ൌ ͲǤͳ ൈ ʹͲͶǤ͸͹ ൌ ʹͲǤͶ͹݇݃ యబఴమ ‫ܯ‬௣ଵ ൌ ʹͲͶǤ͸͹ ቀͳ െ ݁ ିమలరలቁ ൌ ͳͶͲǤͺʹ݇݃ We verify that ‫ܯ‬௦ଵ ൅ ‫ܯ‬௣ଵ ൅ ‫ܯ‬௣௔௬ଵ ൌ ʹͲǤͶ͹ ൅ ͳͶͲǤͺʹ ൅ Ͷ͵Ǥ͵ͻ ൌ ʹͲͶǤ͸ͺ݇݃ d) Thrusts and Firing Times: ‫ܨ‬ଵ ൌ ‫ܯ‬଴ଵ ൈ ͵݃ ൌ ʹͲͶǤ͸͹ ൈ ͵ ൈ ͻǤͺ ൌ ͸ǡͲͳ͹ܰ ‫ܨ‬ଶ ൌ ‫ܯ‬଴ଶ ൈ ͵݃ ൌ Ͷ͵Ǥ͵ͻ ൈ ͵ ൈ ͻǤͺ ൌ ͳǡʹ͹͸ܰ ‫ܨ‬ଷ ൌ ‫ܯ‬଴ଷ ൈ ͵݃ ൌ ͻǤʹͲ ൈ ͵ ൈ ͻǤͺ ൌ ʹ͹Ͳܰ Flow rates are then: ݉ሶ ଵ ൌ ݉ሶ ଶ ൌ ݉ሶ ଵ ൌ ଺଴ଵ଻ ଶ଺ସ଺ ଵଶ଻଺ ଶ଺ସ଺ ଶ଻଴ ଶ଺ସ଺ ௞௚ ൈൈ ௦ ௞௚ ͲǤͶͺʹʹ ௦ ௞௚ ൌ ʹǤʹ͹Ͷ ൌ ൌ ͲǤͳͲʹͲ ௦ Firing times are given by ࢚࢈࢏ ൌ ‫ݐ‬௕ଵ ൌ ‫ݐ‬௕ଵ ൌ ‫ݐ‬௕ଵ ൌ ଵସ଴Ǥ଼ଶ ଶǤଶ଻ସ ଶଽǤ଼ହ ଴Ǥସ଼ଶଶ ହǤଶ଼ ଴Ǥଵ଴ଶ଴ ࢓࢖࢏ ࢓ሶ ࢏ ൌ ͸ͳǤͻ͵‫ݏ‬ ൌ ͸ͳǤͻͳ‫ݏ‬ ൌ ͷͳǤ͹͸‫ݏ‬ 4