Homework 4.1: Solid Propellant Rocke t
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16.50 Propulsion Systems Spring 2012 Homework 4.1: Solid Propellant Rocket 1a) Normal operation mass balance: బ כ ܽߩ ܣ ܲ െ ൌͲ (1) Transient operation after port opening: ௗ ோ் ௗ௧ ൌ ܽߩ ܣ ܲ െ ሺ ା್ ሻ (2) כ Define: ݐ ൌ כ ோ் ್ ௧ ௧ ൌ ು (3) ߬ൌ (4) ߙ (5) ݕൌ (6) బ In terms of these variables, equation (2) becomes: ௗ௬ ௗఛ ൌ ݕ െ ሺͳ ߙሻݕ (7) Multiply equation (7) by ࢟ି : ି ݕ ௗ௬ ௗఛ ൌ ͳ െ ሺͳ ߙሻ ݕଵି (8) ሺͳ െ ݊ሻି ݕ ݀ ݕൌ ݀ሺ ݕଵି ሻ Define: ݕଵି ൌ ݑ ଵ ௗ௨ ଵି ௗఛ (9) ൌ ͳ െ ሺͳ ߙሻݑ (10) This relationship is linear and can be easily solved. The particular solution is ݑൌ homogeneous solution is ݑൌ ݁ ିሺଵିሻሺଵାఈሻఛ . The complete solution is then ݑൌ where A is arbitrary. At ݐൌ Ͳሺ߬ ൌ Ͳሻ, we have ܲ ൌ ܲ ሺ ݕൌ ݑൌ ͳሻ, so ଵ ଵାఈ ఈ ൌ ଵାఈ ͳൌ ܣ ܣ Therefore: 1 ଵ , ଵାఈ ଵ ଵାఈ and the ି ݁ܣሺଵିሻሺଵାఈሻఛ , ݑൌ ଵାఈ షሺభషሻሺభశഀሻഓ ଵାఈ Using ࢟ ൌ ࢛ష : భ ݕൌ ଵାఈ షሺభషሻሺభశഀሻഓ భష ൨ ଵାఈ (11) ଶ ଶ 1b) The combustion stops when ܲ ʹͲܽ ݉ݐቀݕ௫௧௧ ൌ ൌ ቁ. For ݐ՜ λሺ߬ ՜ λሻǡ we obtain from equation (11): ଵ ݕሺλሻ ൌ (12) భ ሺଵାఈሻభష ଶ ଵିǤଶ ቀ ቁ ଶ For this to be equal or less than , ߙ must be more than: ߙ ൌ ቀ ቀ ುቁ ଵି ଵ ቁ ௬ሺஶሻ ൌ ͳǤʹͶ െͳ ൌ െͳ (13) (14) ଶ For values of ߙ ͳǤʹͶ, the extinction limit ݕൌ is reached in a finite time. Solving equation (11) for ߬ gives: ଵ ఈ ߬௫௧ ൌ ሺଵାఈሻሺଵିሻ ݈݊ ቀሺଵାఈሻ௬ భష ቁ ೣ ଵି ൌ Since ݊ ൌ ͲǤʹ and ሺͳ ߙሻݕ௫௧ ߬௫௧ ൌ (15) ିଵ ଵ ఈ ݈݊ ቆ భశഀ ቇ Ǥ଼ሺଵାఈሻ ିଵ ሺଵାఈሻ ሺͳ ሺଵାఈౣ ሻ ଵି ߙ୫୧୬ ሻݕ௫௧ : (16) మǤళమర ݐ௫௧ ൌ ߬௫௧ כ ோ் (17) Assuming a molecular mass ܯൌ ʹͲ כ ோ் ଵൈଵ଼ ൌ ఴǤయభర బǤబమ ൈଷସ ൌ ͳǤʹͶ ൈ ͳͲିଶ ݏ (not specified in problem statement), (18) We can now calculate a few extinction times corresponding to choices of ು above the minimum, shown in Table 1. Table 1: Extinction Times ߙൌ ܣ ܣ௧ ߬௫௧ ݐ௫௧ ሾݏሿ 1.724 2 3 4 λ λ 1.243 1.583e-2 0.5803 7.42e-3 0.3915 4.99e-3 2 Homework 4.2: Monopropellant Hydrazine Rocket ସ ଷ ସ ଶ ܰܪଷ ՜ ܰଶ ଷ ଷ ଵ ଷ ܰଶ ܪସ ՜ ܰܪଷ ܰଶ Rapid disproportionation: Slow ࡺࡴ decomposition: (1) ʹܪଶ (2) Since we assume 40% Ammonia decomposition, form equation (1) + 0.4*equation (2): ସ ଷ ܰଶ ܪସ ՜ ሺͳ െ ͲǤͶሻܰܪଷ ଵାǤ଼ ܰଶ ଷ ͲǤͶ ܪʹ כଶ (3) ܰଶ ܪସ ՜ ͲǤͺܰܪଷ ͲǤܰଶ ͲǤͺܪଶ (4) We can now write the enthalpy balance for the reaction. The enthalpy of the reactants (liquid Hydrazine at 298.2K) is +50.63 kJ/mol, so using the fits provided for gaseous ܰܪଷ ǡ ܰଶ ǡ and ܪଶ : ͷͲǤ͵ ൌ ͲǤͺሺെͲǤͶͲ ͷͳǤͳߠ ͶǤͳͳߠ ଶ ሻ ͲǤሺെͳͳǤͺͶ ͵ʹǤͶʹߠ ͲǤߠ ଶ ሻ ڮ ͲǤͺሺെͺǤʹ͵ ʹǤͳߠ ͳǤ͵Ͷߠ ଶ ሻ ଶ ͶǤͻߠ ͺʹǤͷͲͺߠ െ ͳʹͲǤͶ ൌ Ͳ ߠൌ ି଼ଶǤହ଼ାξ଼ଶǤହ଼మ ାସכସǤଽכଵଶǤସ ଶכସǤଽ ൌ ͳǤ͵ͷʹ ൌ ் ଵ (5) ܶ ൌ ͳǡ͵ͷʹܭ Mean molecular mass: ഥ ൌ Ǥ଼כଵାǤכଶ଼ାǤ଼כଶ ൌ ͳǤͲ ܯ Ǥ଼ାǤାǤ଼ ൌ ͲǤͲͳ Mean specific heat: ෦ ෦ Ǥ଼൫෦ ൯ಿಹ ାǤ൫ ൯ಿ ାǤ଼൫ ൯ಹ ൨൙ య మ మ ሺǤ଼ାǤାǤ଼ሻ ܿ ൌ ൫ܿ෦ ൯ ൫ܿ෦ ൯ ൫ܿ෦ ൯ Ǥଵ ேுయ ேమ ுమ ܿ ൌ ൌ ൌ ൌ డಿಹయ డ் డಿమ డ் డಹమ డ் ൌ ൌ ଵ డಿಹయ ହଵǤ ൌ ଵ డఏ ଵ כ ଵ డಿమ ଷଶǤସଶ ൌ ଵ డఏ ଵ డಹమ ଵ డఏ ൌ ൌ ଵ כ ଶǤଵ ଵ כ ǤସכହଵǤାǤଷכଷଶǤସଶାǤସכଶǤଵ Ǥଵ ൌ ͷͳǤ כ כ ʹǤͳ כ ൌ ͵ʹǤͶʹ ൌ כ ଼Ǥଷଵସ ൌ ʹǡͷͻͲ We could now calculate ܿ௩ ൌ ʹǡͷͻͲ െ Ǥଵ ൌ ʹǤͲͲ כ 3 and so ߛ ൌ ೡ ൌ ͳǤʹͷͳʹ 0,72SHQ&RXUVH:DUH KWWSRFZPLWHGX ,QWURGXFWLRQWR3URSXOVLRQ6\VWHPV 6SULQJ )RULQIRUPDWLRQDERXWFLWLQJWKHVHPDWHULDOVRURXU7HUPVRI8VHYLVLWKWWSRFZPLWHGXWHUPV
