16.50 Practice Mid-Term Quiz (Solutions)

16.50 Propulsion Systems Year Unspecified Quiz 1 One hour, open book, open notes TRUE-FALSE QUESTIONS (50%) Please include a 1-2 line explanation for each of your answers. Statement 1. If one takes a given rocket, with a fixed chamber pressure, and replaces its operating gas by one with half the molecular mass, the thrust does not change. ‫ ܨ‬ൌ ܲ௖ ‫ܣ‬௧ ܿி and ܿி depends on geometry and ߛ, not molecular mass. ‫ܣ‬ 2. For any external pressure, adding segments to a supersonic nozzle (increasing ௘ൗ‫) ܣ‬ ௧ always increases thrust. Maximum thrust when ܲ௘ ൌ ܲ௔ . Extending the nozzle more than that creates suction, reduces thrust. 3. A conventional rocket nozzle operates on the ground with ܲ௘ ൌ ͲǤͷܽ‫݉ݐ‬. A device is proposed that will force flow separation at the point in the nozzle where ܲ ൌ ͳܽ‫݉ݐ‬Ǥ This will increase thrust. Separation will allow back-filling of the negative pressure region past ܲ ൌ ͳܽ‫݉ݐ‬Ǥ 4. A small test rocket is fired in vacuum and measurements are made of the jet flow speed many exhaust diameters downstream. When this is done using different expansion area ratios ‫ܣ‬௘ Τ‫ܣ‬௧ , the downstream speed is found to be invariant. The speed for downstream is ܿ ൌ ‫ܨ‬ൗ݉ሶ , i.e. the specific impulse (times ݃). This does vary with ‫ܣ‬௘ Τ‫ܣ‬௧ . 5. The method of characteristics can be used to design the contour of a nozzle, but only downstream from the throat. The M.O.C. does require ‫ ܯ‬൐ ͳ. 6. In a solid propellant rocket, increasing the throat area increases the thrust. భ ‫ ܨ‬ൌ ܲ௖ ‫ܣ‬௧ ܿி ; ܲ௖ ൌ ஺ భష೙ ቀߩ௣ ܽܿ ‫ ್ כ‬ቁ , ஺೟ so ‫ܣ̱ܨ‬௧ addition, ܿி also decreases (through ஺೐ ஺೟ భ భష೙ ଵି ൌ ‫ܣ‬௧ ష೙ భష೙ ௉೎ ஺೟ √ √ √ √ √ √ , F decreases with ‫ܣ‬௧ . In decreasing). 7. The characteristic damping time of pressure oscillations inside a solid propellant rocket is proportional to the linear dimensions (assuming geometrically similar rockets). ௏ ఘ ௏ఘ ௏ ௏ ̱߬ሺͳ െ ݊ሻ ೎೓ሶ ೎ ൌ ሺͳ െ ݊ሻ ೎ ೎ ܿ ‫ ̱ כ‬೎ For similar geometry, ೎ ̱‫ܮ‬௖ ௠ True False ஺೟ √ ஺೟ 8. In an equilibrium nozzle expansion, dissociated species recombine fairly completely. This means the performance is the same as one would calculate if dissociation were ignored in the chamber. The heat of dissociation is all recovered in the nozzle recombination, but part of it at ܲ ൏ ܲ௖ , so the expansion to ܲ௘ is less efficient than it would be from ܲ௖ . 9. The chemical reactions that are selected for imposing equilibrium in the calculation of chamber temperature must be the ones that actually happen during combustion. They can be selected arbitrarily, as long as they linearly span the space of possible reactions. 1 √ √ 10. Heat flux to the nozzle walls peaks at the throat because that is where stagnation temperature is maximum. Stagnation temperature is ܶ௖ everywhere. What is maximum at the throat is ߩ‫ݑ‬. √ PROBLEM (50%) The gas leaving the combustion chamber of a LOX-LH rocket has the following characteristics: x Molecular mass: ‫ ܯ‬ൌ ͳ͵݃Ȁ݉‫݈݋‬ x Specific heat ratio: ߛ ൌ ͳǤʹ͸ x Temperature: ܶ௖ ൌ ͵͸ͲͲ‫ܭ‬ x Pressure: ܲ௖ ൌ ʹͳͲܽ‫݉ݐ‬ ଴Ǥ଺ ் ቁ ‫݃ܭ‬Ȁ݉Ȁ‫ݏ‬ ଷ଴଴଴ x Viscosity: ߤ௚ ൌ ͵ ൈ ͳͲିହ ቀ x Prandtl’s number: ܲ௥ ൌ ͲǤͻ The nozzle throat has a diameter ‫ܦ‬௧ ൌ ͲǤʹ͹͹݉, and its first wall is a thin Copper shell which is cooled on its back side to ܶ௪௖ ൌ ͵ͲͲ‫ܭ‬. The thermal conductivity of Copper is ݇ ൌ ͵͸ͲܹȀ݉Ȁ‫ܭ‬. What is the maximum Copper thickness such that the hot-side temperature ܶ௪௛ does not exceed ͺͲͲ‫?ܭ‬ Formulas and constants: ߩ௧ ‫ݑ‬௧ ൌ ௉೎ , ௖‫כ‬ with ܿ ‫ כ‬ൌ ඥோ்೎ , ୻ሺఊሻ ംశభ Ȟ ൌ ξߛ ቀ ଶ మሺംషభሻ ቁ ఊାଵ మǤమల ʹ మൈబǤమల Ȟ ൌ ξͳǤʹ͸ ൬ ൰ ൌ ͲǤ͸ͷͻͻ ʹǤʹ͸ ͺǤ͵ͳͶ ‫ܬ‬ ܴൌ ൌ ͸͵ͻǤͷ ൗ݇݃‫ܭ‬ ͲǤͲͳ͵ To calculate: ܿ ‫ כ‬ൌ ʹʹͻͻ ܶ௧ ൌ ܶ௖ ͳ൅ ఊିଵ ଶ ൈ ͳଶ ݉ ʹͳͲ ൈ ͳǤͲͳ͵ ൈ ͳͲହ ݇݃ ՜ ߩ௧ ‫ݑ‬௧ ൌ ൌ ͻʹͷʹ ൗ ଶ ݉ ‫ݏ‬ ‫ݏ‬ ʹʹͻͻ ൌ ͵ͳͺ͸‫ ܭ‬՜ ߤ௧ ൌ ͵ ͵ͳͺ͸ ଴Ǥ଺ ݇݃ ൰ ൌ ͵Ǥͳͳ ൈ ͳͲିହ ൗ݉Ȁ‫ݏ‬ ͵ͲͲͲ ൈ ͳͲିହ ൬ Reynolds number: ߩ௧ ‫ݑ‬௧ ‫ܦ‬௧ ͻʹͷʹ ൈ ͲǤʹ͹͹ ൌ ൌ ͺǤʹͶ ൈ ͳͲ଻ ͵Ǥͳͳ ൈ ͳͲିହ ߤ௧ ͲǤͲͶ͸ ͲǤͲͶ͸ ିଷ ܿ௙ ൎ ଴Ǥଶ ൌ ሺͳǤʹͶ ൈ ͳͲ଻ ሻ଴Ǥଶ ൌ ͳǤʹͲͳ ൈ ͳͲ ܴ݁௧ ܴ݁௧ ൌ Stanton number: ܵ௧ ൌ ܿ௙ ʹܲ௥ ଴Ǥ଺଻ ൌ ͳǤʹͲͳ ൈ ͳͲିଷ ൌ ͸ǤͶͶ ൈ ͳͲିସ ʹ ൈ ͲǤͻ଴Ǥ଺଻ Heat flux at throat: 2 ͳǤʹ͸ ൈ ͸͵ͻǤͷ൰ ሺ͵͸ͲͲ െ ͺͲͲሻ ൈ ͸ǤͶͶ ൈ ͳͲିସ ‫ݍ‬௪ ൌ ߩ௧ ‫ݑ‬௧ ܿ௣ ሺܶ௖ െ ܶ௪௛ ሻܵ௧ ൌ ͻʹͷʹ ൬ ͲǤʹ͸ ‫ݍ‬௪ ൌ ͷǤͳ͹ ൈ ͳͲ଻ ܹȀ݉ଶ Heat conduction through Copper shell: ܶ௪௛ െ ܶ௪௖ ‫ݍ‬௪ ൌ ݄ ߜ௖௨ ܶ௪௛ െ ܶ௪௖ ͺͲͲ െ ͵ͲͲ ߜ௖௨ ൌ ݇ ൌ ͵͸Ͳ ൌ ͵ǤͶͺ ൈ ͳͲିଷ ܹȀ݉ଶ ͷǤͳ͹ ൈ ͳͲ଻ ‫ݍ‬௪ So the shell must be less than 3.5 mm thick. 3