Exam 2 solution

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Purdue University**We aren't endorsed by this school

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AAE 339

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Mechanical Engineering

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Dec 17, 2024

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Uploaded by BarristerOxide15682

Name ______________________ PUID_______________________ Exam 2 Solutions Problem 1 Knowledge-based questions. (four parts, 28 pts total) In class the analogy between the transport of momentum and heat across a boundary layer was used to introduce correlations for convective heat transfer. It was shown that by proper non-dimensionalization of velocity and temperature, the equations that describe the transport of heat and momentum can be made functionally equivalent. Consider a rocket combustor with a regeneratively-cooled wall. Define the non-dimensional variables (u*, T*) and their values at the wall and in the freestream that allow the similarity. For Prandtl number 𝑃𝑟 =𝑐𝑝𝜇𝑘= 0.5, provide a sketch of the thermal and velocity boundary layers from the wall to the free stream. Solution See Slide 8 in Lecture 13 on heat transfer. A main point of the lecture material was the similarity between heat transport across a boundary layer (convective heat transfer) and momentum transport across a boundary later, and the similarity between correlations for cfand Nuthat are an outcome of the analogy. It was expected that students have some knowledge about momentum transport across boundary layers, e.g., 𝑢?∞= 0 at the wall and 𝑢?∞= 0.99 at the edge of the boundary layer. For similarity between solutions, boundary values of non-dimensionalized velocity and non-dimensionalized temperature must be the same, so at the wall T*= 0 and at the edge of the thermal boundary layer T*= 0.99. Certainly you were not expected to memorize an equation for T*, but with a bit of critical thinking combined with knowledge gained from HW 7, one could reason the definition must be 𝑇∗=?𝑤?𝑙𝑙−??𝑤?𝑙𝑙−?∞. The Prandtl number represents the ratio of momentum transport to heat transport. For Pr < 1, it can be reasoned that the thermal boundary layer is thicker, and the sketch should reflect that. Sketches were generously graded.What are the three main types of space electric propulsion? Provide a range of Ispfor each. Identify and explain two advantages of using EP for satellite station-keeping compared to chemical rocket propulsion. Solution See Slide 5 in Lecture 16, also shown in the Exam 2 review package. Three main types are electrothermal, electrostatic, electromagnetic. Main advantages of using EP for station keeping include higher specific impulse allowing less mass of propellant on board the spacecraft, and integration with satellite power systems that use solar energy as a source. Other advantages include finer control with naturally lower-thrust EP system.

Name ______________________ PUID_______________________ From your knowledge of rocket propulsion (calculations are not necessary), draw a line plot of approximate values of Ispv altitude (sea level to 50,000 m) for a high-pressure LOX/H2 engine with a nozzle flow that is optimally expanded at all altitudes. On the same plot, show Ispv altitude for a conventional nozzle with a fixed geometry that is under-expanded at sea level. Name a specific type of altitude-compensating nozzle that can theoretically approach the performance of an optimally-expanded nozzle over a wide range of ambient pressures. Solution A plot similar to Slide 7 in Lecture 18 was expected. Acceptable values for specific impulse of a LOX/H2 engine are between 400 and 500 s. From the reading assignment of HW 7, the aerospike nozzle is a type of altitude-compensating nozzle. Dual-bell nozzles were also accepted. Demonstrate your knowledge of nozzle flows by providing a schematic of the ‘ideal nozzle,’like the one in the notes. Clearly show and label the expansion region and turning/straightening region. Draw a line showing where the nozzle flow becomes one-dimensional and has achieved the design Mach number. Show contour details in the area of the throat. Neatness and clarity count, messy and ambiguous sketches will receive 0 points. Solution A sketch similar to what is shown in Slide 14 in Lecture 18 was expected. One-dimensional flow at the design Mach number begins downstream of the last left-running characteristic. Contour sketches in the area of the throat should show some indication of radii of curvature (values not necessary). Expansion region ends where the wall angle wrt the nozzle axis is a maximum, turning region is downstream of that, where the wall angle is decreasing.

Name ______________________ PUID_______________________ Problem 2 (24 pts) A two-stage rocket launched from Cape Canaveral is used to insert a 10,000 kg spacecraft into low earth orbit with an altitude 200 km above Earth. Simplify the problem by neglecting gravity, drag, and steering losses. The 1ststage provides 30% of the required V, and the 2ndstage provides 70% of the required V. The interstage and payload fairing are jettisoned after the 1ststage burn and before the 2ndstage burn; each has a mass of 1000 kg. Stage performance characteristics are: Stage 1 Stage 2 Isp, SL= 310 s, Isp, vac= 330 s Isp, vac= 440 s Propellant mass fraction = 0.92 Propellant mass fraction = 0.90 a)Calculate the Vprovided by each stage. b)Calculate the propellant required for each stage, and the initial mass of each stage. c)Calculate the thrust and propellant flowrate of the first stage at launch. d)Order the losses neglected in the problem according to their magnitude, from highest to lowest. Solution

Name ______________________ PUID_______________________

Name ______________________ PUID_______________________ Problem 3 (24 pts)Use the CEA output below to size a thrust chamber assembly that produces 250,000 N of thrust at sea level, pa= 0.10 MPa. Let the energy release efficiency ERE= 1.0. The TCA uses an 80% bell nozzle that is over-expanded at sea level according to the Summerfield criterion. (24 pts) a)Calculate oxidizer and fuel flowrates. b)Provide values for throat diameter, nozzle exit diameter, and nozzle length. c)What is a likely application? Be specific, eg ‘first stage’ is not specific.Solution a)Use 𝐹 = 𝑚̇ 𝐼??to find flowrate. First the correct value of Ispmust be calculated, CEA provides values only for pa= peor pa= 0. Knowing the nozzle is over-expanded according to the Summerfield criterion, use the CEA output for pe= 0.4 atm: Ae/At= 23.9 and cf= 1.7593. Characteristic exhaust velocity c*= 1755.9 m/s and is independent of expansion ratio. Now, cfmust be corrected for peNE to paby adding the term ?𝑒−???0,𝑡𝜀to cf. 𝐼??= 𝑐∗𝑐?,𝑐???= 1755.9𝑚?(1.7593 +0.4−1101.3323.922) =2840.5m/s.Calculate propellant mass flowrate from given value of thrust and calculated value of Isp, 𝑚̇ = 88𝑘??.Mixture ratio is provided by CEA, O/F= 2.08. Hence 𝑚?𝑥̇= 59.4𝑘??and 𝑚?̇= 28.6𝑘??. b)Use 𝑐∗=?0,𝑡𝐴𝑡𝑚̇to find throat area, At= 0.01525 m2, dt= 0.139 m. Using the expansion ratio from CEA, Ae= 0.3648 m2and de= 0.682 m. The length of a 80% bell nozzle Ln= 0.80 Lcone,where the conical nozzle has a 15 deg divergence and the same throat and exit area as the bell nozzle. Ln= 0.81 m. c)The CEA output provides information related to application. Obviously it is used for some type of launch application. Ambient temperature propellants are used so it is not a high-performance staged-combustion engine but the pressure is high so it is probably a gas-generator engine. Propellants are N2O4 and MMH, these storable hypergols are used most often for ballistic missiles. 1 bar = 0.1 MPa

Name ______________________ PUID_______________________ Problem 4 (20 pts) A cylindrical stick of solid propellant is used in tests to measure its burning characteristics. The stick is attached on one end to an insulated wall, and is free on the other end. It is placed in a chamber with windowed access so the burning rate rbcan be measured. The initial dimensions of the stick are: length, L0= 0.1 m; and radius, r0 = 10 mm. Propellant density p=1700 kg/m3and c*= 1500 m/s. The burning rate exponent, n, is 0.5. Burning of cylindrical stick of propellant with initial dimensions L0= 0.1 m and r0 = 10 mm is observed through window in combustion chamber. a)Use a control volume analysis to derive the equation used in the lumped-parameter model. What assumptions are used in the model? b)The stick is ignited and the chamber pressure pcand burning rate rbare measured. After a negligibly-short transient period during ignition, the chamber pressure is measured to be 1.0 MPa and the burning rate observed through a window is 0.6 cm/s. Use the steady state lumped parameter model to calculate the chamber pressure at the instant when the stick diameter has been reduced by half (r= 5 mm). Solution a)

Name ______________________ PUID_______________________ For this problem, all terms in the above equation are constant except for pcand Ab. Hence we can write ??2??1= (𝐴?,2𝐴?,1)11−𝑛. The burning rate exponent nwas given, so to calculate pc2it is a simple matter to find the ratio of the burning area at the two times. A longer route to the answer would require finding the burning rate constant aand the throat area.